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Original Definition of Variable Values# - Copy of Value Defined PreviouslyICoordinates of the Photo Principle Point in the Shuttle Coordinate Systemn 1) We begin by selecting the Unit Photo Pointing Vector from the Shuttle Location to one of the Photo PointsCShuttle Location (Perspective Center) in Shuttle Rectangular SystemPCompute the Linear Distance between the Shuttle Location and the Auxiliary Point[Compute the Distance from the Shuttle to the Projection of the Auxiliary Point on the PhotoSGo back to the map, draw a horizontal or vertical line through the center point andPa positive value for the rotation will be computed rather then a negative value.UThis list contains the lists for the Drop-down menus on the "Input-Output" worksheet.Selected Hand Entered in mm+Value to be used in subsequent calculationsA little bit of House-Keeping k_gi_p3 = -Spherical to Rectangular Conversion Equations j_gi_p2 = k_gi_p2 = Unit Vector for Point "p3" i_gi_p3 = j_gi_p3 = 9 of the point in the Global Rectangular Coordinate System.X=Radius of Earth*Cos(Latitude)*Cos(Longitude).Y=Radius of Earth*Cos(Latitude)*Sin(Longitude) Long_sn Greenwich Meridian{Compute the difference between the Deflection angle based on the Auxiliary Point Location and the User Input Rotation angle And applying the following rules/The Deflection angle relative to -X axis equalscwhich lies in a single plane. The third side of the triangle always consists of the Earth's RadiusShuttle Coordinates uCompute the Angles between the adjacent vectors to boundary photo points and computed the Earth Surface Arc distance.cby calculating the angle between vectors from the earth's center to a pair of photo points and then Longitude = CGlobal Rectangular Coordinates of Graphical North Pole (Point "np")X_np = Y_np = Z_np = 1Linear Distance from Earth Center to North Pole =KConvert the Shuttle Rectangular Coordinates of the Photo Points Back to theL1) LATITUDE - Use the computed Latitude directly without any modification. ,we get the angle by which the photo must be Delta Angle =@We then convert it to a 0 to 360 degree clockwise rotation angle\RConvert the Global Rectangular Coordinates for Point "p8" to Spherical CoordinatesFor Point "p8"8Correlate Photo Point Numbers to Top and Bottom of Image Top RightAngle Between Point "p4 and "p6Angle Between Point "p1 and "p4Unit Vector for "p4Rotation #1: We rotate about the +Z axis of the Global Rectangular Coordinates System by an angle equal to the Longitude of the Shuttle Nadir (Long_sn) ETo align the two coordinate systems we apply three separate rotationsNCompute the Normalized Vector from the Shuttle Location to the Auxiliary PointBShuttle Centered Rectangular Coordinates for Auxiliary Photo PointX_Lap XapY_Lap Yap +]To prevent mathematical complexities later in the process (specifically in calculations whichRotation #3: We rotate about the +Z axis of the Global Rectangular Coordinates System by the angle "Theta" defined as being the line ofdCompute the Angle between the two Normal Vectors which equals the Theta angle between the two planesZAngle between 2 vectors = arcos (Vector A " Vector B / (|| Vector A || * || Vector B || ))Vector A * Vector B =|| Vector A || =|| Vector B || =Wto align the global +Z axis with the vector from the earth's center through point "sn".QIf either case 1 or case 2 is true modify the Theta angle by the respective rule.<Is the absolute value of the Original Longitude of "sn" and ! TOP OF PHOTOBthe Unit Photo Vector does not intersect the surface of the earth. X_ei_p8 = Y_ei_p8 = Z_ei_p8 =  np, +X axis in Shuttle Coordinate System sn esn - Newly Calculated Value-Existing Global Rectangular Coordinate SystemrDue to the way I set up the different coordinate systems throughout this program Photo Point #1 is the Upper LeftGat the time of exposure will create uncertainty in the locator ellipse.The Above Rules work with one exception: "sn" and "pc "are within 10 degrees of AND on different sides of the Greenwich Meridian5"pc" both within 10 degrees of the Greenwich MeridianKIf the "pc" has a negative Longitude the "sn" has a positive Longitude theiThis relationship will be employed when the results are transferred back to the "input-output' worksheet. Top RightPoint 6Point 1Point 4Point 7Point 2Point 8Point 3Point 5Phase VIRCompute the Arc Distance on the Spheroid Between Each of the Parameter Photo Poin< t!Angle Between Point "p6" and "p7"Unit Vector for "p6"Unit Vector for "p7"Z_ce = For completeness it should be stated that the coordinates for the "Center of the Earth" in the Global Rectangular Coordinate System are:X_ce = mY_ce = Unit Vector for Point "p5" i_gi_p5 = j_gi_p5 = k_gi_p5 = Unit Vector for Point "p6" i_gi_p6 = j_gi_p6 = k_gi_p6 = Unit Vector for Point "p7" i_gi_p7 = (Direction~Given the Latitude and Longitude of a point (and the radius of the earth) the following equations will compute the coordinatesUnit Vector for "p2"Angle CalculationsSurface DistanceAngle "Epsilon" esnC2) Compute Angle "Gamma" from vector algebra using the Unit Photo Z_gi_p3 = <Compute the Global Earth Centered Coordinates for Point "p4" X_gi_p4 = Y_gi_pp = Z_gi_pp = 2There are NO User Inputs on the Computation SheetsCof a point at the geographical North Pole of the Earth (Point "np")0Spherical Coordinates of Geographical North Pole Latitude = <Compute the Global Earth Centered Coordinates for Point "p8" X_gi_p8 = Y_gi_p8 = Z_gi_p8 = Phase V/computing the arc distance using the following:BThe shuttle centered rectangular coordinate system was defined as:bof a point on the Equator which has the same Longitude as the Shuttle Nadir Location (Point "esn")Yax/Radius of Earth =Error Message #3k_ap =5Global (Earth Centered) Rectangular Coordinate System+Compute the Location of the Principle PointUnit Vector for "p6of Shuttle Nadir Location Origin2) LONGITUDE - Use the angle computed by the Longitude#1 equation and apply to it the SIGN of the angle computed by the Longitude#2 equation\Compute the Vector Normal to the Plane defined by the points: Earth Center, "np" and "esn".LWhen the computed value exceeds 1.0 then the Asin of it does not exist whichb to the X_local, Y_local, Z_local axes of the Local Shuttle Centered Rectangular Coordinate SystemAmeans that the vector does not intersect with the earth's surfacejSpherical Coordinates (i.e. Latitude and Longitude) of the point in the Global Spherical Coordinate SystemJ Longitude- Measured in the Equatorial Plane from the Greenwich Meridian- North Pole +Z Axis Earth CenterZ_Intersection_Coordiante =FMagnitude of Photo Point Vector * k_element of Unit Photo Point VectoraCompute Unit Vector from Earth Center to the Rectangular Coordinates of the Auxiliary Photo Pointi_ap =Xax/Radius of Earth =j_ap =j_np2 = k_np2 = 3 ) Original Earth Pointingi_RSX_p1 = -1.0 *j_RSX_p1k_RSX_p12)Compute the "Gamma" AngleGamma = Photo Principle Point LocationWTranspose of the Rotation matrix from the Shuttle Coordinate System to the Photo SystemnReversing the vector from the shuttle location to point "pc" gives the direction vector to the Principle PointV(the point on the photo plane at its intersection with the central axis of the photo). Photo CornersPrinciple Point Local +Y Local +X Local +YL means that the vector does not intersect with the earth's surfaceCalculation Message + Latitude #1- Longitude Side+ Longitude Side Greenwich Greenwich Meridian Meridian - LATITUDE+ Longitude 1 &2PCompute the Earth Intersection Point with the Earth Pointing Photo Point Vectorsand Point CoordinatesEThe Global Earth Centered Spherical Coordinate System was defined as: 0 Deg. LongitudesHand Corner of the photo. The Following Chart will relate the Photo Point Numbers to the orientation of the Photo.6Global Rectangular Coordinates for Center of the EarthGlobal Rectangular CoordinatesQIF Both TESTS 1 and 2 show False the two Latitude values are not within 5 degreesExceptions to the General RulehWe can also immediately compute the unit vector from the Shuttle Location to each respective Photo Point and Magnitude AngleKnown) ZetaRadius of Earth Y_gi_p4 = Z_gi_p4 =  Angle "Gamma" Unit Photo Point Shuttle Location Y_gi_p1 = Z_gi_p1 = !Angle Between Point "p1" and "p2"Unit Vector for "p1"LPhoto Point Vector Magnitude * k vector element of Unit Photo Point Vector =gCompute the Earth Intersection Coordinates for Photo Point Vector "p2" in the Shuttle Coordinate SystemError Message # 4Error Message # 5i 2) If user input Longitude is between 0 and -180 apply a rotation of 360 - user specified Longitude.+Z Axial Rotation Angle =" Rotation Matrix = Rotation #2=Rotate about the Global +Y axis by 90-Latitude of Point "sn". Photo CenterIEnter a Negative value for the angle if it was measured Counter Clockwise Y_gi_p7 = Z_gi_p7 = X_gi_p6 = Y_gi_p6 = Z_gi_p6 = <Compute the Global Earth Centered Coordinates for Point "p7" X_gi_p7 = 6If Auxiliary Point Information does Not exist, is the 6Top Width of the Format Different from the Side Height)Rotation Matrix from Shuttle Rectangular Unit Vector for Point "p1"Global Coordinates Linear DistanceBetween Earth Center i_gi_p1 = QCompute Unit Vector from Earth Center to the North Geographical Pole (Point "np")i_np =X_np/Radius of Earth =j_np =uWe do this by computing the angular rotations which align the X-Y-Z axes of the Global Rectangular Coordinate System ELongitude#2 = arsin( Y_Coordiante /(Radius of Earth * Cos(Latitude) )Once the Latitude and Longitude is calculated we must correct the angle for the +/- range of each angle. The equations produce the following anglesIReverse the Direction of the Photo Point Vector so it points to the EarthLTo achieve this the photo must be rotated 20 degrees counter clockwise whichQTEST - is the ORIGINAL value for the Longitude of the "pc" on the - side of the D12) You are now ready to load the data into the Photo Calculator.Programmers NoteWorksheet (E198-G200).g"AUX_PT" worksheet and reset the above reference to the rotation matrix in the middle of the"Rec_Photo"Error Message #1 & 2 i_Lss-Lap =In Shuttle Coordinate System?Correct Theta Angle for Positive or Negative Rotation directionAThus, when this computation yields a value greater then 1.0000000%Revision: 3.01 Released: 11/01/02G If more precise results are desired, follow the steps in theb geographic coordinates using appropriate image analysis/remote sensing software.gCompute the Earth Intersection Coordinates for Pixel Point Vector "pr" in the Shuttle Coordinate SystemgCompute the Earth Intersection Coordinates for Pixel Point Vector "pb" in the Shuttle Coordinate SystemgCompute the Earth Intersection Coordinates for Pixel Point Vector "pl" in the Shuttle Coordinate SystemgCompute the Earth Intersection Coordinates for Pixel Point Vector "pt" in the Shuttle Coordinate SystemhCompute the Shuttle Rectangular Coordinates of the Pixel Points Projected to the Spherical Earth SurfaceUnit Vector for Point "pt"Unit Vector for Point "pl"Unit Vector for Point "pr"Unit Vector for Point "pb"PURPOSE This program will ESTIMATE the Latitude and Longitude of the visible corners of a Low Oblique Earth Photo taken from lowRotation AngleJComputation Worksheet - Do not change or enter any data on this worksheet. k_SZ_pt = i_SX_pl = KConvert the Shuttle Rectangular Coordinates of the Pixel Points Back to theEquivalent Ground SizeEquivalent Ground Size SX_pt = i_RSX_ptj_RSX_ptk_RSX_pt<Compute the Global Earth Centered Coordinates for Point "pt"VThe Transformation from the Shuttle Coordinate System to the Global Coordinate Systems<Compute the Global Earth Centered Coordinates for Point "pl"<Compute the Global Earth Centered Coordinates for Point "pr"<Compute the Global Earth Centered Coordinates for Point "pb" X_gi_pb = Y_gi_pb = D center point. For approximate calculations, values from NASA's< G as listed for the photo in NASA's Astronaut Photograph Database.Unit Vector for "pt"Unit Vector for "pb" i_gi_pt = j_gi_pt = k_gi_pt = i_gi_pl = j_gi_pl = k_gi_pl = i_gi_pr = G negative counter clockwise angle) about the center of the phototowards the horizon.@ identified Auxiliary Point visible on the photo for which#! ProtractorsGiven the Rectangular Coordinates of a point (and the radius of the earth) the following equations will compute the!Angle Between Point "p2" and "p3"Unit Vector for "p3"!Angle Between Point "p3" and "p5"Unit Vector for "p5"E for each photo from NASA's Astronaut Photography Database.;Auxiliary Point Input - Which provides more precise resultsIResults are displayed in the bottom half of the "Input-Output" Worksheet.d1) The latitude equation directly computes the correct latitude in the range of -90 to +90 degrees.nSince Longitude rotation is measured with a +/- angle from the +X axis we can use the Longitude value directly according to the following rulesX 1) If user input Longitude is between 0 and +180 apply the user specified LongitudeZ=Radius of Earth*Sin(Latitude)AEnter a Positive value for the angle if it was measured ClockwiseWAngle between 2 vectors = across (Vector A " Vector B)/ || Vector A || * || Vector B ||Where .Vector A - Shuttle to Earth Center Unit Vector X_p5 =  Y_p5 = RConvert the Global Rectangular Coordinates for Point "pb" to Spherical CoordinatesACompute the Arc Distance on the Spheroid Between Each Pixel Point[Convert the Global (Earth Centered) Rectangular Coordinates of the Pixel Points Back to the!Shuttle Coordinate for Point "pt"!Shuttle Coordinate for Point "pl"!Shuttle Coordinate for Point "pr"!Shuttle Coordinate for Point "pb"X_pt = Y_pt = Z_pt = X_pl = Y_pl = KNOTE: All of these values (accurate to +/- 0.5 degrees ) can be obtained A the user should verify the coordinates of the photo center.Bb) The orbital altitude of the Shuttle in Kilometers or Nautical#(9The worksheets following the "Input-Output" sheet contain orbiting space vehicles. Users needing more exact photo corner coordinates should attempt to register their photo to RConvert the Global Rectangular Coordinates for Point "p4" to Spherical CoordinatesFor Point "p4"WConvert the Global Rectangular Coordinates for Principle Point to Spherical CoordinatesFor Principle Point Photo CenterD earth's surface over which the vehicle was located at the time "+Y axis of Photo Coordinate System Top of Photo-X axis of Photo Coord System Top of Photo 4 Angle = asin(Y value/ Linear distance to Aux Pt) Longitude2 = Corrected Spherical CoordinatesSpherical Earth Model X_ei_pp = Y_ei_pp = Z_ei_pp = gCompute the Earth Intersection Coordinates for Photo Point Vector "p5" in the Shuttle Coordinate System Space Vehicle Nadir Point+ Compute the Angle between the two vectorsMWhen the Unit Photo Vector is perfectly tangent with the surface of the EarthUnit Photo Vector BWhen the Unit Photo Vector does not intersect the earth it becomes3 Sin(Gamma) * Dist from Shuttle to Earth CenterYsn/Radius of Earth =Error Message #1k_sn =Zsn/Radius of Earth = Side Height TOP WidthOGreenwich Meridian. One will have a modified Longitude value in the 350 to 359>c) Geographic Location (Latitude and Longitude) of the Photo#2# j_SY_pl = k_SZ_pl = i_SX_pr = j_SY_pr = k_SZ_pr = j_SY_pb = k_SZ_pb = i_SX_pb= SZ_pb = SY_pb = SX_pb = SZ_pl = SY_pl = SX_pl = SX_pr = SY_pr = SZ_pr = SZ_pt = SY_pt = L of photo exposure), as listed in NASA's Astronaut Photograph Database.M Miles as listed for the photo in NASA's Astronaut Photograph Database.EIs the Original value of "sn" within 10 degrees of the 180th meridianOGreenwich Meridian. One will have a modified Longitude value in the 354 to 359;Is the Original value of "pc" within 5 degrees of GreenwichIA similar Problem occurs if the "pc" and "an" are within 5 degrees of theDIs the Original value of "pc" within 5 degrees of the 180th meridian&Does Auxiliary Point Information Exist!Angle Between Point "p4" and "pp"Unit Vector for "p4"Unit Vector for "pp"!Angle Between Point "pp" and "p5"!Angle Between Point "pp" and "p8"Unit Vector for "p8") Rotate Photo about the +Z axis%Are all three of the above tests TRUE1Vector B - Earth Pointing Photo Point Unit Vector_Rotation about the +Z axis is measured in angular units from the plane defined by the +X and +Z4axes in the direction given by the right hand rule. $The difference will exceed 5 degreesYCheck if Longitude of Photo Center and Auxiliary Point are within 5 Degrees of each other#value should be less then 5 degreesvalues less then 5 degrees Theta Long_pc sn esn + X Axis pc Earth Center esnband Equator Point of Shuttle Nadir "esn" the Local Shuttle Centered Rectangular Coordinates SystemI Astronaut Photograph Database my be used. For more precise resultsI relative to a line from the photo center to the top of the image.J) eCompute the Locations of a Pixel at the Center of the Image, Relative to the Photo Coordinate SystemspixelsX_pr=Y_pr=X_pt=Y_pt=Y_pb=X_pb=XCompute the Coordinates of the Pixel Points in the Shuttle Rectangular Coordinate System@ Shuttle's Nadir Point (i.e. the geographical point on the# Longitude1 = -Input and output error messages are containedin the "Input-Output" Worksheet\I.e. A "pc" a -1 degree (+359 degrees) is greater then the "sn" at +1 degree so Theta wouldA be given a positive rotation rather then a negative value.ComputedaCompute the Magnitude of the Photo Point Vector to the Surface of the Earth by the Law of CosinesError Message # 3For Point "p1"|to align the global +X axis with a point on the equator which has the same Longitude as point "sn" which is the point "esn".Product of Two Rotations<So the rotation angle entered by the user is relative to the"+X axis in Photo Coordinate System9 +Y axis in Photo Coordinate SystemSystem +Y axis of4 Angle = acos(x value/ Linear distance to Aux Pt)*Cross Product (Vector Normal to Plane "B")) Vector #3 Vector #4i_np2 =  (Y_Lsn - Y_Lss )/ LD_sn =k_Lsn =  (Z_Lsn - Z_Lss )/ LD_sn =3)gCompute the Unit Vector from the Shuttle Location to Point "pc" in the Local Shuttle Coordinate System.X_Lss = Y_Lss = jThe Normalized vector from the Shuttle's Location to Point "pc" was computed at the top of this worksheet.Vector A%Vector #4 Earth Center to Point "pc",Earth Centered Rectangular Coordinate SystemcSince the Shuttle Rectangular and Global Rectangular Coordinate Systems are orthogonal the rotationVWe create a Local Photo Rectangular Coordinate System based on the following criteria:<Is the Original value of "sn" within 10 degrees of GreenwichJA similar Problem occurs if the "sn" and "pc" are within 10 degrees of the" "How-To-Use" worksheet.=d) The focal length of the camera lens in millimeters (mm) #(4e) The size of the film image in millimeters (mm)# +Space Vehicle Location (Perspective Center)# =See the "How-To-Use" worksheet for step by step instructions Phase IbThe Ground Distance, which is actually an arc along the surface of the Spheroid Earth, is computedgCompute the Earth Intersection Coordinates for Photo Point Vector "p8" in the Shuttle Coordinate SystemError Message # 25Error Message # 26Error Message # 27RNote: Corner Photo Points Valid Only if Accurate Auxiliary Point Data is Includedm ) Rotation #< 1?Rotate about the Global +Z axis by the Longitude of Point "sn". Standard Rotation Equations for Earth Z value - Y Axis Center +Y Axis Y Value8Rectangular to Spherical Coordinate Conversion Equationsc When the computed angle is NEGATIVE, the point is on the NEGATIVE Longitude side of the Earth- Longitude #2 + LATITUDE + Longitude #1 + Latitude #2*3) Rotation about the Z axis of the Photofor Rotation About the Z axis R_Z(Angle) = Cos(Angle) Sin(Angle) -Sin(Angle)Fdegree range. The other will have a modified value in the 0 to 5 rangeO Perform the more difficult calculation within this local coordinate system andA then convert the solutions back to the global coordinate system.6Definition of Local Shuttle Centered Coordinate SystemNWhen "pc" and "ap" are both within 5 degrees of, but on opposite sides of the *180 degrees - Angle "Gamma" - Angle "Zeta"+Y Rotation Angle = 360-Alpha= L5) Compute the magnitude of the Photo Point Vector from the Law of Cosinesh(Magnitude of Photo Point Vector)^2 = (Distance from Shuttle to Earth Center)^2 +(Radius of Earth) ^2 - #F Longitude of the vehicle's Nadir point at the time of photo exposure.(Basic Assumptions Used in the Math Model-in the top half the "Input-Output" worksheet.REQUIRED user input include:EXAMPLE Computed Values Latitude =Corrected Values will Be Longitude#1 = Longitude#2 =Radians4)eNote: The North Pole and point "esn" were chosen to define this plane since they create a large wellq defined plane which has it normal vector pointed in the positive rotation direction about the +Z axis.1Rectangular Coordinates for Auxiliary Photo PointXap =Yap =Zap =aTo Facilitate Later Calculations it is Necessary to Compute the Global Rectangular Coordinates of*Cross Product (Vector Normal to Plane "A"))ijk Vector #1 Vector #2i_np1 = j_np1 = 3Spherical Coordinates of Point on Equator which hask_np1 = 2 ) RConvert the Global Rectangular Coordinates for Point "p2" to Spherical CoordinatesFor Point "p2"RConvert the Global Rectangular Coordinates for Point "p3" to Spherical CoordinatesFor Point "p3"Zpc/Radius of Earth =3 zero we will reduce the problem to two dimensions.0Since all point on the Photo have a "Z" value of Y axis value X axis value3The deflection angle left of right of the "-X" axis$"-X" axis of Photo Coordinate System Photo CoordDspherical earth surface in the Shuttle Rectangular Coordinate System+ +Y axis in Shuttle Coordinate System%Shuttle Location (Perspective Center)^Compute the Relationship Between the Shuttle Rectangular Coordinates System back to the Global|Composite Rotation Matrix can be used to apply the specified rotations to a set of coordinates using the following equation: X_rotated X_original Y_rotated = Rotation Matrix Multiplied by Y_original Z_rotated Z_originali_RSX_p2kDraw a line from nadir to the photo center point extending it past the center to the far edge of the photo.& image and mark it on the acetate.P Database, this value is only accurate to about +/- 0.5 degrees. We Photo Pointsi_sn * Scale Factor =Yss = j_sn * Scale Factor =3Convert Spherical Coordinates for Shuttle Nadir andkchange in the Z coordinate equal to the radius of the earth plus the altitude of the shuttle (which equals:$Error Checking and Detection SummaryErrorStatus Message #Code Total ErrorsPhase IIEConvert From the Earth Centered Global Rectangular Coordinate System 9to a Local Shuttle Centered Rectangular Coordinate System Comments:zThe Center of a photo must be within 10 degrees of Latitude and Longitude of the Shuttle Nadir Location to be classified, Center (Angle "Gamma") X Value! '+ X Axis$ 0 Deg Longitude&by NASA, as a low oblique space photo.Leave Blank if not Used[Compute the Local Shuttle Rectangular Coordinates for Points: "sc", "pc", "Center of Earth"2 South Pole -Z AxisbZnew = Xold * 0 + Yold * 0 + Zold * 1Standard 3-D Rotation Matrix 4) Compute Angle "Epsilon" based on the fact that the sum of the three interior angles of a plainer triangle must sum to 180 degreesAngle "Epsilon" =) Vector from the Photo Center to the ShuttleZWe know the distance from the Shuttle to the Photo Principle Point equals the Focal Length5to Shuttle vector with the Photo using the following:Distance to PhotoADistance to Photo = Focal Length / Cos(Angle Between the Vectors)gCompute the Earth Intersection Coordinates for Photo Point Vector "p4" in the Shuttle Coordinate SystemError Message # 10Uand the Central Axis of the Photo) in the Local Shuttle Rectangular Coordinate System (X_Lsn - X_Lss )/ LD_sn =j_Lsn = UCompute the +Y Rotation Matrix between the Local Shuttle and Photo Coordinate Systems+Y Rotation Angle=Error Message #9ZCompute the Relationship between the Local Shuttle and Photo Coordinate Systems using the Rotation about the Z axis X_gi_sn = Y_gi_sn = Z_gi_sn = X_gi_pc = Y_gi_pc = Z_gi_pc = Latitude EquationTEST6Shuttle Centered Rectangular Coordinates for "Shuttle"X_Lss  XssACompute the Inverse Relationship from the Photo Coordinate SystemnReversing the direction of the Normalized Vector (by changing the sign of each vector element) and MultiplyingFit by the Focal Length produces the Coordinates of the Principle PointReversed Normalized Vector i_RLpc = i_Lpc *-1 = j_RLpc = j_Lpc *-1 = k_RLpc = k_Lpc *-1 = Camera Focal Length = mm =  metersPhase IV Phase III Photo +Y AxisX_p1 = Y_Lss =Z_Lss =Error Message # 3 - 5X_Lpp = vThis is done my multiplying the three individual rotations matrices together in the opposite order to their creation :Rotation order^we can reduce the mathematics for computing the earth intersection point to a simple triangle soRadius of the Earthfeet = @ 39.37 ft/12 m =XCompute the Coordinates of the Photo Points in the Shuttle Rectangular Coordinate System7And their Unit Vectors Relative to the Shuttle Location Unit VectorSX_p1 = i_SX_p1 = ! BOTTOM OF PHOTO_The normal vector is computed by taking the cross product of two vectors which lie in the Plane2Vector #1 Earth Center to North Pole (Point "np")&Vector #2 Earth Center to Point "esn"Compute the Alpha angle between the vector from the Shuttle Location to Point "pc" and the vector from the Shuttle Location to Point "sc".scpcV +Z_local is aligned with the global vector from the earth's center through point: snShuttle Nadir (sn) Z_translation; +Y_local is mutually perpendicular to the other two axes.+ X axis +Z_Local Photo Center (pc)\As a result of the Projection through the Shuttle Location (Perspective Center of the Image)-Modified by Auxiliary Point data if Available4Test: To verify proper reference of Rotation MatrixbCompute Unit Vector from Earth Center to the Rectangular Coordinates of the Shuttle Nadir Locationi_sn = Photo "Footprint" onJ2) The translation between the Shuttle Location and the Principle Point.(Distance Known) Center ofbCompute the Vector from the Shuttle Location to Point "sn" in the Local Shuttle Coordinate System.Coordinates of Shuttle LocationEThe Remaining Angle of the Triangle equals 180 degrees - Gamma - Zeta5 Questions or Problems Should be E-mailed to:Error Message #5k_esn =i_esn =X_esn/Radius of Earth =j_esn =Across the Top WidthAlong the Side HeightY_esn/Radius of Earth =* Photo Point Vector Magnitude =6)SY_p1 = DThe above equations only yield a value between 0 and +180 degrees. eIf Longitude of point "pc" is < greater then Longitude of point "sn" then Rotation equals Theta Angle.>a) Geographic Location (Latitude and Longitude) of the Space#2#:the computations and a discussion of the mathematics used.Altitude of Shuttle = km =aScale Factor = Total Distance from Earth Center to Shuttle = Radius of Earth + Shuttle Attitude = Matrix Multiplication Order-1) Rotate about +Z by Longitude of Point "pc"%1) Rotate about +Z by the Theta Angle02) Rotate about +Y by 90-Latitude of Point "pc".%3) Rotate about +Z by the Theta Angle-3) Rotate about +Z by Longitude of Point "pc")1) Rotate about +Z by 180 +/- Theta Angle Mult|Using the Location of the Shuttle and the Photo Center (Point "pc") we can compute the vector in the Local Coordinate SystemShuttle LocationCoordinates of Point "pc"?Linear Distance from Shuttle Location to Point "pc" = LD_pc =IXnew = Xold * Cos (Rotation Angle) - Yold * Sin(Rotation Angle) +Zold * 0Delta Difference?Required to estimate the ground size of the center photo pixel.3Number of Pixels Across the Top Width of the Photo 5Number of Pixels Along the Side Height of the Photo 7f) Geographic Location (Latitude and Longitude) of an#2#If this transformation and rotation is correct we should be able to plug in the values of the "sn" and "pc" in the Shuttle Rectangular Coordinate system andhCompute the Shuttle Rectangular Coordinates of the Photo Points Projected to the Spherical Earth SurfaceRadius of Earth =X_Lpc =Y_Lpc =Error Message #6Z_Lpc =Z_Lap ZapX_Lap =Y_Lap =Z_Lap =Y_Lpp = Z_Lpp = 7Shuttle Centered Rectangular Coordinates for Point "pc"X_Lpc XpcY_Lpc Ypc +Z_Lpc ZpcJXnew = Xold * Cos (Rotation Angle) + Yold * 0 - Zold * Sin(Rotation Angle)7Shuttle Centered Rectangular Coordinates for Point "sn"X_Lsn  XsnY_Lsn Ysn +Z_Lsn ZsnX_Lsn =Y_Lsn =Y_p4 = Y_pp = m Y_p5 = X_p6 = X_p7 = X_p8 = Y_p6 = Y_p7 = Y_p8 = LUnless auxiliary point data is provided, all computed coordinates assume theG"Top Width" of the photo is looking away from the Shuttle's Nadir PointeUse the features on the map which surrounding the Shuttle Nadir Pont to locate the nadir point on theIEarth Sciences and Image Analysis Laboratory - NASA, Johnson Space CenterdFor User Instructions click on the "Introduction" or "How-To-Use" Tab at the bottom of the workbook. (+ North, - South Hemisphere)   (+ East, - West Hemisphere)   (Left Deflection -180 to 0 ) (Right Deflection 0 to +180 )_This calculator is for use with Low Oblique Photos taken from low Earth orbiting space vehicles Vector from Shuttle Location Vector from towards Earth np (Direction Known)to Earth CenterData - Explanation Coordinate System<Test: Vector A " Vector B/ (||Vector A|| * ||Vector B||)<=1Test: Z value must be positiveTest: Z must = 0Digitized Image SizeXsn/Radius of Earth =j_sn =3Vector A " Vector B/ (||Vector A|| * ||Vector B||)=Theta = PThe computed value must be corrected to the full 0 to 360 degree rotation range.Z_np/Radius of Earth =PCompute Unit Vector from Center of Earth through the Point on the Equator "esn"Program Updates:earthweb@ems.jsc.nasa.gov E-Mail: N equivalent ground size of the center image pixel_Optional The number of pixels across the top and down the side of the digital image.5http://eol.jsc.nasa.gov/sseop/FootprintCalculator.xls3Software Updates can be Obtained from the Web Site:CPlace a protractor on the photo with the center index mark over theGphoto center and align the zero angle mark with the line connecting the%photo center to the top of the photo.GMeasure the angle left or right to the line connecting the photo centerto the Auxiliary Point.D6) On the acetate covered photo, draw a line from the Photo Center- to the midpoint of the Top edge of the PhotoV7) On the map, locate and mark the Shuttle Nadir Point (as listed in the Astronaut . Photography Database) on the Map8) On the acetate covered photo, locate the Shuttle Nadir Point on the Photo and connect it to the Photo Center with a straight line.N9) Select and mark an Auxiliary Point Location on both the Photo and the MapM10) Using the map, compute the Latitude / Longitude of the Auxiliary Point.  No User Input Below this Line> See "How-To-Use" Worksheet for Auxiliary Point InstructionsShuttle Location (nadir) (sn) Surface of the Earth)The "Top of Photograph" looks towards the9interested users to follow the mathematical calculations.produce their values in the Earth Centered Rectangular Coordinate System as was computed originally in the "SPH_PEC" worksheet."Earth Radius = 6,370,332.74 meters/Angles measured clockwise have a positive value7Angles measured counter-clockwise have a negative valueImage Format Size (in mm))OPTIONAL - Digitized Image Size of Photo Cg) The rotation angle (measured as a positive clockwise angle or #Y_Lss Yss +Z_Lss ZssX_Lss =xConvert the Global Rectangular Coordinates for the Earth Center, Shuttle Nadir "sn", Photo Center "pc", North Pole "np" Z_Lsn ="1x1 Digitization Error Message "V"ABCDEFGHIJKLMNOPQRSTUVP11) On the Photo, measure the Deflection Angle of the Auxiliary Point relativea8) On the map, connect the Shuttle Nadir Point and the Photo Center Point with a straight line.H5) Using the map, compute the Latitude / Longitude of the Photo Center.4) Find the center of the photo on the map.=about how to compute the Auxiliary Point and Rotation Angle.Developed By Lockheed Martin Space Operations for The Earth Sciences and Image Analysis Laboratory, NASA - Johnson Space Center.Z(Space Shuttle, Skylab, etc.); which have a photo center within 10 degrees of Latitude and9Image and Equivalent Ground Size of Center Pixel of ImageImage Pixel Size Image ScaleCentral Image Pixel Size Image SCALE fLow Oblique Space Photo Footprint Calculator , 2002, Lockheed Martin Corporation, All Rights Reserved-,fLow Oblique Space Photo Footprint Calculator , 2002, Lockheed Martin Corporation, All Rights ReservedfLow Oblique Space Photo Footprint Calculator , 2002, Lockheed Martin Corporation, All Rights Reserved-0fLow Oblique Space Photo Footprint Calculator , 2002, Lockheed Martin Corporation, All Rights Reserved,3-0Copyright InformationFNOTE: If you are working with a non-square image format, make sure toN enter the format size into the "Input-Output" worksheet as you haveP oriented the photo on the work surface ("TOP width" by "Side Height") OR<to the line from the photo center to the "TOP" of the photo.sDue to the way I set up the different coordinate systems throughout this program Photo Point #1 is the Upper Right?Digital Image Size - Which allows the calculator to compute theCh) Enter the number of pixels across the "Top Width" and down the" "Side Height" of the image.U Pt7 to Pt 2 line and the Vertical Pixel size (pl to pr) is along the Pt4 to Pt5 line7Angle Between Point "pl" and "pr" (Vertical Pixel Size)Vertical Pixel Surface Distance!Horizontal Pixel Surface DistanceThese calculations computed the Earth surface distance between additional points to aid in checking the Pixel Scale measurments added in worksheet "Pixel_Scale"Vertical Scale at Image Center Horizontal Scale at Image Center pixelsj Spatial Resolution", International Journal of Remote Sensing, Vol. 23, No.20; Oct. 20, 2002, P. 4403-4438The mathematical technique and this Excel based calculator were developed by: Donn A. Liddle, Lockheed Ma< rtin Space Operations, The Earth Sciences and Image Analysis Laboratory /SX3, NASA - Johnson Space Center, Houston, Texas, 77058 TOP OF PHOTO/ BOTTOM OF PHOTO PP to Pt 7 Pt 2 to PPSum Pt 4 to PP PP to Pt 5=X_plTest Data for 1x1 digitizationcomparison with pt 2Should equal Zerocomparison with pt 4comparison with pt 5comparison with pt 7FError Checking and Detection Summary for 1x1 pixel Image Digitization "1x1 Digitization Error Message "A""1x1 Digitization Error Message "B""1x1 Digitization Error Message "C""1x1 Digitization Error Message "E""1x1 Digitization Error Message "F""1x1 Digitization Error Message "G""1x1 Digitization Error Message "H""1x1 Digitization Error Message "I""1x1 Digitization Error Message "J""1x1 Digitization Error Message "K""1x1 Digitization Error Message "L""1x1 Digitization Error Message "M""1x1 Digitization Error Message "N""1x1 Digitization Error Message "O""1x1 Digitization Error Message "P""1x1 Digitization Error Message "Q""1x1 Digitization Error Message "R""1x1 Digitization Error Message "S""1x1 Digitization Error Message "T""1x1 Digitization Error Message "U"*data values to: earthweb@ems.jsc.nasa.govQ/The program has been extensively tested and has4IF an error is encountered, please E-mail your input4been equipped with numerous error checking routines.1All User Input are Entered in the Dark Blue CellsHLoad data into the dark blue cells on the "Input-Output" worksheet page.#Enter Data into the Dark Blue CellsReference Information: Robinson, Amsbury, Liddle, Evans; "Astronaut-acquired Orbital Photographs as Digital Data for Remote Sensing:4No correction for atmospheric distortion is applied.03-D orthogonal rotations / translations to allowC by accounting for the roll angleC about the cameras optical axis. J the location (Latitude and Longitude) is known or can be calculatedRobinson, Julie A.; Amsbury, David L.; Liddle, Donn A., Evans, Cynthia A.; "Astronaut-acquired Orbital Photographs as Digital Data for Remote Sensing: Spatial Resolution", International Journal of Remote Sensing, Vol 23 No.20, Oct 20, 2002, P. 4403-4438 0There are no user inputs in any other worksheet. Rotation Angle YResults are displayed directly below the inputs on the same "Input-Output" worksheet pageGDo not use any results which display any type of input or output error.=The "Input-Output" page is pre-formatted to print on one page]This page is pre-formatted to print the input parameter summary and results on a single page.Input Parameter Summary;The "Input-Output" page is pre-formatted to print the inputsummary and results on one pageFocal Length (in mm) Film Format Size Total 1x1 Digitization Errors/comparison with sum of Pt4 to pp and pp to pt 5/comparison with sum of Pt2 to pp and pp to pt 7:Angle Between Point "pt" and "pb" (Horizontal Pixel Size)UAdding the Pixel Points we see that the Horizontal Pixel size (pb to pt) is along the3A complete guide to the use of Astronaut Photograph8 for Remote Sensing is contained in the following paper:JInstruction Worksheet - Do not change or enter any data on this worksheet.AWe have the normal vector from the Auxiliary Point to the ShuttleFinal Theta Rotation Angle = fCompute the Relationship between the Photo Coordinate System and Shuttle Rectangular Coordinate SystemModify the Shuttle Rectangular Coordinate system to Photo Coordinate system relationship by adding a rotation about the Photo's Z axis]by the computed distance we get the coordinates of the projected Auxiliary Point on the PhotorCompute the coordinates of the Auxiliary Point projected to the Photo in the Shuttle Rectangular Coordinate SystemError Message # 16X(E531 through G533). To remove the Auxiliary Point computations the user can delete thewCompute the Normal Vector from the Auxiliary Point to the Shuttle Location in the Shuttle Rectangular Coordinate System-Auxiliary Point in Shuttle Rectangular System/Convention for Auxiliary Point Deflection AngleKThe point where the line reaches the photo edge is the "top" of the photo. $ map and mark it with a pencil. Lin reference to the way the Shuttle and Photo Coordinate systems are defined1 North Pole +90 Latitude Arc AnglelTo aid in photo Analysis it is necessary to know the earth surface distance between each of the photo points TOP Width = Side Height =SCompute the Locations of the Photo Corners Relative to the Photo Coordinate Systems Longitude =M2) if Computed Zeta is less then 90, the True Zeta Angle = 180-Computed Value True Zeta =3Linear Distance from Earth Center to Photo Center =6Linear Distance from Earth Center to Auxiliary Point =FMagnitude of Photo Point Vector * j_element of Unit Photo Point VectorvCompute the Earth Intersection Coordinates for the Principle Point Photo Point Vector in the Shuttle Coordinate SystemError Message # 13Error Message # 14Error Message # 15 i_SX_p8 = SY_p8 = j_SY_p8 = Z_p8 = SZ_p8 = k_SZ_p8 = Error Message # 18 X_ei_p5 = Y_ei_p5 = Z_ei_p5 = ELongitude#1 = arcos( X_Coordiante /(Radius of Earth * Cos(Latitude) )or j_Lss_Lce Y_Lss - Y_Lce)/LD_ss_ce= k_Lss_LceNormal of Vector "A" =TCompute the Vector Normal to the Plane defined by the points: Earth Center, sn, pc.%Vector #3 Earth Center to Point "sn"Error Message #2k_pc =gCompute the Earth Intersection Coordinates for Photo Point Vector "p6" in the Shuttle Coordinate SystemError Message # 19Error Message # 17 Y_ei_p7 = Z_ei_p7 = vthe shuttle location towards the earth. Using these earth pointing vectors we can compute their intersection with theqConvert Auxiliary Point Rotation out of the range of -180 to +180 to the new range of 0 to 360 Clockwise Rotation Phase VII Phase VIIIXCheck if Longitude of Shuttle Nadir and Photo Center are within 10 Degrees of each otherPhoto Center (PC)%The difference will exceed 10 degreesk_RSX_p2i_RSX_p3j_RSX_p3k_RSX_p3i_RSX_p4j_RSX_p4k_RSX_p4i_RSX_ppj_RSX_ppk_RSX_ppi_RSX_p5 in the Shuttle Coordinate System Shuttle Locationj_RSX_p2C2) The origin is located at the principle point of the photograph.oUsing this distance data and the corresponding distance measured on the image, the Photo to Ground scale Factor[can be computed allowing the user to compute the ground distance of an photographed object.Computed Ground Distance Between Skylab S190BZ_esn/Radius of Earth = Unit Photo Vector Instructions 1) Photo 2) North PoleXss = breverse its direction so it points towards the earth (by changing the sign on each vector element)qUsing the "Earth Pointing" Unit Photo Point Vector and the Vector from the Shuttle Location to the Earth's CenterDPoint Vector and the Vector from the Shuttle Location to the Earth's^Enforcing this criteria, we can be sure that the earth's horizon does not appear in the image.lThis allows us to simplify the mathematical calculations by defining a local rectangular coordinate system. Must Be Within +/-10 DegTheta Angle = GLongitude out of the range of -180 to +180 to a new range of 0 to 360.Original Value7Angle between 2 vectors = acos( (Vector A " Vector B)/ Z_Lss = !|| Vector A || * || Vector B || )I 3) Compute Angle "Zeta" from the Law of Sins using the magnitude of theM vector from the Shuttle Location to the Earth's Center and the Radius of theEarth Sin(Zeta) Sin (Gamma) = Distance fromShuttle to Earth Center'Southern Hemisphere - Negative Latitude'Eastern Hemisphere - Positive Longitude'Western Hemisphere - Negative Longitude+Shuttle Altitude (in Km or Nautical Miles)9Auxiliary Point Latitude and Longitude in Decimal Degrees < Lat_sn -Longitude +Longitude Top Center Top Right Middle Left Photo Center Middle Right Bottom Left DistanceTauCoordinates of Center of EarthRConvert the Global Rectangular Coordinates for Point "p5" to Spherical CoordinatesFor Point "p5"5Arc Length = Angle Tau (in Radians) * Radius of Earth3Horizon away from Shuttle Nadir Point (see diagram)4Linear Distance from Earth Center to Shuttle Nadir =.Rectangular Coordinates for Photo Center PointXpc =Ypc =Zpc =RConvert the Global Rectangular Coordinates for Point "p7" to Spherical CoordinatesFor Point "p7"gCompute the Earth Intersection Coordinates for Photo Point Vector "p3" in the Shuttle Coordinate SystemError Message # 12 X_ei_p4 = Y_ei_p4 = Z_ei_p4 = If this relationship is correct then entering the Principle Point location in the Local Shuttle Rectangular Coordinate System should yield values of X=0, Y=0, Z=0$in the Local Photo Coordinate SystemX_Ppp =Y_Ppp =Error Message #10Z_Ppp =Where Radius of the Earth = and the plane defined by the points: Earth Center, "sn" and "pc" (measured with a right handed rotation {even in the southern hemisphere] about the "Z" axis)AThe rules for correcting the computed Theta Value are shown below4Corrected Theta Rotation Angle using the above RulesN10) Draw a line on the acetate from the photo center to the Auxiliary Point.gCompute the Earth Intersection Coordinates for Photo Point Vector "p1" in the Shuttle Coordinate System1)TAnd the translation from the Photo to the Shuttle Coordinate System is equal to the Error Message # 1/Compute the "Zeta" Angle using the Law of SinesZeta = 'Should equal X=0.0 Y= 0.0 Z=0.0.Should equal X=0 Y=0 Z = - Focal LengthDecimal Lens Focal Length (in mm)7Photo Location Data from Astronaut Photography DatabaseJNote: Although you can use the value listed in the Astronaut Photography'In this situation the computed value of=Compute Earth Centered Rectangular Coordinates for point "sn"(Local Shuttle Coordinates for point "sn"values less then 10 degreesj_RSX_p5k_RSX_p5i_RSX_p6ZSince it was originally computed from the Shuttle Location to the photo points, we need toZ3) +Y axis is aligned with the +Y axis of the Local Shuttle Rectangular Coordinate System+Must be Within +/-5 Degrees of Photo CenterRoloflexLinhof Skylab S190A4 +Y in Photo Coordinate System+X in Photo Coordinate SystemRectangular Coordinate System`And the translation from the Shuttle to the Global Coordinate System is equal to the CoordinatesCof the Shuttle Location in the Global Rectangular Coordinate SystemZss = k_sn * Scale Factor =CIf Auxiliary Point information does not exist and the TOP width of 'Ethe format is different the side height, uncertain camera orientationD180 degree Meridian. One will have an original value of 170 to 1802while the other will have a value of -170 to -180.( The difference will exceed 10 degrees.INPUT ERROR MESSAGESShuttle Nadir LocationLatitudeDecimal Degrees LongitudeShuttle Altitudein KmCenter of Photo Camera Data,Ruler (as long as the diagonal of the photo)SUsing the ruler draw lines on the acetate connecting opposite corners of the photosModified ValueAuxiliary Point Rotation Shuttle "X" Axis Photo Center7Shuttle Nadir Latitude and Longitude in Decimal Degrees,Lat/ Long. convention for all program inputs6Photo Center Latitude and Longitude in Decimal DegreesFocal Length (in mm)Optional'Northern Hemisphere - Positive Latitude'Does the Auxiliary Point Latitude Exist(Does the Auxiliary Point Longitude Exist-Does the Auxiliary Point Rotation Angle Exist8IF FALSE do not perform the Auxiliary Point ComputationsIF an Auxiliary Point ExistsoProject the Auxiliary Point to the Photo and Compute its location in the Shuttle Rectangular Coordinate System.' Auxiliary Point Projected to Photo Bottom CenterGround Surface DistanceshCompute the Unit Normal Vectors for each of the Photo Points in the Global Rectangular Coordinate SystemCustom Camera Lens Image FormatRConvert the Global Rectangular Coordinates for Point "p6" to Spherical CoordinatesFor Point "p6" Photo CenterXproduce the coordinates of the earth intersection point in the Shuttle Coordinate SystemX_Intersection_Coordiante =FMagnitude of Photo Point Vector * i_element of Unit Photo Point VectorY_Intersection_Coordiante =]Draw a line along this angle through the photo center and extend it to the edge of the photo KWhere this line intersects the edge of the photo is the "top" of the image.Rotate about the Global +Z axis by the angle "Theta" defined as the angle between the plane formed by points: Earth Center, "np and "sn"Q map and the image and mark it on both the map and the photo. This could be aPhotoAcetate (enough to cover photo)TapeErasable MarkerPencilNotepadDetailed Map of Photo Area! Photo Center (Lat / Long)# Shuttle Nadir (Lat / Long) Shuttle Altitude1) X-Y Axes of the Photo Coordinate System lies in the plane of the Photograph - which is normal to the vector from the Principle Point to Point "pc".nThe following provides basic instructions for preparing a photo for the Low Oblique Space Footprint CalculatorEUsing the features on the image, locate the photo center point on theDegreesMinutesSecondsLIf deflection angle is negative then ADD the negative value to 360 degrees.4The sign of the deflection (- for Left, + for right)/is obtained from the sign of the value computed! recognizable spot on the map.bThis Yields the Transformation from the Shuttle Coordinate System to the Global Coordinate Systems!180 - Angle computed using "acos"$value should be less then 10 degreesj_RSX_p6k_RSX_p6i_RSX_p7j_RSX_p7k_RSX_p7i_RSX_p8j_RSX_p8k_RSX_p8Pin Hole Camera Model5Do not use any results containing an error message !!x4) +X axis lies in the plane defined by the vectors from the Principle Point to Point "sc" and Point "pc" respectively.) Photo "Footprint" on Surface of Earth,Questions or Problems Should be Refereed to:dWarning: For Use With Photo Centers Within 10 Degrees (Latitude & Longitude) of Shuttle Nadir PointEnter Photo ID # -------->8All computations are completed using vector algebra and HCompute the Auxiliary Point lat./long. and convert it to Decimal DegreesANote: Decimal Degrees = Degrees + (Minutes/60) + (Seconds /3600) Shuttle "Y" Axis' Angle between the Vector from theNEarth Centered Rectangular Coordinates for Point "sn" from "SPH_REC" worksheet:Original User Entered Spherical Coordinates for Point "sn"Phase IX? Greenwich Meridian while the "sn" is on the + sideKIf the "pc" has a positive Longitude the "sn" has a negative Longitude theGthe negative compliment of the positive rotation value will be computed? Greenwich Meridian while the "sn" is on the - side_Using the features, transfer the horizontal or vertical line you drew on the map to the acetateSRelative to this horizontal or vertical line, lay out the map angle on the acetate.`Project the Auxiliary Point to the Photo and compute its location in the Photo Coordinate SystemHVerify that all three elements of the Auxiliary Point Information ExistsNRotation Matrix which accounts for the Theta Angle Rotation about the +Y axisHCompute the Single Axis Rotation Matrix for a Rotation about the +Z axis,Coordinate System to Photo Coordinate SystemdThe above cells access the composite rotation matrix computed at the bottom of the "AUX_PT Worksheet Auxiliary Point3Auxiliary Point Deflection Angle in Decimal DegreesBetween Photo Points (meters) Bottom Right TOP OF PHOTO* Away From Shuttle NadirBottomLeftCenterRight Hasselblad35mm SLR mm Drop Lists7 Latitude - Measured + < or - from the Equatorial Plane Top Center>is a positive rotation angle about the Z axis of either system{ constant Longitude which passes through point "sn" and the plane defined by the points "sn", "pc" and center of the earth.1IF the nadir point falls outside the image area. V measure the angle between that line and the nadir/center line with the protractor.J mountain peak, a bend in the river, a large building or just an easily!Angle Between Point "p7" and "p8"Unit Vector for Point "p2" i_gi_p2 = ^The Top center of the Photo has a negative "X" coordinate value in the Photo Coordinate System&+Y axis in the Photo Coordinate System, +X axis in the Photo Coordinate System("-X" axis of the Photo Coordinate System$ +X axis of Photo Coordinate System[Compute the Clockwise Deflection Angle of the Auxiliary Point from the -X Coordinate System8Coordinate of Auxiliary Point in Photo Coordinate Systemis computed by:* +X axis of Photo Coordinate System/ Camera Type (or image format size in mm)B) A) C) D) E) F) G) H) I) 1) Assemble needed materials:+2) Tape down the photo to a flat surface.*3) Tape down the acetate over the photo.LIf deflection angle is positive then use it as the clockwise rotation angle|or Longitude) it is possible that the unit photo vector may not intersect the earth (I.e. the horizon appears in the photo) Zeta = 90 degreesTangent to Earth's Surface Top WidthL1) if Computed Zeta is greater then 90, the True Zeta Angle = Computed Valueusing the "asin" computed angleAcos Computed Angle =Asin Computed Angle =decimal DegreesZEarth Centered Rectangular Coordinates for point "sn" computed in the "SPH_REC" worksheet=Compute Earth Centered Rectangular Coordinates for point "pc"6If TRUE, the Longitude difference between the original" Radius of Earth Angle "Epsilon"Gdegree range. The other will have a modified value in the 0 to 10 rangeZEarth Centered Rectangular Coordinates computed for these point in the "SPH_REC" worksheetNEarth Centered Rectangular Coordinates for Point "pc" from "SPH_REC" worksheet= recommend that you compute a more accurate value.%Compute the Composite Rotation Matrix1) Rotate about +Z2) Rotate about +Y[Compute the Relationship between the Local Shuttle and Photo Rectangular Coordinate Systems=Auxiliary Point Rotation Angle - User Measured Rotation AngleJThe user entered latitude and longitude indicates that the auxiliary pointKThe user measured auxiliary point has a 20 degree left deflection relative GTo have the Auxiliary Point in this location the camera must have been Efor the Auxiliary Point so it is aligned with the True Point Location STS-30-93-44Xthe Zeta angle between it and the radius of the Earth equals 90 degrees (as shown below)QTEST - is the ORIGINAL value for the Longitude of the "pc" on the + side of the 0Modifications to Theta based on Case 1 or Case 2#Photo ID # ----------------------->0RESULTS -- Locator Ellipse and Scale InformationE180 degree Meridian. One will have an original value of 1175 to 180>IF the dimensions differ we must reduce the format size to thexFrom this information we can correct the computed angles to the correct Latitude and Longitude using the following rulesXTranspose of the Rotation matrix from the Global Coordinate System to the Shuttle System j_gi_p1 = k_gi_p1 = Umatrix from the Shuttle Coordinate System back to the Global Coordinate System is the k_gi_pp = j_SY_p1 = -Photo Center Point To Rectangular CoordinatesLegion-All Mathematical Computations are Color CodedRConvert the Global Rectangular Coordinates for Point "p1" to Spherical Coordinates (in Decimal Degrees) From Nadir!Shuttle Coordinate for Point "p1"X_Photo Y_PhotoPhoto CoordinatesLPhoto Point Vector Magnitude * i vector element of Unit Photo Point Vector = Y_ei_p1 = LPhoto Point Vector Magnitude * j vector element of Unit Photo Point Vector = Z_ei_p1 =  Arc DistanceOUTPUT ERROR MESSAGESPoint # Tilt Angle[cross over the 180 degree meridian of Longitude or which cross the Equator) we must convert^Latitude out of the range of -90 to +90 to a new range of 270 to 360 and 0 to 90 respectivelyAngle Between Point "p2 and "ppAngle Between Point "pp and "p7Unit Vector for "p7AltitudeNautical Miles KilometersError Message # 6 X_ei_p2 = Y_ei_p2 = Z_ei_p2 = +Shuttle Coordinate for Principle Point "pp"SX_pp = i_SX_pp = SY_pp = Mof each other and an error must be generated on the "Input-Output" worksheet.<Compute the Global Earth Centered Coordinates for Point "p5" X_gi_p5 = Y_gi_p5 = Z_gi_p5 = <Compute the Global Earth Centered Coordinates for Point "p6";If auxiliary point information exists then the above format/values will be used for subsequent calculations$If less then zero add to 360 degrees6relative the Shuttle Coordinate System as shown below: Image SizeJPoint to the Shuttle to computed the Coordinates of the Point on the Photo(should be at the top center of the image i_gi_p8 = j_gi_p8 = k_gi_p8 = V +Z axis is perpendicular to the Equatorial Plane through the North Geographical PoleAuxiliary Point#OPTIONAL - Auxiliary Point on Photo Shuttle Nadir j_gi_p7 = k_gi_p7 = Zax/Radius of Earth =8Global Rectangular Coordinates of Point on Equator whichY_np/Radius of Earth =Error Message #4k_np = Radians = 4 ) 9Convert the Photo Center Lat. / Long. to Decimal Degrees.X_esn = Y_esn = Z_esn = .Latitude=arsin(Z_Coordinate / Radius of Earth)4Linear Distance from Earth Center to Equator Point ==impossible to create a triangle which has the computed Gamma :angle and a leg distance equal to the radius of the earth.8exceeds the 0.000 to 1.000 range of the "Asin" function.@km = nm * (1/0.86897624 miles/nm) * (1/0.6213711922 km/mile)In this case we want to compute the new local coordinates for points "sc", "pc" and "Earth Center" by applying the composite rotation matrix and Q+Z translation equal to the radius of the earth plus the altitude of the shuttle.>Shuttle Centered Rectangular Coordinates for "Center of Earth"Local CoordinatesGlobal CoordinatesX_Lce  XceY_Lce j_Lss-Lap = k_Lss-Lap = but only in the range of 0 to +/- 90 degrees. When the computed angle is POSITIVE the point is on the POSITIVE Longitude side of the EarthParsin( (sin(Gamma)*(Distance from Shuttle to Earth Center) ) / Radius of Earth )B (SIN(Gamma)*Dist Earth to Shuttle)/ Radius of Earth = BNOTE: IF the angle exceeds 1.0 then the asin does not exist which j_Lap-Lss = k_Lap-Lss =/Multiple each element of the above vector by -1/Normalized Vector from "pc" to Shuttle Location pcGCompute the Coordinates for the Corners of the Photo Image Plane in theZ_p1 = SZ_p1 = k_SZ_p1 = ,is computed according to the following rulesyCompute the Coordinates of the Photo Point Vector's intersection with the Earth Spheroid in the Shuttle Coordinate System X_ei_p1 = Linear Distance=decimal degreesShuttle Location in Global(Unit Vector for the Principle Point "pp" i_gi_pp = j_gi_pp = ' The difference will exceed 5 degrees. j_SY_pp = y=0Case 1Case 2 Top Height Side WidthOutput Error MessagesInput Error MessagesY_p5 = SY_p5 = j_SY_p5 = Z_p5 = ^Special Rule - if Longitude of "pc" is <= 0 and Longitude of "sn" >= 0 then Theta = -1* Theta Z_ei_p3 = %Which will be written in the form of: Translationm = m +Top Left3Compute Global Spherical Coordinates for point "pc"For Point "pc"3Compute Global Spherical Coordinates for point "sn"For Point "sn"'In Global Rectangular Coordinate System j_SY_p6 = Z_p6 = SZ_p6 = k_SZ_p6 = !Shuttle Coordinate for Point "p7"SX_p7 = i_SX_p7 = SY_p7 = j_SY_p7 = Z_p7 = SZ_p7 = k_SZ_p7 = !Shuttle Coordi< nate for Point "p8"SX_p8 = ACompute the Normalized Vector from Shuttle Location to Point "pc"i_Lpc =  (X_Lpc - X_Lss )/ LD_pc =j_Lpc = 'Was altitude entered in Nautical Miles? km = nm *  (must be a Positive Value)RIF Both TESTS 1 and 2 show False the two Latitude values are not within 10 degrees<TEST2 - Is the absolute difference in the MODIFIED Longitude=TEST 1 - Is the absolute difference in the ORIGINAL LongitudeAuxiliary Point on Photo (ap)True Format ValuesLIf altitude was entered in Nautical Miles it must be converted to Kilometers1If TRUE Convert to Kilometers using the following9Altitude in Kilometers for use in subsequent calculations$Convert Altitude Units to KilometersClockwise Rotation Angle to theX_app = Y_app = Z_app = j_Lap_Lss*D_ss_appDistance = D_ss_app =to the top of the photo7turned 20 degrees to the Left when the photo was taken.=To compute the correct latitude and longitude for the corner Cpoints of the actual photo we must rotate the mathematical location8the same Longitude as the Shuttle Nadir Location ("esn")tThis equation provides an easy way to determine if the Unit Photo Vector does not intersect the Surface of the Earth Gammab 1) If user input Latitude is between 0 and +90 apply a rotation of (90 - user input Latitude)1 which is opposite the direction of the nadir.nLatitude rotation is measured with a +/- angle from the +X and +Y plane we can use the Latitude value directly@Compute "Epsilon" Angle using the Two Previously Computed Anglesb 2) If user input Latitude is between 0 and -90 apply a rotation of (90 - user input Latitude)XWe can then any needed translations by adding an additional vector to the above equation X_translation9 +X in Local Shuttle Rectangular Coordinate System O Rotation Angle About the +Y axis of the Shuttle Rectangular Coordinate System Alpha Angle<Compute the Global Earth Centered Coordinates for Point "p2" X_gi_p2 = Y_gi_p2 = Z_gi_p2 = = PThis angle must be obtuse (i.e. greater then 90 degrees) so the True Zeta Angle + Y_translationaThis Yields the Transformation from the Photo Coordinate System to the Shuttle Coordinate SystemsShuttle CoordinatesPhoto Coordinates @(Vector B) Normalized Vector from Shuttle Location to Point "sn" i_Lap-Lss =DLinear Distance from Shuttle Location to Point "sn" LD_sn = i_Lsn = qSo with this relationship, if we know the Coordinates for the corners of the photo in the Photo Coordinate SystemYwe can compute their coordinates in the Shuttle Coordinate System with the above equation!Shuttle Coordinate for Point "p2"SX_p2 = i_SX_p2 = SY_p2 = j_SY_p2 = Z_p2 = SZ_p2 = k_SZ_p2 = !Shuttle Coordinate for Point "p3"SX_p3 = i_SX_p3 = smaller of the two dimensions.0Values to be used in all subsequent calculations Focal Length2while the other will have a value of -175 to -180.Z_pp = SZ_pp = k_SZ_pp = !Shuttle Coordinate for Point "p5"X_p5 = SX_p5 = i_SX_p5 = @ +X axis in the Equatorial Plane through the Greenwich MeridianError Message # 7Error Message # 8Error Message # 9 X_ei_p3 = Y_ei_p3 = uWith the three individual rotation matrices computed we can now combine them into a single composite rotation matrix._Special Rule - if Longitude of "pc" is >= 0 and Longitude of "sn" <= 0 then Theta = 360- ThetaError Message # 11!Shuttle Coordinate for Point "p6"SX_p6 = i_SX_p6 = SY_p6 = dCompute Unit Vector from the Shuttle Location to the Earth's Center in the Shuttle Coordinate SystemWThis Coordinates for the Shuttle Location and Earth Center were previously computed as:Earth Center =F Linear Distance between Shuttle and Earth Center = LD_ss_ce = (Y_Lpc - Y_Lss )/ LD_pc =k_Lpc =  (Z_Lpc - Z_Lss )/ LD_pc =Error Message # 20Error Message # 21 X_ei_p6 = Y_ei_p6 = Z_ei_p6 = gCompute the Earth Intersection Coordinates for Photo Point Vector "p7" in the Shuttle Coordinate SystemError Message # 22Error Message # 23Error Message # 24 X_ei_p7 = uFrom the previous computations, we have the unit vectors from the shuttle location to each of the photo points in the%Shuttle Rectangular Coordinate SystemBy reversing the direction of each of these unit vectors (by changing the sign of each element) we produce the unit vectors from  + Y_localSEstablish a Photo Coordinates System based on the Principle Point Location of PhotoRotation about the Y axis\Znew = Xold * 0 + Yold * 1 + Zold * 0IZnew = Xold * Sin (Rotation Angle) + Yold * 0 + Zold *Cos(Rotation Angle)for Rotation About the Y axisi_Lap_Lss*D_ss_appk_Lap_Lss*D_ss_app6Compute the photo center point's lat./long on the map.@ the intersection of the two lines is the center of the photo.Given the Format Size it is easy to computed to coordinates of the photo corners and other key photo points on the image in the Photo Coordinate SystemInput Format Size Multiplied by Yce +Z_Lce Shuttle LocationWe then translate the origin of the coordinate system from the center of the earth to the shuttle location by applying a single  +X/Definition of Global Earth Centered Rectangular+Local Shuttle Rectangular Coordinate System k_SZ_p3 = !Shuttle Coordinate for Point "p4"SX_p4 = i_SX_p4 = SY_p4 = j_SY_p4 = y = 0Z_p4 = SZ_p4 = k_SZ_p4 =  sn pcWCompute the Location of the Principle Point (intersection between the Photo Image PlaneSZ_p5 = k_SZ_p5 = pc Earth Center of Earth (Magnitude Unknown)WAngle between 2 vectors = across (Vector A * Vector B)/ || Vector A || * || Vector B ||Alpha = Error Message #7.Compute the +Y axis Rotation Angle = 360-AlphaError Message #8Coordinates of Point "sn"Q(2 * (Distance from Shuttle to Earth Center) * (Radius of Earth) * Cos(Epsilon) )IYnew = Xold * Sin (Rotation Angle) + Yold * Cos(Rotation Angle) + Zold *0If this relationship is correct then entering the Shuttle location in the Local Shuttle Rectangular Coordinate System should yield values of X=0, Y=0, Z=-Focal LengthError Message #11{Using the Photo to Shuttle Coordinate Transformation developed above we compute the Shuttle Coordinates of the Photo Points6) The magnate of the Photo Point Vector is then used as a scale factor and multiplied by each element of the Unit Photo Point Vector to [Convert the Global (Earth Centered) Rectangular Coordinates of the Photo Points Back to theShuttle Nadir PointAuxiliary Point Z_p3 = SZ_p3 = SGlobal (Earth Centered) Spherical Coordinate System ( i.e. Latitude and Longitude)Coordinate SystemSY_p3 = j_SY_p3 = GPrinciple Point Location in Local Shuttle Rectangular Coordinate System+Y axis Rotation Matrix and the Translation between the Shuttle Location and Principle Point which equals = - Focal Length in the Z axis.9Point Locations in a Global Rectangular Coordinate System/Rectangular Coordinates for Shuttle Nadir PointXsn =Ysn =Zsn =1 ) KThe Local Shuttle and Photo Rectangular Coordinate System are separated by:1) A single rotation about the +Y axis of the Shuttle Coordinate System by the angle 360-ALPHA (Angle between the Vector from theZSince the Photo and the Shuttle Rectangular Coordinate systems are orthogonal the rotationJWe define the new shuttle centered rectangular coordinate system in which:O +X_local is aligned with the plane defined by the point: sn, pc, Earth Center9Unit Vector from Shuttle to Earth's Center - Vector "A" = i_Lss_Lce X_Lss - X_Lce)/LD_ss_ce=!Product of Previous Two RotationsComposite Rotation MatrixTmatrix from the Photo Coordinate System back to the Shuttle Coordinate System is the5 into the Local Shuttle Rectangular Coordinate System - X_local np<GThe Global Earth Centered Rectangular coordinate System was defined as:: +Y axis is mutually perpendicular to the other two axes.2 North Pole +Z AxisPThis earth radius value is based on the Clark 1866 Spheroid Model for the Earth. -Y_local +X_local Lat_sn +Y axis  np - Y Axis Lat_pc Long_snX_Lce =Y_Lce == Shuttle Location to Points "sc" and "pc" respectively._Rotation about the +Y axis is measured in angular units from the plane defined by the +X and +YZ_Lce = ZceError Message # 27Which has the same Longitude as the Shuttle Nadir Point Remaining Angle =5)X_p2 = X_p3 = Y_p1 = Y_p2 = Y_p3 =  Photo -Y Axis Photo +X AxisX_p4 = X_pp = m X_p5 = +Y Axial Rotation Angle = Rotation #3Compute the Angle Theta3 +Y in Local Shuttle Rectangular Coordinate System tIf Longitude of point "pc" is less then the Longitude of point "sn" then Rotation equals 360 degrees - Theta Angle. Rotation Angle ="Compute Composite Rotation Matrix@Compute Shuttle Location in Global Rectangular Coordinate System*The Earth Center to Shuttle Nadir Location Local +X -Y +Y j_gi_pr = k_gi_pr = i_gi_pb = j_gi_pb = k_gi_pb = Angle Between Point "p2 and "p7Unit Vector for "pl"Unit Vector for "pr"Angle Between Point "p4 and "p5Unit Vector for "p5Z_pl = X_pr = Y_pr = Z_pr = X_pb = Y_pb = Z_pb =  Multiplied by i_SX_pt = j_SY_pt = ft2) Longitude Equation #1 computes the angle in the equatorial plane from the Greenwich meridian, but only in the range of 0 to 180 degrees POSITIVE Z_gi_pb = X_gi_pr = Y_gi_pr = Z_gi_pr = X_gi_pl = Y_gi_pl = Z_gi_pl = X_gi_pt = Y_gi_pt = Z_gi_pt = Known Radius of the EarthRConvert the Global Rectangular Coordinates for Point "pt" to Spherical CoordinatesRConvert the Global Rectangular Coordinates for Point "pl" to Spherical CoordinatesRConvert the Global Rectangular Coordinates for Point "pr" to Spherical Coordinatesgf G|&KL9RO!S VWwY:[`]x`*bvdXe'ghGjwoHq r at vQy*|}M_N+u{gq |' $Aͪs  dMbP?_*+%1  &A&RPage &P&Q?'q= ףp?(zG?)333333?"FzG?zG?cU} I} m } $ }  m } } m  ;; @  @ h h@  @   ;                K" "<<<<<<<<<< UUUUUUUUUUU I ! !U  K Q    N      !O  %   , (     (   (  (  (  ,g  (      (X !  ,     ,    ,~ , ( J, (J, @`D44". ,,, ,<D*2.64,&TT@ : ! h" # ;$ ;% & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ?   , !c! 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" $b%&'()*+ - . / 234 45 56 6u7 8 9 : ; = > >  >  > ? ?"zc?'MD? D? D? B$"D?D?<D?B:O   8@ABCDEFGHIJKLMQ S T V W X Y \ ^ h@_ h@ @.#A.B. CsC.D Dt E. F. G. H. I. J. K. L. M.Q Sx T V W XI Y \y ^"_ 4N ` h@a h@ b h@ e f g h i j k l m n o qruw;x;z{|}~ `` a a  a  aa b b"zcb'?MDb Db Db B$"DbDb<DbBb ef g hi jn ko lpm nq orq q r u.@ w! x z>X z{>|> |}> }~> ~}>6Q"L    > ~> > >W > >Y > >Z > >[ > >\   >]  6  . .& >^ k   0b     **  PB$ K8l  #(  8 *8 <Z  Ԕ" <Q]*`Z: wH<x 1) Identify the geometric center of the photo by intersecting lines through opposite photo corners. 2) Re-estimate the Latitude and Longitude of the photo center (database coordinates are accurate to +/- 0.5 degrees). 3) Identify the "TOP" of the Photo which is the edge that appears to be more towards the horizon. NOTE: The "TOP" of the photo has no relationship to North, but will simply be the direction TOP OF PHOTO away from the shuttle nadir point. 4) Draw a line from the center of the photo to the middle of the "TOP" edge of the photo. 5) Identify an Auxiliary point on the Photo for which its Geographic Coordinates (Latitude and Longitude) are known. 6) Draw a line from the center of the photo to the Auxiliary Point. 7) Place a protractor on the photo placing the center index mark over the photo center and aligning the zero angle mark with the line connecting the photo center to the top of the photo. 8) Read the angle between the line to the top of the photo and the line to the Auxiliary Point. Enter a Positive value for the angle if it was measured Clockwise Enter a Negative value for the angle if it was measured Counter Clockwise<Hl l> 1 > I  -| w 58 6$[  @" Zi]5`$[: 8<9Computing the Rotation Angle of an Auxiliary Photo Point< 8a 68 6t[  @" sx]6`t[< <The optional input allows users to compute more precise approximations for the photo corners by accounting for the effects of camera rotation at the time of exposure<06tNd M% 783 9ZN-] 7`<^V 88 S A@M%]8`0=d\B 98 c $D@]9`>d\B :8 c $D@]:`t>d\2 ;8 c $A@jJ];`D?|tB <8 <D jJ]<`?d\2 =8 c $AjJ]=` @pp >8 0P\ A@]>`P\@ <Aux Pt< `` G8 S \  @f7@8]G`\A  < Top of Photo<G n^ ]] 8# s ] `,Bpp Y8 0]  @]Y`]B  < Top Width<; zB Z8 BD@V ]]Z`CzB [8 BD@V][`CzB ]8 BD@Vt t_]]`DzB ^8B BD@Vrkr]^`Evv \8 6] A@]Zq]\`]F  < Side Height<C d\ R8 c $A@ ]]R``Hn^ [ 8# Ci P<] `Hd\ b8 c $A@[]b`@Ijj P8 s *^  @']P`^ J  < Top of Photo<Q jbB e8 s *D@<=]e`JjbB f8 s *D@1H]f`\Kjb2 g8 s *A@jJ9@]g`KzB k8 BD@8c9]k`L|| w8 <^ A@?1Qe]w`^L  < Photo Center<L nh^ |LM! 8# . :] `Md\ @8 c $A@|LM!]@`NjbB B8 s *D@}LM!]B`OjbB C8 s *D@]C`OjbB D8 s *D@]D`TPjb2 E8 s *A@jJ]E`,Q F8 6p_ A@"  ]F`p_Q  <Photo Center Lat, Long< u jbB A8 s *D@}NM]A`@<d @cV0e2d@<1 Y XY  ( Q 2 d@<1 yCtnih a.A ;A"tsorantua-qciuer drOibat lhPtogoarhp ssaD gitilaD ta aof reRometS neisgn :pStaai leRosulitno,"I tnreanitnolaJ uonrlao feRometS neisgn ,oV l32N .o02 ,cO t02 ,0220 ,.P4 04-34483 /fci hsit ehe gd e t ah tpaepra sotb eomert worasdt ehh rozino . N TO:E hT eT"PO "fot ehp ohoth san oeralitnohspi t ooNtr,hb tuw li lispmylb eht eidertcoi n T POO FHPTO O wayaf or mht ehstult eanid ropni.t 4 ) D ar w ailenf or mht eectnreo fht ehpto oott eh m dild efot eh" OT"Pe gd efot ehp ohot . )5 dIneityfa nuAixilra yopni tnot ehP ohotf row ihhc i stG oergpaih coCroidanet sL(tatidu ena doLgntidu)e a erk onnw . )6 rDwaa l ni erfmot ehc neet rfot ehp ohott oht e A xuliaiyrP iotn . )7 lPca e arptoartcroo nht ehpto olpcani ght eectnre nied xamkro ev rht ehpto oectnrea dna ilngni ght e ezora gnelm ra kiwhtt ehl ni eocnnceitgnt eh p ohotc neet rott eht poo fht ehpto.o 8 ) eRdat eha gnelb teewnet ehl ni eott eht poo fht e hpto ona dht eilent oht euAixilra yoPni.t E tnrea P sotivi eavul eof rht enalg efii taw semsarudeC olkciwes E tnrea N getavi eavul eof rht enalg efii taw semsarudeC uotnreC olkciwesiwllE TSMITA Eht eaLituteda dnL noigutedo fht eivislb eocnrre sfoa L woO lbqieuE rahtP ohott kanef or molwoRatitnoA gnelJCmoupatitnoW roskehte- D oon thcnaego rneet rna yadato nhtsiw roskehte ._kZSp_ t = iS__Xlp= K oCvnre tht ehStult eeRtcnaugal roCroidanet sfot ehP xileP iotn saBkct oht*eeRivisno :3 0.-1EBAT eRelsade :110//120Euqvilane trGuodnS zieqEiuaveltnG ornu diSez XSp_ t =iR_XSp_t_jSR_XtpkR_XSp_>>>>>>>TT  '<Z(%Revision: 3.01 Released: 11/01/02  %&&&&&&&' U  LU  U          i <<= 1 2 =~ @ 3 4Mk <(UD AZ  None9/Latitude Must Range Between -90 and +90 DegreesBNone  56 `~  7 4Nn H(?XD A  None<2Longitude Must Range Between -180 and +180 DegreesBNone   56687 |||| 9 ::@@ 4%ZT(<DD  None+!Shuttle Altitude must be PositiveBNone  uuuu 12 =~ @ 3 4Mv`( `DD A  None>4Photo Center Must Be Within 10 Degs of Shuttle NadirBNone  n `~ ^ 7 4Nl(+iZQ Z_ B$ ;4Photo Center Must Be Within 10 Degs of Shuttle NadirNoneBNone  ::;|||| V8666||||   |||| 5 |||| 56 `~ C@ 7 4Mx(D  NonehDDA  None?5Auxiliary Point Must Be Within 5 Degs of Photo Center"BNone  56 `~  7 4N(D  NoneqZu Z B$ <5Auxiliary Point Must Be Within 5 Degs of Photo CenterNone"BNone  566 k7|||| 5 `6~ ׿ 7 4O(yD  None`DA  None=3Rotation Angle Must Range From -180 to +180 Degrees"BNone  | 4P(9DDDB$  NoneiD D D B$  None0&Incomplete Auxiliary Point Information"BNone  |||| 1 22]M(G#DBCustom  Enter Custom Focal Length B3|||| 5 X~ @ 7A_O(GI#DBCustom " (must be a Positive Value) B(#DBCustom RD 2+Enter Custom Focal Length Greater Then ZeroNone"CD  None#Delete Value in Green Box"BNone  5 N~ @^@Z@H#DBCustom  Top Width#D"B`^@J#DBCustom  Side Height#D"Be4(O#DBCustom ( TOP Width x Side Height in mmB! TOP Width x Side Height in mm|||| 56q(v#DBCustom (!Enter Custom Formt Size in mm -->.$Do not Enter Values in these Cells->B'$Do not Enter Values in these Cells-> b4H)L#DBCustom " (must be a Positive Value) BL) #DBCustom PD D B$  Must be Greater Then ZeroNone"UD D B$ "Delete Values in White BoxsNone"BNone  Dl0k ( 4 R&2&F2: h4! h4" h4 # h4 $ h4% h4 & h4 ' h4 ( h4 ) h4 * h4 + h4 , h4 - h4 . h4 / h4 0 4 1 4 2 h43 h44 h45 @ 6 !7 ,!8 ,!9 !: !; ,"< h"= "> "? "r X)A \Z] C<Due to uncertain camera orientation, these calculations will B   s!\)0]Z] D+be completed using a square format size of Zc mm.BP B!!|||| q"`) [Z] B;Enter auxiliary point data to eliminate this approximation. B""|||| #z8qr{#|||| $$$|||| % %%%|||| &8 &~ &p@&d)'lD&  NoneSD& 0)Number of Pixels must be a Positive ValueNone"BNone & '8 '~ 'p@'p)lD'  NoneSD' 0)Number of Pixels must be a Positive ValueNone"BNone ' ( (((|||| )z8qr{*} **~ ++|)CD None D None DNone DNone DNone DNone DNone DNone DNone DNone DNone D&None D'None B $ @None5+INPUT DATA ERROR - See Input Error MessagesBNone+&f+)DPZ @None1'Error in Worksheet 'Intersection_Calc' BNone+ ,\,)-FZt @None'Error in Worksheet 'SPH_REC' BNone,&c,)/MZ( @None.$Error in Worksheet 'ReTrans_Origin' BNone, -a-)KZ @None,"Error in Worksheet 'Trans_Origin' BNone-&\-),FZW @None'Error in Worksheet 'REC_SPH' BNone- .^.)p HZ  @None)Error in Worksheet 'Rec_Photo' BNone.&^.)6 HZ @None)Error in Worksheet 'Surf_Dist' BNone. /[/)m EZ . @None&Error in Worksheet 'AUX_PT' BNone/&`/)0JZ @None+!Error in Worksheet 'Pixel_Scale' BNone/ 0`0)RJZ @None+!Error in Worksheet 'Photo_Setup' BNone0&q0*,[Z @None<21x1 Digitization Error in Worksheet 'Pixel_Scale' BNone0 1vwxvvvv 2z8 2r{ 3z8 3r{ 3 4z8 4r{5oppp 5{5 66 6 'D(%Revision: 3.01 Released: 11/01/02 77 8'8( 8)8*̌C@D  8+8, 8-8. 8)P8 *C@: :DDDB$  $ B 8  9/0 919*̌^8D  92 93094d*8 pDDDB$ BDNone "Incomplete Data Not Used"B 91P9 *^ :DDDB$  $ B 9 :/0063337 :1VP: *q= ף`T9 :DDDB$  $ B :  ;8 ;99;*Qr@9DH;2h*.2D Nautical Milesin KmBin Km;9:;::< <8999<t*-#D  :3 B8<Bx*;"D   Km B<9 <9_<;;=> =::::9: =@=:G= p@&1D&D'B$  $& B = A >8>9 >1>?fffff&C@;D >2 >99 >B>;G> p@= 1D&D'B$  $' B > 5 ?8: ?1?*^>D ?2?9::::< D0l&2<:4   *6u,Q@ Y"A Y"B "C "D "E "F "G w"H I !J K L M N O P Q R S T U V W X Y Z ![ \ ] ^ _ @8:1C29 @9@: @1@ D@Z@DZc @ 5A A8d A;;AD@o@"ZI A2AA9@A|**D; Bf Film FormatBPLinhof Film FormatA A1A D^@BZc A 5AB8:1C29::ZB*CDZ] +$Due to uncertain camera orientation, B B  C8 C0:lC*|VD+None @None5+INPUT DATA ERROR - See Input Error MessagesBNoneC:ZC*@ DZ] +$these calculations will be completed B C  D8 D0:D*RD,None D-None D.None D/None D0None D+None D,None D-None D.None @D/None D0None B $ @None7-OUTPUT DATA ERROR - See Output Error MessagesBNoneD:_D*!IZ] 0using a square format of D@  mm.BP B D  EEFGHHHHFFFI FF GJ GKLM7G *!D   $ B STS-30-93-44G  GNGOPPQ HR HNHQ HNHQ HS HT7HU==V IW IXIY IXIY IZI[ I\ I==V J] J B@_ZJJ ·rGc^`ZJJyKL<@ZZJ[]J^*fGZ  1' Away From Shuttle NadirB J00_ K`[Kb2@c Z K0K cn[@\ Z K _ LdLZKB@_ZLLʝ5^`ZLL~<@TZwL[0f0g M`[000_ NdN;TB@_ZNN>'+^`ZNN8?@@PZN[0h0i Oj[0h0i PdPC.C@lZPPQj^mZPPF1@+Z'P[000_ Q`aQkL`@KZ Q0Ql=r@K Z Q 0Q m_@XZ RdRkffff&C@mZRR^nZRRnY"|T1@cZR[000_ Sde[f00_ TdTE|PC@lZTT@}^mZTT kjr9@VZT[000_ U`[000_ VdVut@vC@xZ%VVLp^yZ&VVCk @JZSV[Vo,7e@QZ V0V oo@QZ V _W`[000_ XdX ]tpC@xZXXlo^yZXX^cR@LZX[000_ Yj[000_ Zp8Z3[hC@xZZZ,r ^yZZZ>3@NZKZ[000_ [q [r[...s00[lK=^2U@V Z [ 0_ \td\^*JNZ  8. Away From Shuttle NadirB\\\\i\uۣC@VZ \000 \ mLP75@Q Z ]t^\\\i0f00_ ^t ^v  ^vD^\ ^2o^i0000_ T_B@V>DJNo Earth Intersection No Earth $JB_T_ZKB@X>DLNo Earth Intersection No Earth $LB_T_;TB@X>DNNo Earth Intersection No Earth $NB_0000_ Dlk ,lpn ] T` a b c d e f g h !i ;j k l !m !n o !p !q r !s !t !u v !w !x !y !z !{ !| !} Y"~ ! X`·rGc^XBDJNo Earth Intersection  Intersection $JB`X`ʝ5^JBDLNo Earth Intersection  Intersection $LB`X`>'+^TBDNNo Earth Intersection  Intersection $NB`0000_ a[0000_0000_ b[0000_00h0i cx0000y0cb):r@\Z c0c c@ +@[Z c _d[0000_0000_ e[0000_ ee000_ f[0000_uf^*A_Z  I? Towards Shuttle NadirBf000_ g[0000_zzzz{ h[0000_||||} i[0000_ ii j[00^0_[000_ k~k000v k k[0 k k TlC.C@L>DPNo Earth Intersection No Earth $PB l00 l^%l0TlE|PC@P>DTNo Earth Intersection No Earth $TB l[00 l vl _XmQj^JBDPNo Earth Intersection  Intersection $PB m00Tmkffff&C@V>DRNo Earth Intersection No Earth $RBm0Xm@}^ZBDTNo Earth Intersection  Intersection $TB m[00Gm Q?w 1Z; Z;D&B m _J n[00Xn^RBDRNo Earth Intersection  Intersection $RBn0_[000_ o[0000_[00 o v[o _p[0000_[00;p jF6@s %Zx  :x B p _q[0000_[000_ r[000_[00 r vr _s[000_[00~s *thDm  SDp  71 / Dp Dm  ###,###,###.0A0BP"B1 / 1,301,290.9s t[000_[0t+y |Dm  gDp  K 1 Image mm = Dp Dm  ###,###,###.0A0 Earth MetersBP"B$!1 Image mm = 1,301.3 Earth Meters t  u[000_[000_ v[0000_[0 v v w[ wv ww w8 w_[0 wIw {Gz?.3Z< Z<Z 'B w _JTxut@vC@Z>DVNo Earth Intersection No Earth $VBxTx ]tpC@N>DXNo Earth Intersection No Earth $XBxTx3[hC@T>DZNo Earth Intersection No Earth $ZBx[000_ XyLp^PBDVNo Earth Intersection  Intersection $VByXylo^NBDXNo Earth Intersection  Intersection $XByXy,r ^LBDZNo Earth Intersection  Intersection $ZB y[0 y\;y 9@{ %Zn  :n B y _z[vw[000_ {[ {{000_[0 {~{ T+|hDw  SDy  71 / Dy Dw  ###,###,###.0A0BP"B1 / 1,253,284.3{ |[d|^t+\NZ  8. Towards Shuttle NadirB|000_||x+/|Dw  gDy  K 1 Image mm = Dy Dw  ###,###,###.0A0 Earth MetersBP"B$!1 Image mm = 1,253.3 Earth Meters |  }.}  Dql> n 6 6 V4i 4 42 T,   @  h    +> Zearthweb@ems.jsc.nasa.gov g  ' +Z >;http://eol.jsc.nasa.gov/sseop/FootprintCalculator.xls  <,,450Nd  4(  p  6 A 55]trj  0 A@s5M5]@txp N 6@A?)]bNu 5  $ # Drj O 0@A ]bOv 5 $ #Dxp w 6@A? ]bwv 6 4$ 4#Dh X =  L a] `w^VB ? S  1 B|x|]?Dwjb2 H s * @B =K]Hxph2 I 0@A B Ez]I4yphB J 0D 1 ]J`zphB K 0D 1 0]K`zphB LB 0D 1 ]L`{vnB S 6D jJ EE]S`|vnB T 6D jJ D0D]T`}vnB U 6D jJ CD]U`~vnB V 6D jJ {{]V`lvnB W 6D jJ y2y]W`TvnB X 6D jJ yy]X`<vnB Y 6D jJ E{]Y`$vnB Z 6D jJ @w]Z` jb2 x s * @B ]xjb2 y s * @B q]yXjb2 z s * @B =tK]z0jb2 { s * @B u]{jb2 | s * @B ru]|ljb2 } s * @B ?M]}Іjb2 ~ s * @B  ]~jb2  s * @B t ] hX \l  dav] `䈛jbB s s *D  e]s``ph2 b 0@A B es]b8jbB d s *D  s]d`vnB l 6D jJ dvd]l`|vnB o 6D jJ s]o`dvnB q 6D jJ e]q`LvnB r 6D jJ `ss]r`4d\2  c $ @Bm{]d\2  c $ @Bm{]d\2  c $ @B\liz]Pd\2  c $ @B^k ]d\2  c $ @B ]d\2  c $ @B ]葛d\2  c $ @B]d\2  c $ @B]k]d\2  c $ @B]쓛  Z0Ge+H@ @>?#" `)jk]`0P <Central Image Pixel<qhX    kCy] `B  HD jJ?#" ` '(]`B  HD jJ?#" `XY]`$  H  @V?#" ``]`,f(V     ] `42  B @>?#" `Nc3]`<vn2  6Z@A8clU{y]@d\B  c $Z8caynz]@$vn2  6Z@A8cYzc]@vn2  6@A8c6UDz]@pd\B  c $8cCzN{]@Xvn2 B 6@A8cO{X]@  s pZ0e0e    B9CDE(F  @  8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||   29 @   c"$` )6T]`<  s p0e0e    B9CDE(F  @  8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||   29 @   c"$`{)W]`У<  s p0e0e    B CDE(F  A@  8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||    @   c"$`Xq]`,<f V \a  \a] `<ph2  0@A8c]@\<^VB  S 8c]@<<ph2  0@A8c(]@<ph2  0@A8c=K]@<^VB  S 8c2=]@`<ph2 B 0@A8c(1]@IJ<  s p0e0e    B9CDE(F  @  8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||   29 @   c"$`\Ka]`<  s p0e0e    B9CDE(F  @  8c 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||   29 @   c"$`]]`и<nJ^ ,   # #O] `,<  T )+yd m,  ]`< 8<A complete guide to the use of Astronaut Photography for Remote Sensing is contained in the following paper: Questions or Problems Should be Emailed to: Software Updates can be Obtained from the Web Site: <8l|$(T  <  Jĕ  @E-mail" =s  ]`ĕ<  ":<earthweb@ems.jsc.nasa.gov<<<  ND  @?#" `<  ]`D,<  5\b:<6http://eol.jsc.nasa.gov/sseop/FootprintCalculator.xls<<5<  Nl  @?#" `) iI ]`l<  "pb:}<The mathematical technique and this Excel based calculator were developed by: Donn A. Liddle, Lockheed Martin Space Operations, The Earth Sciences and Image Analysis Laboratory /SX3, NASA - Johnson Space Center, Houston, Texas, 77058<<  N  @?#" `/ O ]`,›< <Copyright Infomation Low Oblique Space Photo Footprint Calculator , 2002, Lockheed Martin Corporation, All Rights Reserved<1-?<  N  @?#" `-_  ]`<Û< <Robinson, Julie A.; Amsbury, David L.; Liddle, Donn A., Evans, Cynthia A.; "Astronaut-acquired Orbital Photographs as Digital Data for Remote Sensing: Spatial Resolution", International Journal of Remote Sensing, Vol 23 No.20, Oct 20, 2002, P. 4403-4438 <GE <>@@cV0e2d@<,&p= ף?'zG?(333333?)\(\?"7??RTTRRPPNNVVXX GGJJLLLLJJ((++NNPPRRTT BB CC DD ,,--77 00** ++       FF CC____&& ''   ..  ``DD__``   ,, -- //..// wwxxxx``ss ii tt !!""00 ZZZZXXVVAA xxyy}} yyyy|| {{  7v D CD         &' +0+- ;$CD       JNone;4       @ @ hd  B94       hd  B.0 .0 ;$.0       None;$       d  B<<<<;4<<       @ @ hl  B Sheet2 |' @u$.Z TJ",9#EP$X  dMbP?_*+% &A&RPage &P'zG?(RQ?)Gz?"D??cU} m } } I } m }  } m }  } m } } m u   V@ @ h@ ,@ V@ h h h   ;                       h h       W@  @@ @@ ?@@  "( " " j   ,  , }Y ,Z  y= ,> ', z= ,? {, |= ,    ##  C B<X*,,  (4444$(((( ! " # $ % & ' ( ) * + , - . / 0 1 2 3  4  5 6 ;7 ;8 ;9 : ; < = ,> ?   D !,"?#$%,&'() )) * * * * *+ +W+, - -X- . ../01 2 2D23 4 5 6${6 7$7 8$ 9 9,9 : :,p: ; < <,q< <,< = =`}=6 =6I=M̌C@=3Z  Z :0yE>: B =6==ďC@'D=  hD= $=B =6= >66 >6>M̌^KZ  >6=>dm@_'D>  hD> $>B >6> ? Dl22H2222$$00$88$N@ A ,B C  D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _  @ A A`KA6 A6AMfffff&C@AZ  A6=Adfffff&C@'DA  hDA $AB A6A B66 B6BM^BZ  B6=Bdm@'DB  hDB $BB B6B C D DJD E F F"F G GG H HrH I ILI J K K  K6K\Q D>A  BK  L M MoM N NN O P P(P Q Q Q<Q]Y&D>DBA  BQ  Rx Sx T T!Tx U UUx V VVx W WWx Xx Y Y Y:Y]>$D>A  BY  Z^ [ [o[ \ \\ ] ^ ^'^ _ _ _<_]G&D>DBA  B_  D l$$:$::::$|$::$:$$::::$$::$:` a b c  d e ,f g , h i j k l m n o p q r s t u v w x y z { | } ~   `^ a a&a^ b bb^ c_ d e ec)e8 e6;eMC@f%Z  : B e6iedC@SDeDfB$ .De  hDe $e"B e6e f6 f6;fM^f%Z  : B f6ifd33333m@eSDeDfB$ .Df  hDf $f"B f6f g`J88a8e6 h`J8 hh i`J8 j`J8 jIj k`J8 kk l`J8 lEl m`J8 mm n`J8 o`J8 o oRo\/><De 'DBA  "Bo  p`J8 q`J8 qoq r`J8 rr s`J8 t`J8 t(t u`J8 u uXu]/oBDe -DBDfA  "Bu  v`J8x w`J8x x`J8 xxx y`J8 yyx z`J8 zezx {`J8 {{x |`J8x }`J8 } }V}]/u@De +DBA  "B}  ~`J8^ `J8 o Dn l$::$$ $:$::::$$::$:$$::::$$        ,       ,     J J              `J8  `J8 `J8 ' `J8  X]/}BDe -DBDfA  "B  `J8^ `J8  ^JJJJ `J8 ^JJJJ 6_ 66a8e6  ,G66a8e666a8e6  ,6;Mq= ף`TV%Z  : B 6Ydp= gq@7CD .D  hD $"B 666a8e6::b: A A[  m   D@D\l:$:$::$$0((00002                    ;           h h h h   3  S6 4   5 ) 7   8    D   $       0~ h   0~ h   0~ h    +N-- # # D # T   U DXl2>X22222$$0$8$PPP$$0$88$           ,   h                         V     "~  *2g/MXAD(\C@   AJJC  BJJC  66a6  # , 03.Xo@?DDD=AVAD>AVA  03.)聃ODD=AVAD>AVA  0'.PnV_NADD=AVA   K2g/MXA85DDDDDDA  , 03.,9^Q~DDDAAVADBAVA  03.EPDDAAVADBAVA  0'.Т@TSNADDAAVA  & K2g/MXA=5DDDDDDA  ,[ 0\O.= <_TaD9De $DDeAVADfAVAB  0]O.&c|6P9De $DDeAVADfAVAB  0^C. :MA-De DDeAVAB  ' g2g/MXAQD <DDDDDDAB D l::$$:$$(0qq0qq0                                     _    '~ QV@  '`~ L  a  0b3.n0=DDAVADAVA  0c3.DDAVADAVA  0d'.2g/MXADDAVA 0  e K2g/MXA5DDDDDDA 0 _  h  '~ Q  9  '`Lm@D>     03.W1IDDAVADAVA  9  03.1TDDAVADAVA  0'.DDAVA 0   K2g/MXA 5DDDDDDA 0D l((R>qe((Rce                                                      0 %`%%wpbwڿ=ZZ  0a %%%>ZZ  <=  0 %%%v.??ZZclh,nMDDDA  NoneNot Unit VectorBNone m    8     09 %: %% ‹`zڿ ZZ    0; %< %% @. ZZ  ? <=   0@ %o %% ? :? ZZ c lt,oMD D D A  NoneNot Unit VectorBNone m        0 %%C$;Nڿ-Z ZZB  0 %%CpY-Z ZZB  <=  0 %%C9ɽ?-Z ZZBl,pmD  NoneTDDDA  NoneNot Unit Vector"BNone m 0%% 0%%  6   07 %8%!\3&< DD  09 %%! DD  <=  0 %e%!? DDcl,qMDDDA  NoneNot Unit VectorBNone m 0%% 0%%  f     D l*m*m*#*i}**   !  "  #  $  % & ;' ;( @) @* @+ @, @- @. @/ @0 @1 @2 @ 3 @ 4 @5 6 7 8 9 : ; < = > ?   0 % %! 7QR ! DD  ! !0 !%!%!! Ӱ" DD! ! !<= " "0 "%a"%!"" DD"c"l,rMD D!D"A  NoneNot Unit VectorBNone "m #NOOP % &$ & '$ ($ (,.(, )$ )D+) *$D +$ +D# +5+y0Z   B + ,$, -$ -,,- .$, /$, // 0$, 0$30lʡ?3-yB?E?0 1$, 2$, 3$ 3,- 3B3Qr@8,D+ Z D0: B 3 3 4$, 5, 6 6 6 7 7" 77u2g/MXA:Z 77 8 8 8uQr@eD3 8 8uA D8 88J 9 : :, :!:K2g/gnYA D7D8 :: ; < = =0h =%=%!=.d ڔE DD: == > >0 >%>%!>.HP DD: >> ? ?0 ?%?%!?.^g& PA DD: ?? D li}*24u44i2giuu@  A B  C w@D E F G H I J K L M N O P Q R S T U  V  W X Y Z [ \  ]  ^ _ @ AB CILC D DE F FH FIFG G,dYGQ@o@IC#Z BCustom  Custom#Z "B;GR,G%Z  : B GJGH I I,K I?IS@o@)DGCustom  $G $GB IJ IJK L LHL LILM M, MP MP MT MT M N_NQ@Z@NI#Z BCustom  Custom#Z "B_NQ^@NI#Z BCustom  Custom#Z "B;NU,Q%Z  : B;NU,Q%Z  : B NOP,,, P P P, Q Q,* Q,,?Qy@Z@N)DNCustom  $N $NB?Qy^@])DNCustom  $N $NB Q,JQRtt S SSJ T TTJUJ V V V9VsN#DeDfB$DB$ VWJ X XXtt Y YY Z ZZ[ \ \.\ ] ]/ ]F]s,c0DV DQDQB$  B]^ _ __D2 l((>F`b>22w2222` a b c d e @ f @g @h j ;k  l ;m J n ;o ;p;q;rJ s tJ ` `b`ab b bb c c,c cXcy@Z@BBD] )DQDQ  $Q $Q" $QBXcy^@cBD] )DQDQ  $Q $Q" $QB cJd,efgh j j l l l mm m~ n?Bn,n,DNone  NoneErrorBNone9n #DnNone  B~ o@Bo,o,D None  NoneErrorBNone9o#DoNone  B~ p@Bp,p,DNone  NoneErrorBNone9pt#DpNone  B~ q@Bq,q,DNone  NoneErrorBNone9qA#DqNone  B~ r@Br,r,D"None  NoneErrorBNone9r#DrNone  B tt#t %nr,?|2<*&H@0 H(   J J <TJ s x2] sʛtLDB [ #  (<(][8˛jb2 Z  s *@A B:#5]Z˛LDB \ B #  rN&7]\L̛LDB ] #  OO|(]]̛XH S:H ^ S:H] ^`͛fX$H S:% _ S:%] _͛bjb2 ` B s *@A BS4%]`Λjb2 a  s *@A BS:%]aXϛX$H 8':H b 8':H] bϛejb2 c s *@A B8'nH]cLЛjb2 d s *@A B4':H]d$ћjb2 e B s *@A BSn04]eћd\2 f  c $@A!)4]f`қLDB g # !$(]gқRJB h  3 !|(*04]htӛX^H +W- o +W-] oӛpjb2 i s *@A!B+,]ihԛXPB k C !B++]k@՛LDB l # !+W-]l՛XjH W@"( p W@"(] pT֛qjb2 j  s *@A!BW@"(]j֛XPB m  C !BW[!P]mכXPB n  C !BWP !]n ؛^V2 q  S  @D$%]q@؛XPB r  C  O(OTJ]r0ٛld2 t s *@A B<]tٛZ&J ~2;L v ] v@ڛyjb2 w B s *@A B~nL]wڛjb2 x  s *@A B!~2;K]xۛZ&J ;H y x] yۛ|jb2 z s *@A BnH]ztܛjb2 { s *@A Bm;H]{Lݛld2 | @ s *@A B!x]|ݛf^B } c $ z]}ޛf^B ~ @ c $ m]~ޛf^B  c $ LL]ߛZRB  C  LL]ߛZ`J $K  L] Hjb2   s *@A#B$K]ߛXPB   C #B]9$K]LLDB  # # ]9]Z6J  "&hL  i] `LDB  B # ! 7&K]LDB  # ! hL hL]@LDB  # ! " hL]`X2  S  @q]@T` a c d e f i j  w  x  z  {  |  >@@cV0e2d@< &A&RPage &P'zG?(RQ?)Gz?"D??LLLL7 Sheet3 |' Hssz T-p   dMbP?_*+% &A&RPage &P'zG?(p= ף?":??cU} $} m } }  } $} I} m } $ } I } } m} } m    V@ @ {@ ,@ ,@ h  @ @ ;            h         W@@@@@@@@@@@   @@@@@@@@@@@  @@@@@@@@@@@ ?@@@@@@@@@@@  "" / / / ! R 4 4 44 44 4l4 44 4m4 4F4 4G4 # AH              ?           @D.88$ "D"888N"N ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 9  :  ; < = > ?      ! " # # #  $ % % % & ' ( (( ( ( ( (  ) )? ) )) * ** *  *  + ++ + +  , ,,, - -- -  -  . ..@ / / /  0 0S 0  1 2 3 4 4  4  5 6 7 8 9 ; < < = = > ? ?:? DzlL""8"6""bLNN8N888"""8"""""".."@ A B C D E F G H I J K L M N O P  Q  R S T U V W X Y Z [ \ ] ^ _ @ @@ A B B,tB C D DsD E EE F G G-G H HH I J J|J K KFK L M N N,N O O,OO u2g/gnYAK Z: O /O P R S S,0 S1S T U U2 U UU V V V VV W WW X Y YY Z Z@ Z ZA ZfB ZgC~ ZhZ [ [D [hB~ [h[ \\fhh?\ ] ^ _ __ Dl6"6"68"68"68""6i""B"LL8"6pPB""` a b c d e f g h i j  k  l m n o p q r s t u v w x y z { | } ~  ` `` a aa b bb c cc d d d e f f*!ffsm@fZ> ff f"f!fr7QR # DfAVA!f r Ӱ# DfAVA~ f rf g"gr Ӱ?$ DfAVA!g r7QR $ DfAVA~ g rg hhrrr? h i j l m m,# m$m n o o2 o o5o p p p pp q qq r s ss t t t tA tfB~ tg tgDt uufh?hu v vC~ vf vhBv w x y yy z zz { {>{ | || } }<} ~ ~@~  D l66666"}@"""B"LL8"6pBP""666666                              B B B B B B *!s33333sI@ ZZ=  "!rv.? DAVA~ r" rPF_ DAVA rr?r  !rPF_? DAVA~ r! rv.? DAVA      ,  ,  ,  I      V  V V Y V Z V          07\3&<Z   09Z   0?ZD       07QR Z  CDDDD 0 ӰZ!DDEE CCDDDD 0Z"DDEE CCDDDD0JDDEE CCD D` DD Wa Xb Yc DEE CCDDD Dd\3&<DD?D DEE CCDDD De7QR D ӰDD DEE D l@|""""B8"6""B"88"8"mWW"mWW"f B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B BCCDDDD0JDDEE CCDDDD 0f/Z Ӱ?DDDDDDEE CCDDDD 0g1Z7QR DDDDDDEE CCDDDD 0i/Z&8tލDDDDDDEE CCDDDD0JDDEE CCDDDDJDDEE C Vj =EE C EE C  EE C EE C  >  0wpbwڿZEE C  0aZEE C  0v.?ZDEE C EE C    09‹`zڿZ EE CDDDD 0;@.Z DDEE CCDDDD 0@? :?Z DDEE CCDDDD0JDDEE CCD D DD Wa Xb Yc DEE CCDDD DwpbwڿDDv.?D DEE CCDDD D‹`zڿD@.D? :?D DEE CCDDDD0JDDEE CCDDDD 0/ZAI[?DDDDDDEE CCDDDD 01ZDDDDDDEE CCDDDD 0/ZqjDDDDDEE CCDDDD0JDDEE C V D}DDD0JDDEE C[DDDD0JDDEE CCD D~DD0JDDEE CCDDDD0JDDEE CCD\D \=^*?'DDDDDDJDDEE CCDDD \6^? DDDAJE D l"iki""B"8"mWW"mWW"f"iki"B"8"w B B B JB  B  B B B B B B B B B B B B B B ;B ;B B B ;B B B B B B B B BCCDDD \6^7ˍ? DDDAJE CCD kb \L^#KQn?6DD DDDBEE CCDDDD0JDDEE CC &cI^vbS?3DD  @D@AcB J_BADh0@DAW D n oE CCDDDD0Jil,SD 70Error - Value Greater then 1. Cant Compute ACOSNoneBNone mmE CCDDDD0JDDEE C V DGDDD0JDDEE CCDDDD0JDDEE CC DDDD0JDDEE CC DdDDD0JDDEE CCDDDD0JDDEE CCDDDD0JDDEE CC DDDD0JDDEE CCDDDD0JDDEE CCD DDD0JDDEE CCDDDD0JDDEE CCD DDD0JDDEE CCDDDD0JDDEE CCDD,,D DW D EE CCD &DD0D{]N@ eDD  @EZBZ>  :A@Z@"B _K~yu@ DAW EE CCDaDD0JDDEE CC  DDD0JDDEE CCD DDD0JDDEE CC&&DJDDDEE CCD DDD0JDDEE CCD DDD0JD8 + "Z>A ZBA B$ EE CCDDDD0JDDEE CCD D DD0JDDEE CCDD DFD0JDDEE CCDDDD0JDDEE CCDD DD0JDDEE CCDD DD0JDDEE D lp""B"88""8"8"8"D"88"8|"D8"8 B B B B B B B B B B B B B B B B B B B B  JB  B B  B B       CCDDDD0JDDEE CCDD DD0JDDEE CCDDDDDDDD E ECCDD ?D0JDK +,5D  ZB Z> B$ BG d-1D  DAWNONEBNONE ECCDD D)D0JDEE CCDDDD0JDEE CCDDDD0JDEE CCD D D*D0JDEE CCDD D+D0JDEE CCDDDD0JDEE CCDD DtD0JDEE CCDDDD0JDD E ECCDD D0JDK + -5D  ZB Z> B$ BF d- 0D  ADAWNONEBNONE ECCDD D,DDDDDEE CCDDDDDDDDEE CCD DDD0JDbEE CCDDDD0JDbEE CCD D DD 0o]- uD   NONE]D  D6D  ADNONE""BNONE JB _(-,DNONE  NONEDAWBNONE E ECCDDDD0JEE CCDDDD0JDDEE CC , e[=cK~yu@'DNONE  $  $ B DDDDEE CCDDDD0JDDEE CFGGGGGGGGHE CDDDDDDDDDDE CDDDDDDDDDDE  2             @  A fB gC~ h  D hB~ h D l"868""D8"868"8"K""""""LL8"6p                 h                      fhh?     ,pK~yu@ D  "!r#KQn? DAVA! r:ҿ DAVA~  r "r:? DAVA! r#KQn? DAVA~  r errr?      #      s       ,  ,                      q#KQn?  Lq:ҿ#qpv.? LeppPF_   q:? q#KQn?$qf pp?p    qqq?pPF_?%p%pv.?%     Db lB"""}@"""."6"6"N"N"N"N""N"  ! " # $ % & ' ( ) * +  ,  - . /  0  1 2 ;3 4 5 6 7 8 9 : ; < = > ?   ! " "% "%% " "  ##dq!M\?##-#% #LL@LL@LL@#d:ҿ###dzqЁ+###p7QR g##% LC# p Ӱg ## p$ ## $$d%+?$#$d#KQn?$#$d+̿,# $$p Ӱ?%#$ p7QR % #$ p% #$ %%dPF_?#%d #%dv.?# #%%p-#% p-#% p?-#% & ' ( ) * *%*%% + +? +9+dK+#D#D#@D#D$@D#D%@+d<{!׿++-+-#LL#@LL$@LL%@+dzqЁ+ +  ,k,l4-Ue+-A  None5+Composite Matrix Determinate not Equal to 1BNone ,m9, ?,#D$D#@D$D$@D$D%@,d/7j.,+,d+̿,+ ,  -9-dwpbwڿf #D%D#@D%D$@D%D%@-dU+-dv.?+ -  . / 1 2$2 3 4 4{4 5 6 7 7~|7 7~} 7  8 8~~ 88 88 8 8~ 8  9 9~9 9~ 9  : ; < <A< = > ? ?~|? ?~}?  ? ~B? D l""N$""""8#"""."6""NxN""6""@ A B C D E F G H I J K L M N       !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuwxyz{|}~O P Q R S T U V W X Y  Z  [ \ ] ^ _  @ @~~ @@ @@ @ @~ @ L @ ~M@  A A~A A~A  A ~A B C D E EE F FF G H I Ih|IILKD+IL<{!׿ID+ILzqЁID+I Ih}I ~ I h I  J Jh~ JJL ?ID,JL/7j.JD,JL+̿JD, J Jh J L~ J h J  K KhKKLwpbwڿD-KLKD-KLv.? D-K KhK K v2g/gnYU DO  K L M N O O,O P Q QQ Q Q  R S0 Sw SSLKSD+SL<{!׿SD+SLzqЁXD+SSvTZ S ~ S t S  T0 Tw TJTL ?TD,TL/7j.TD,TL+̿YD, TTvUZ T ~ T t T  U0 Uw UULwpbwڿUD-ULUD-ULv.?O D-UUvZZ U U v2g/gnYSDK U V W X Xx@X.T*DSDSDSDTDSDUDS  XX Y Yx@Y.Z*DTDSDTDTDTDUDT  Y YnYo Z Zx@Z.2g/gnYc*DUDSDUDTDUDUDU  ZZl@-DXDYDZDO B  None\RError: C of Earth Not at Correct Local Position X=0, Y=0, Z=-Earth Radius+AltitudeBNoneZm [ \ ] ],] ^ _ __ _ _  DAl`"""66"""""6"N"""?""6"` a b c d e f g  h  i j k l m n o p q r s t u  v  w x y z { | } ~  ` a0 aw aaLKaD+aL<{!׿aD+aLzqЁfD+aavd ڔEbZ= a ~ a t a  b0 bw bJbL ?bD,bL/7j.bD,bL+̿gD, bbvHPcZ> b ~ b t b  c0 cw ccLwpbwڿcD-cLcD-cLv.?c D-ccv^g& PAhZ? c c v2g/gnYaDK c d e f fx@f.>b*DaDaDaDbDaDcDa  ff g gx@g.h*DbDaDbDbDbDcDb  g gg h hx@h.q*DcDaDcDbDcDcDc  hhlL-xDfDgDhB  NoneD:Error: Shuttle Not at Correct Local Position X=0, Y=0, Z=0BNonehm i j k k,6k l m mm m m  n o0 ow7 ooLKoD+oL<{!׿oD+oLzqЁtD+oovXo@?DpZ o 8~ o t o  p0 pw9 pJpL ?pD,pL/7j.pD,pL+̿uD, ppv)聃OqZ p :~ p t p  q0 qw; qqLwpbwڿqD-qLqD-qLv.?q D-qqvPnV_NAvZ q <q v2g/gnYoDK q r s t tx=@t.p*DoDoDoDpDoDqDo  tt u ux>@u.v*DpDoDpDpDpDqDp  u uu v vx@v.K*DqDoDqDpDqDqDq  vvlX-DtDuDvZ8B  NoneWMError: Point 'sn' Not at Correct Local Position X=0, Y=0, Z=-Shuttle AltitudeBNonevm w x y y,.y z { {{ { {  | }0 }w/ }}LK}D+}L<{!׿}D+}LzqЁD+}}v,9^Q~DZ } 0~ } t }  ~0 ~w1 ~J~L ?~D,~L/7j.~D,~L+̿D, ~~vEP}Z ~ 2~ ~ t ~  0 w3 LwpbwڿD-LD-Lv.?D-vТ@TSNA~Z  4 v2g/gnYDK  Dl"""!""6"N"""<""6"N"                         ;   ; J  ; ; ; ; ;   x#@.4!@J*D}D}D}D~D}DD}    x$@**D~D}D~D~D~DD~   %o  x&@.Eo K*DD}DD~DDD  |ld-fD B   None>4Error: Point 'pc' Not at Correct Local Position Y=0BNone    ,v        0 ww LKD+L<{!׿D+LzqЁD+v= <_TaDZ  x~  t   0 wy JL ?D,L/7j.D,L+̿D, v&c|6PZ  z~  t   0 w' LwpbwڿD-LD-Lv.?D-v :MAZ  ( v2g/gnY DK     x)\.ɾV@FD 1DDDDDDD B   x*\. @}FD 1DDDDDDD B   x+\. x](FD 1DDDDDDD B          ~ ?Bp-,DNone  NoneErrorBNone-#LNone  B~ @B|-,D,None  NoneErrorBNoneS~ @B-,DZNone  NoneErrorBNonea~ @B-,DhNone  NoneErrorBNoneo~ @B-,DvNone  NoneErrorBNone~@ D""""6"N"""""*&~~~J  J ~ @B-,DNone  NoneErrorBNone # % (~PB( 3f3333 HJ <B geYi]i] gfRJB ` 3 @<<B]`@ RJB b 3 @<||]b@RJB c 3 @<BB]c@(Z<J |E fYxG]] fhRJB a 3 @|E]a@\LDB dB # @||]d@LDB eB # @EgE]e@pZHJ &E he$i] hlRJB i 3 @&&E]i@PRJB j 3 @&}W}]j@RJB k 3 @&EE]k@lZ<J | E l ex i] l$pRJB m 3 @ | E]m@LDB nB # @||]n@LDB oB # @EAE]o@ZHJ <B pesiwi] ptRJB q 3 @<<B]q@RJB r 3 @<||]r@LRJB s 3 @<BB]s@Z<J 0|E tsx<w] thxRJB u 3 @|E]u@LDB vB # @0|5|]v@HLDB wB # @0EE]w@ZHJ &E x$] x\|RJB y 3 @&&E]y@RJB z 3 @&}W}]z@RJB { 3 @&EE]{@Z<J | E | x ] |RJB } 3 @ | E]}@(LDB ~B # @||]~@LDB B # @EAE]@<ZHJ AB \ii] RJB  3 @AAB]@RJB  3 @A||]@RJB  3 @ABB]@Z<J <F S] ,RJB  3 @F]@LDB B # @<]@`LDB B # @x*Bx]  L.<RJB  3 @Q|QC]@.<RJB  3 @QC]@,/<RJB  3 @QCC]@/<Z<J VC >iSBx] H0%<RJB  3 @VVC]@0<LDB B # @G]@|1<LDB B # @C9C]@1<ZHJ  E % > HB] %2)<RJB & 3 @E]&@ 3<RJB ' 3 @}D}]'@p3<RJB ( 3 @E E](@(4<Z<J <}H ) => B] )4-<RJB * 3 @}H]*@5<LDB +B # @<} }]+@5<LDB ,B # @<H H],@$6<ZHJ Q|C -Hx*Lx] -61<RJB . 3 @Q|QC].@P7<RJB / 3 @QC]/@7<RJB 0 3 @QCC]0@l8<Z<J VC 1HiSLx] 185<RJB 2 3 @VVC]2@L9<LDB 3B # @G]3@:<LDB 4B # @C9C]4@h:<ZHJ ^-F 5HkL] 5;9<RJB 6 3 @^^F]6@;<RJB 7 3 @^}E}]7@;<RJB 8 3 @^F-F]8@<<Z<J lF 9 H IL] 9==<RJB : 3 @llF]:@=<LDB ;B # @Z];@H><LDB <B # @FZF]<@><ZHJ |$C =Hx0Lx] =\?A<RJB > 3 @|C]>@?<RJB ? 3 @]?@<@<RJB @ 3 @C$C]@@@<Z<J }H AH0L] AXAE<RJB B 3 @}H]B@A<LDB CB # @}}]C@B<LDB DB # @HAH]D@B<ZHJ  E E H HL] ECI<RJB F 3 @E]F@D<RJB G 3 @}D}]G@D<RJB H 3 @E E]H@8E<Z<J !}H I H L] IEU<RJB J 3 @}H]J@F<LDB KB # @!}}]K@F<LDB LB # @!HH]L@4G<ZHJ Q|C URx*Vx] UGY<RJB V 3 @Q|QC]V@`H<RJB W 3 @QC]W@H<RJB X 3 @QCC]X@|I<Z<J VC YRiSVx] YI]<RJB Z 3 @VVC]Z@\J<LDB [B # @G][@K<LDB \B # @C9C]\@xK<ZHJ K}E ]RSV] ](La<RJB ^ 3 @K}KE]^@L<RJB _ 3 @K]_@M<RJB ` 3 @KEE]`@M<Z<J |E a Rx IV] a$Ni<RJB b 3 @|E]b@N<LDB cB # @||]c@XO<LDB dB # @EE]d@O<Z<J }H iR0V] ilPm<RJB j 3 @}H]j@P<LDB kB # @}}]k@LQ<LDB lB # @HAH]l@Q<ZHJ  E m R HV] m`Rq<RJB n 3 @E]n@R<RJB o 3 @}D}]o@S<RJB p 3 @E E]p@S<Z<J !}H q R V] qT<RJB r 3 @}H]r@,U<LDB sB # @!}}]s@U<LDB tB # @!HH]t@@V<ZHJ  E  R HV] V<RJB  3 @E]@ W<RJB  3 @}D}]@W<RJB  3 @E E]@iBi] 8]<RJB  3 @TTB]@]<RJB  3 @T||]@^<RJB  3 @TB9B]@^<Z<J |aE >x 1B] 4_<RJB  3 @a|aE]@_<LDB B # @||]@h`<LDB B # @E0E]@`<ZHJ F| C >xBx] |a<RJB  3 @F|FC]@a<RJB  3 @FD]@\b<RJB  3 @FC C]@c<Z<J }H >0B] xc<RJB  3 @}H]@c<LDB B # @}}]@d<LDB B # @HAH]@e<ZHJ Q|C `x*dx] e<RJB  3 @Q|QC]@ 3 @ssE]>@<RJB ? 3 @s}}]?@X<RJB @ 3 @sEE]@@<Z<J }H An0r] AtE<RJB B 3 @}H]B@<LDB CB # @}}]C@<LDB DB # @H_H]D@ <ZHJ Q|C Ex*x] EI<RJB F 3 @Q|QC]F@8<RJB G 3 @QC]G@<RJB H 3 @QCC]H@T<Z<J VC IiSx] IM<RJB J 3 @VVC]J@4<LDB KB # @G]K@쨄<LDB LB # @C9C]L@P<ZHJ K}E MS] MQ<RJB N 3 @K}KE]N@|<RJB O 3 @K]O@઄<RJB P 3 @KEE]P@<Z<J |E Q x I] QU<RJB R 3 @|E]R@x<LDB SB # @||]S@0<LDB TB # @EE]T@<Z<J }H U0] UDY<RJB V 3 @}H]V@<LDB WB # @}}]W@$<LDB XB # @HAH]X@ԯ<ZHJ  E Y  H] Y8]<RJB Z 3 @E]Z@<RJB [ 3 @}D}][@l<RJB \ 3 @E E]\@б<Z<J !}H ]  ] ]a<RJB ^ 3 @}H]^@<LDB _B # @!}}]_@h<LDB `B # @!HH]`@<ZHJ  E a  H] a|e<RJB b 3 @E]b@<RJB c 3 @}D}]c@<RJB d 3 @E E]d@<ZHJ sE e] e̶i<RJB f 3 @ssE]f@H<RJB g 3 @s}}]g@<RJB h 3 @sEE]h@d<Z<J }H i0] iȸm<RJB j 3 @}H]j@D<LDB kB # @}}]k@<LDB lB # @H_H]l@`<Z<J } F m K" &] mq<RJB n 3 @  F]n@<LDB oB # @}e ]o@<LDB pB # @}F F]p@<Z<J `|E qex?] q<RJB r 3 @|E]r@<LDB sB # @`||]s@8<LDB tB # @`EE]t@<ZJ h?WM K0] L<jb2  s *@A B(90>]ȿ<LDB  #  2h?2],<LDB  B #  (#%@] <LDB   #  2] @<X$H K92 K92] <jb2 B s *@A B2]l„<jb2  s *@A BK9n2]DÄ<X$H y3K9WM y3K9WM] Ä <jb2  s *@A By3NWM]8Ą<jb2   s *@A B3K9WM] ń<jb2  B s *@A BNg=] tń<^VB 3B S #,a2&]3LƄ<jbB 5 s *!X1 S#]5Ƅ<XPB 6 C B## ,']6DŽ<d\2 = c $@A#6%9]=DŽ<d\2 ? c $@A#)%6]?Ȅ<XH v0 ; :v0 ;] : ɄA<XH v0 ; v0 ;] Ʉ9<d\2  c $@A#Nv0 ;]@ʄ<LDB  # #',2]˄<RJB 9 3 #2v0 ;]9t˄<LDB AB # #]%6%-8]A,̄<LDB BB # ##6]%"7]B̄<RJB E 3 #( )!)]E@̈́<RJB F 3 #) *]F̈́<XH v0< @v0<] @\΄C<XH v0< 8v0<] 8΄<<XH v0< v0<] Tτ7<d\2  c $@A!N(<]τ<LDB  # !v02]Є<RJB 7 3 !2(<]7ф<d\2 < c $@A!56]<ф<d\2 > c $@A!6?.5]> ҄<LDB CB # !X56]C҄<LDB DB # !55]DTӄ<RJB G 3 !6?..]GԄ<RJB H 3 !p?./]HhԄ<LDB vB # @z+z+Z]v@ Մ<d\2 w c $@A@375]w@Մ<RJB x 3 @b,-7]x@Tք<RJB y 3 @56]y@ք<d\ z c $@A@+1N]z@pׄ<RJB { 3 @z+Z6Z]{@ׄ<^V | S @4z4]|@؄<RJB } 3 @z+b,]}@؄<RJB ~ 3 @-7O47]~@ل<d\2  c $@A@Y6{7]@ ڄ<d\2  c $@A@6{7Z]@ڄ<RJB  3 @-7-C]@@ۄ<RJB  3 @g37g3C]@ۄ<RJB  3 @-Cg3C]@\܄<ZbJ BgM  8] ݄<jb2  s *@A B+<>]݄<XH ,=gM ,=gM] ݄"<X$H < 5 < 5] ބ<jb2 B s *@A B 5]߄<jb2  s *@A B<4]߄<X$H 5,=gM 5,=gM] P<jb2  s *@A B5 gM]<jb2  s *@A B 5,=gM]<jb2 B s *@A B >]<XH  6< " 6<] "K<d\2 # c $@A# 6<]#p<LDB $ # #*-24]$@<X4H B%A KB%A] K\<LDB L #  4B4]L4<LDB MB #  +(%A]M<LDB N #  4]NH<XPB \ C `(#1*]\<XPB ] C `]->?/]]\<LDB ^ # @*? ]^@<LDB _ # !L6;]_<jbB  s *H#*r,*]0<^VB B S W0j!'5(]<d\2  c $@A,(W0*]\<XH L6> L6>] <XH L6> L6>] P<d\2   c $@A!N G.>] <LDB ! # !L64]!<RJB  3 ! h4G.=]<XH 0T|< 0T|<] <LDB B # @0T0]@H<d\2  c $@A@8:]@<RJB  3 @}1T2]@|<RJB  3 @:;7]@<d\  c $@A@076]@<RJB  3 @0;]@<^V  S @99]@<RJB  3 @0T}1T]@<RJB  3 @2G9]@<d\2  c $@A@W;7|<]@4<d\2  c $@A@;|<]@<RJB  3 @33]@h<RJB  3 @\8\8]@ <RJB  3 @3\8]@<  =?  < > w # >@@cV0e2d@<# I II  &A&RPage &P'zG?(p= ף?":??7 Sheet4 |' LHo#-)5.ESTTTTTT:U  dMbP?_*+%"??cU} m } m } m } m }  } }  m } } m    ,@ {@ h@ ,@ , h h h h   @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ W@@@@@@@@@@@  @@@@@@@@@@@@ @@@@@@@@@@@@ @@@@@@@@@@@@  " " " " + C  C  C  #    #  # # , # # #* #  #   # # # # # ,3 # # , Dl.00$  "..""88"8"""86""""8" @! @" @# @$ @% @& @' @( @) @* @+ @, @ - @. @ / @0 @1 @2 @3 @4 @5 @6 @7 @8 @9 @: @; @< @= @> @ ? @  # !# ! ! "# "  " " " ## $# %# &# && &,& '# '4' (#( )# ), )  *# +# , -"/ 0$0 1$1 2 3 33 4 5 5, 5 5,5 6 66n>7Zf 6 6#6m4!@7Z 66 7 77n6Zg 7 7$7m8Z 77 8 88nZh 8 8&8mEo ;Z 88 9 : ; ; ;K;lhg5A>5D6D6D7D7D8D8A ;; < = == > >0  >?!>(>k&>b?GD6D6D;> ? ?0" ???(?kF|HD7D7D;? ? ? <= D/ l"6L"""N8*8""" ".."6"N"""8v@ @ A @B @C @D @E @F @G @ H @ I @ J @K @L @M @ N @ O @ P @ Q @ R @S @ U  V @W @X @Y @Z @[ @\ @] @^ @_ @ @ @0 @?@(@kLID8D8D;@c@l-MD>D?D@A  NoneNot Unit VectorBNone @ m A B C CC D DD E F FF G G0 G%Gk&>bӿ?D>G H H0 H%HkF|<@D?H H? H <= I I0 I%IkL?ID@IcIl-MDGDHDIA  NoneNot Unit VectorBNone I m J0 K KK0KK@o@KZI KK?O DK K K J LJ M M,MM N N N <= O O0 O%O"Ok&>bO DGDK OJsOl-]DOeABeA  None* X value must be Negative or ZeroBNone O  P P0, P$P!PkF|<P DHDK PWPl-ADPB  NoneY must equal ZeroBNone P  Q Q0- Q#Q!QkL?Q DIDK QJQl-4DQ  NoneZ mut be PositiveBNone Q  R S U V W$W X$ Y$ YY Z$ ZZ [$ [[[ \$ \\ ]$ ]] ^$ _$ Bb X""88"8k"":8""""."68888"` @a @b @c @d @e @f @g @h @i @j @k @l @m @n @o @p @q @r @s ;t u v @w @x @y @z @{ @ | @} @ ~ @ @ `$ ` ` a$ b$ b? b c$ d$ dd e$ f$ g$ g,g h$ i$ j$ k$ kCk l$ m$ n$ nDn o$ p$ pEp q r rr s ss t u uu v w ww x xx y yy z { | ~ $y Dl8"8"8""8"""8""8"8"68"6"888"""" @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @                      , 0 &>b?Z >  0"F|Z ?  0LZ @     j  , ,  , ,   0u>Zf   , 0uZg   , 0su6Zh   , ,   =uZt   DDDD >uZu   DDDDD uZv   DDDDD0JD DD DD S KA5DDDDDDA   DD DD ,Q DD 0T ?(s-Oܼ DDD DD 0 ?(s-ODDD % <= DD 0 ?(DDDc l-MDDDA  NoneNot Unit VectorBNone m DDDDDJD D l"6"B"8"kW]""B"8"ycc"ycc"""8v           J  J  ; ; J  J  ; ; ; ;            DDDDDJD    , , DDD0JDD ,DDD0JDD ,\D \:^L?$DDDDDDJDD ,DD \6^? DDDAJ ,DD \6^? DDDAJ  , kb \(^L?DDD ,DDD0JDD  &I^ [?3DD  @D@AcB J_ҊY"|T1@DAW D noo &JDil.SD  None:0Error - Value Greater then 1. Cant Compute ACOSBNonem &JD  D&JD D&J noo  K&Sg=ju@ hD Dml .WD  None>4Error - +Y Rotation Can Not Be Less then 180 DegreesBNonem &DD D&JD  D&JD D&JD  2  5             A fB~ g gD  fh?h   fC~ f hB      *sSg=ju@D  "!L? DAVA~ " 7>b? DAVA D& l"B"8"tnnj""68""6"NN8"8rBP"""                    ; ; ;      ;      ; ; ;?    !7>bӿ DAVA~ ! L? DAVA  kl.Ue A  None5+Composite Matrix Determinate not Equal to 1BNonem     ?         O     ~| L? L7>b? ~} ~   ~~  ?  ~  ~   ~ 7>bӿL? ~ " п ZI Z   &  D&JD  D&JD D&JD   ~| L? L7>b?&>bZ O  Z Z   ~~  ? F|<Z P   Z [   ~ 7>bӿL?L?Z Q  пD  D&JD D @.<*DDDDDDD   D D @.F|< *DDDDDDD    D @.*DDDDDDD  l$.DDDB  NoneLBError: Principle Point Not at Correct Local Position X=0, Y=0, Z=0BNonem D&JD  a'D&JD  D&JD DKl@"66"8"t"""666"""")"6 ; ;      ;      ; J ; ;J ;;;;;;;;;;; D&JD D&JD   ~| L? L7>b?Z ^  Z h   ~~  ? Z _   Z i   ~ 7>bӿ8L?K Z `  пD  D&JD D @.*DDDDDDD   D D @.*DDDDDDD    D @.п*DDDDDDD  l0.DDDD B  NoneLBError: Principle Point Not at Correct Local Position X=0, Y=0, Z=0BNonem D&JD GPG       ~ ?DY<..Z @None  NoneErrorBNone- #LNone  B~ @D5H..Z INone  NoneErrorBNone~ @D5T..Z ONone  NoneErrorBNoneP~ @D5`..Z PNone  NoneErrorBNoneQ~ @B5l.,DQNone  NoneErrorBNone~ @D5x..Z None  NoneErrorBNone~ @D5..Z None  NoneErrorBNone~  @D5..Z None  NoneErrorBNone~ "@D5..Z None  NoneErrorBNone~ $@D5..Z None  NoneErrorBNone ~ &@D5..Z None  NoneErrorBNone@<6""""0""*&~ J #@ %Og @ @ @ @ @dP:U`3`S( @A@A 0^X AM 1 r)] jb2 1 s *@A BFn+B;?]0f<V F;M 1 F;M]  f2V F;a5 1 F;a5] jb2 1B s *@A BFa5]t jb2 1 s *@A B;4] f2V F6;M 1 F6;M]   jb2 1 s *@A BF6M]@ jb2 1 s *@A B6;M] jb2 1B s *@A Bb>]| fV ?4,< 2 ?4,<] T d\2 2 c $@A#?4,<] LDB 2 # #v*"04]fBV RADA 2 RADA] LDB 2 #  4A4]8LDB 2B #  n+%DA]LDB 2 #  R4]LXPB 2 C `%l#/v*]XPB 2 C !`+_?]\LDB  2 # !*= ] LDB  2 # !}47] PfV }4>  2 }4>]  ,fV }4>  2 }4>]  *d\2  2 c $@A!D,>] dLDB 2 # !}44]RJB 2 3 !h4,=]xLDB 2 # @/5v*]@^V2 2 S  @4l6M]@f(V 3. :~ 2 3. :~] LDB 2B # @3. 3.~]@d\2 2 c $@A@68]@(RJB 2 3 @*/ 0]@RJB 2 3 @89 ]@\d\ 2 c $@A@q. 4]@RJB 2 3 @3.~ :~]@x^V 2 S @78]@0RJB 2 3 @3. */ ]@RJB 2 3 @0d7]@Ld\2 2 c $@A@9 :]@d\2 2 c $@A@9:~]@RJB 2 3 @00]@RJB 2 3 @l6l6]@RJB  2 3 @0l6] @ZRB !2 C !`&  {]!NFB "2 # ! ]"ZRB #2 C jJ  ]#XZRB $2 C jJ &&D]$NFB %2 # @ (]%@(`X2 &2 S  @ MD Zm]&@`XB /2@ S 0 s s]/nR^  24v 02# ` D] 0 _XPB 12 C ` p!v]1p XPB 22 C `/q+p]2!LDB 32 # S i(]3H"LDB 42 # "/24v]4"LDB 52 #  5]5\#XPB 62 C ` 3 ]6 $n0^  &$ <2#  M] <p$LDB =2B # @  $]=@$d\2 >2 c $@A@R Y"."]>@%RJB ?2 3 @R ]?@&RJB @2 3 @Y"."$"]@@&d\ A2 c $@A@%"DX#]A@,'RJB B2 3 @ $$$]B@'^V C2 S @ !."]C@`(RJB D2 3 @ ]D@()RJB E2 3 @R R ]E@)d\2 F2 c $@A@#"&#]F@D*d\2 G2 c $@A@$#&$]G@*RJB H2 3 @R !]H@x+RJB I2 3 @R !]I@+RJB J2 3 @!!]J@,ldB K2 s *DjJ ` ]K`,H@ L2  B0CQDELF$ 05%0'     !&+2DGMPQN@      sm 7] L`H-XP M2  BbCkDEXF( 05%<60*" &  !&,2>M Ub,i<kHjWdbd@       m ] M`-80 N2  B)CADE@F  05%  %%'' )'&;AA @       7] N`h0hVX  -@ u2  Y] u2RJB v2 3 @ -@]v@D3RJB w2 3 @rr]w@4RJB x2 3 @-@-@]x@D5hVX B y2 x@r] y5<RJB z2 3 @B]z@$6<RJB {2 3 @<<]{@6<RJB |2 3 @BB]|@H7<hJX @\D }2  x .] }8<RJB ~2 3 @@@\D]~@|8<LDB 2B # @ ]@49<LDB 2B # @\D\D]@9<hJX KB 2 x fx] H:<RJB 2 3 @KKB]@:<LDB 2B # @<hJX !<RJB 2 3 @2UF2VG2_2`2a3b3c3d3e3f3>@@cV0e2d@<" "??<7 |' X/?!MK $`-7l@fHO@X/bn:r  dMbP?_*+%"??<cU} m }  }  } } m } $ } m /   ,@ h@ {@ ,@ ,@ h h h w  @ @ @ @ @ @ @ @ @ @ @ @ @ @ @  @   @ @ @ @ @ W@@@@@@@@@@@  @@@@@@@@@@@@ @@@@@@@@@@@@ @@@@@@@@@@@@  " " "/ " + $0   #  #  5 Ze  B  # #  5gZf  B # #  5 Z  B # # #  @l*$ $$B$  B # #  #    $ $ $  $  D#l.00$ "."w"w"w"""8""""."6 @! @" @# @$ @% @& @' @( @) @* @+ @, @- @. @/ @0 @1 @2 @3 @4 @5 @6 @7 @8 @9 @: @ ; V@ < @= @> @? @ $ [  !$ !! "$ "" #$ $$ %$ %% &$ '$ '?' ($ )$ )) *$ +$ ,$ ,,, -$ .$ /$ 0$ 0C0 1$ 2$ 3$ 4$ 5$ 6 7 77 8 88 9 : <+< == > ?C? Dl888""8"8"8""8"""8""""""68"".. @ @A @B @C @D @E @F @G @H @I @J @K @L @M @N @O @P @Q @R @S @T @U @V @W @X @Y @Z @[ @\ @] @^ @ _ @ @C@      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwyz{|}~ A B# C# CC D# DD E# EE, F# G# H#* I# II J# J J K# L# M# N# O# O,3O P# Q# Q,Q R# S# S S T# T  T T T U# V# W# X# XX X,X Y# Y4Y Z# Z Z 5Z [# [, [  \# ]# ^ _ Dl.""888"""86""""8"8"6L"""N8>8"""` @ a @b @c @d @e @f @g @h @i @j @k @l @m @n @o @p @q @r @s @t @u @v @w @x @y @z @{ @| @} @~ @  @ a bb c d e e e eBe f gg.Z X_Lap =gɾV@hZg.gZmgg.gZf X_Lss =g!>hZfg.gZfmg hh.gZ Y_Lap =h @iZh/hZmhh/hZg Y_Lss =h!Zgh/hZgmh ii /hZ Z_Lap =i x](Zi4/iZmii8/iZh Z_Lss =i!gZhiL/iZhmi j k l lClflZAArPD 9DgDgDhDhDiDiA B l l  m n nun o p q qECq fr ?{-D DgDgDlBq r rCrL#bվ?q-D DhDhDlBr s sCs r|-D DiDiDlBs t u v v,v w x xx y z zRAz fr ׿+Dq DqBz { {A{L#bվ+Dq DrB{ | |A| r?+Dq DsB| } ~ D l".""L""""6""}}}""6"8"{{{"" @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @  @  @ @  @ @  @ @ @ @ @ @ .      /  P/iZ > i_Lpc = &>bӿZ G d/L i_Lap-Lss =  fr ׿ |/Z ? j_Lpc = F|<Z H / j_Lap-Lss = L#bվs /Z @ k_Lpc = L?Z I / k_Lap-Lss =  r?z       DDD0JDD DDD0JDD \D \V^ɴ?@D +DDDDDDBJDD DD \6^? DDDAJ DD \R^?<D 'DDDABJ  kb \D^ɴ?.D DDDB DDD0JD D\  &e^xB?OD :DD  @D@Ac"B J:_$ A@$D DAWB D noo &JDl/*sD  NoneZD  None:0Error - Value Greater then 1. Cant Compute ACOS"BNonem    0 1 2    DQ l".""L"""6"6"p8"""D."" @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @  @  @ @ @ @ @ @  @  @ @   $  5           0 d  %   6            7 ,           D   4GeI(?1D ZIDAB    Dgl".6..""""6""J8888L"V."".""6"" @ @ @ @ @ @  @  @  @ @       ;      ;      ;                  00 >k,mQ(D DzDB J ]   01 3>k*E<"(D D{DB  ?<=  02 >kI !%?(D D|DB Jl/+4D  NoneZ mut be PositiveBNone      D&JD   ~| L?Z Z 7>b?Z ,mQZ   Z   ~~  iZ ?Z Z  *E<"Z    Z   ~ 7>bӿZ Z L?Z I !%?Z   пZ  D&JD D \,(ԑ0FD 1DDDDDDD B  ^D D \*E<"1FD 1DDDDDDD B   D \FD 1DDDDDDD B l/,D  NoneDj@Bj@  NoneJ@Error: Projected Auxillary Point Not on Photo Plane 'Z' must = 0"BNonem D&JD    M   Dn l66""&"""."""""b""".."                                 N      O                               Dl88"88""""""""""""""""""""""""""                 h                              P                      ?            Dl""""""""..""8""""""""""88""""""  ! " # $ % & '  )  * + , - . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ?   ! " # $ % %Q% & ' * + R+ +*u +  , - . .S. / 00/ 002 L X_Ppp =0,(ԑ00 000m 0 0r0 11010 Y_Ppp =1*E<"I01$020m1 22(020 Z_Ppp =2202<000m2 3 4 4q4 4s 4  5 5p5 6 7 7  8 8t8 8 v9 9T9 9  : ; ;; < << = > >kZ> ? B X"""""8"""D""6""L6".BB"66"6@ A B C D E F G H I J K L M N O P Q R S T U V W X  Y  Z [ \  ]  ^ _ @ @0 @[@ A0 AA B0 C0 D D0 D D E E E EU E  F FiF G H I IGI]D*?K1D D0D1AB II J K KjKBKSG]@M,D D0DIAcAWB KlK LL M MkMBMZqNP,D D1DIAbAWB MlM N O P PD PhPZqNYRD >DM DKDK"B PP Q, R, S SDjS T, U, UcU V, V V W, X, Y Y,/ YYYՆfѼr@uCDP .DP  $PhDP"B YY Z Z,Z [ \ ^ _Y_ D lB8""BN8"""*""""6"88""6"""` a b c d e f g h i j k l m n o p q r s t  u  v w x y z { | } ~  ` a b c d dd e ee f fhf g g g h hh ii j k l m n o o o p q q qzq r rr s s ss t, tzt u uirucO5@\D 0GDYZ hDYZDYZ"B uu v w x xx y yy z { | } }{} ~ ~~  Dl""""66666*"""""6"B4<8""66"""66                                             |  5      }  6  7  8  ~       >    E  Dl""""""""""""66"(((66666(""((666                           @ @ @ @ @ @ @                       $  $  $ [ $  $  $ $ BFX(((((((((((((((((""""."68888" @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @        $ $  $ $   $ $ ?  $ $  $ $ $ , $ $ $ $ C $ $ $ D $ $ E                  ?  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J   & & ( ( ( )) )~ *?B*6@0*,DNone  NoneErrorBNone9*.#D*None  B~ +@B+6L0+,DNone  NoneErrorBNone9+#D+None  B~ ,@B,6X0,,DNone  NoneErrorBNone9,#D,None  B ..#.2 %*,Ok"*&p46( ( 8I[ 4^X AM 4 VD[] \jb2 4 s *@A BFn+B;?]ؗf<V F;M 4 F;M]  f2V F;a5 4 F;a5] Hjb2 4B s *@A BFa5]jb2 4 s *@A B;4]f2V F6;M 4 F6;M] X jb2 4 s *@A BF6M]蛅jb2  4 s *@A B6;M] jb2  4B s *@A Bb>] $fV ?4,<  4 ?4,<]  d\2  4 c $@A#?4,<] LDB  4 # #v*"04] \fBV RADA 4 RADA] LDB 4 #  4A4]LDB 4B #  n+%DA]LDB 4 #  R4]XPB 4 C `%l#/v*] XPB 4 C !`+_?]̢LDB 4 # !*= ]0LDB 4 # !}47]ࣅfV }4> 4 }4>] D,fV }4> 4 }4>] *d\2 4 c $@A!D,>]LDB 4 # !}44]xRJB 4 3 !h4,=]ܦLDB 4 # @/5v*]@^V2 4 S  @4l6M]@f(V 3. :~ 4 3. :~] LDB 4B # @3. 3.~]@Pd\2 4 c $@A@68]@RJB  4 3 @*/ 0] @dRJB !4 3 @89 ]!@d\ "4 c $@A@q. 4]"@RJB #4 3 @3.~ :~]#@P^V $4 S @78]$@RJB %4 3 @3. */ ]%@|RJB &4 3 @0d7]&@୅d\2 '4 c $@A@9 :]'@d\2 (4 c $@A@9:~](@RJB )4 3 @00])@̯RJB *4 3 @l6l6]*@0RJB +4 3 @0l6]+@谅n ^ ; ,4# 3Em Z] ,`LXPB -4 C !``|]-ȱLDB .4 # !_s].XPB /4 C jJA]/XPB 04 C jJ;<]0LDB 14 # @]1@ lP\  24v 24# ] 2д_XPB 34 C ` p!v]3`XPB 44 C `/q+p]4 LDB 54 # S i(]5LDB 64 # "/24v]64LDB 74 #  5]7XPB 84 C ` 3 ]8Hl.\  &$ 94# +I] 9LDB :4B # @  $]:@d\2 ;4 c $@A@R Y"."];@칅RJB <4 3 @R ]<@RJB =4 3 @Y"."$"]=@ d\ >4 c $@A@%"DX#]>@ػRJB ?4 3 @ $$$]?@<^V @4 S @ !."]@@RJB A4 3 @ ]A@XRJB B4 3 @R R ]B@d\2 C4 c $@A@#"&#]C@td\2 D4 c $@A@$#&$]D@DRJB E4 3 @R !]E@RJB F4 3 @R !]F@`RJB G4 3 @!!]G@jbB H4 s *DjJ]H`|phB I4 0D jJ]I`jbB J4 s *D@c]J`…  K4  B"C DEF @jJ "@  ] K`$Å  L4  B"C DEF @jJ "@   ] L`Ån ^ ; M4#  s] M`ŅXPB N4 C !``|]N4ƅLDB O4 # !_s]ODžXPB P4 C jJA]P0ȅXPB Q4 C jJ;<]QȅLDB R4 # @]R@TɅlP\  24v S4# ] Sʅ_XPB T4 C ` p!v]TʅXPB U4 C `/q+p]UT˅<LDB V4 # S i(]V˅<LDB W4 # "/24v]Wh̅<LDB X4 #  5]X̅<XPB Y4 C ` 3 ]Y|ͅ<l.\  &$ Z4# +I] Zͅ<LDB [4B # @  $][@΅<d\2 \4 c $@A@R Y"."]\@ υ<RJB ]4 3 @R ]]@υ<RJB ^4 3 @Y"."$"]^@TЅ<d\ _4 c $@A@%"DX#]_@ х<RJB `4 3 @ $$$]`@pх<^V a4 S @ !."]a@(҅<RJB b4 3 @ ]b@҅<RJB c4 3 @R R ]c@DӅ<d\2 d4 c $@A@#"&#]d@Ӆ<d\2 e4 c $@A@$#&$]e@xԅ<RJB f4 3 @R !]f@ԅ<RJB g4 3 @R !]g@Յ<RJB h4 3 @!!]h@Յ<jbB i4 s *DjJ]i`օ<phB j4 0D jJ]j`ׅ<jbB k4 s *D@c]k`ׅ<  l4  B"C DEF @jJ "@  ] l`X؅<  m4  B"C DEF @jJ "@   ] m`؅<NFB n4 # @1]n@څ<nR^  24v o4# `{] o܅_<XPB p4 C ` p!v]p<ۅ<XPB q4 C `/q+p]q|݅<LDB r4 # S i(]r݅<LDB s4 # "/24v]sޅ<LDB t4 #  5]tޅ<XPB u4 C ` 3 ]u߅<rjB v4 0D jJD]v`<  w4  B"C DEF @jJ "@  `=] w`X<nf x4  ,BCUDEdF, 0jJ  /089BBLQ US TT@        b`] x`<^V y4  B*CPDEXF(  8c  $ * -2 73;? 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