class 19: 3d cartesian coordinate computations gisc-3325 26 march 2009
TRANSCRIPT
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Class 19: 3D Cartesian Coordinate Computations
GISC-332526 March 2009
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Class Update
Remember Article Reviews (2) are due 16 April 2009.
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Coordinates
Geodetic reference systems use curvilinear and Cartesian (rectangular) coordinate systems that are referred to the ellipsoid. Curvilinear values: geodetic latitude, longitude and
ellipsoid surface. Right-handed, earth-fixed, 3-D coordinate system. Cartesian [ X;Y;Z ]
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Cartesian Coordinates
Earth-Centered-Earth-Fixed (ECEF) Orientation of axes is identical to spherical
earth model. X lies in equatorial plane intersecting Greenwich Y in equatorial plane at 90deg E longitude Z coincident with earth's spin axis
Origin Earth Center of Mass (COM) corresponds to the center of the ellipsoid.
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GPS vectors
Represent differences in geocentric coordinates.
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GPS Vector
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Local Geodetic Horizon Coordinates LGH is an earth-fixed, right-handed, orthogonal,
3-D coordinates system having its origin at any point specified.
N axis in meridian plane Positive north
U axis along the normal to the ellipsoid. Positive up
E axis right-handed system perpendicular to meridian plane Positive east
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Local Geodetic Coordinates
Referred to origin of the local geodetic system using Geodetic azimuth Vertical angle or zenith angle Mark-to-mark sland range from the origin
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Geocentric ↔ Geodetic
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Radius of curvature of prime vertical
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Algorithms
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Geocentric to Geodetic
• XYZ to Lat, Lon, ellipsoid height
• Longitude is computed as in spherical case.
• Both height above ellipsoid and latitude must be iterated.
– Ellipsoid height requires we know latitude and radius of curvature of prime vertical
– Latitude requires we know radius of curvature of prime vertical and ellipsoid height
– Radius of curvature of prime vertical requires we know latitude.
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Iteration required• Required because latitude and ellipsoid
height are dependent upon one another.
• One approach (used in text) is to first set ellipsoid height to 0 then solve for latitude.
• Then solve for h then lat again... then again
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Coordinate transformations by Molodensky (translation)
The values DX, DY, DZ above show the difference in origin between datums and WGS 84. Also shown are ellipsoid parameter differences.
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Bursa-Wolf Transformation
• For geographic transformations between two geocentric datums.
• Applied to geocentric coordinates.
– X axis points to Greenwich
– Y is 90 deg east
– Z is to north
• Consists of three translation, three rotations and scale change.
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Rotations
• Rotations preserve the length of a vector.
• When we rotate the point stays fixed by the coordinate axes are moved resulting in new coordinate values.
– A counter-clockwise (CCW) rotation is considered positive with respect to the old system.
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Rotations
• Coordinates for a point in 2-D space are computed using plane trigonometry.
• Where r is length and theta is azimuth
– X = r * cos(theta)
– Y = r * sin(theta)
• Rotations either add or subtract an angle (gamma) from the azimuth (theta).
– X' = r * cos(gamma – theta)
– Y' = r * sin(gamma - theta)
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Underlying trigonometry
• X' = r * cos(gamma – theta).
• We apply the difference formula (see below)
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Rotation angles (detail) • X' = r * cos(gamma – theta)
– = r(cos(gamma)*cos(theta)+sin(gamma)sin(theta))
– = r cos(gamma)cos(theta) + r sin(gamma)sin(theta)
– Since X = r * cos(gamma) and Y = r * sin(gamma)
• X' = X cos(theta) + Y sin(theta)
• Y' = r*sin(gamma – theta)
– = r(sin(gamma)cos(theta)-cos(gamma)sin(theta))
– = r(sin(gamma)cos(theta) – r cos(gamma)sin(theta)
• Y' = -X*sin(theta) + Y*cos(theta)
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Matrix form• We can translate the previous page into matrix
form:
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Transformations
• Translation: coordinates are derived by merely subtracting the translations from their corresponding coordinates.
– X' = X – T where T is translation vector
• Scale change
– Can be used for feet to meter conversions.
– X' = s*X where s is scale factor
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Transformations• Four parameter (2D)- Apply translations,
rotations, and scale in two dimensions only.
• Seven-parameter transformation (3D) – apply three translations, three rotations and one scale change.
– X' = s*R*X+T' (in alternate form below)
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Matrix rotations
• A single axis rotation matrix is the rotation matrix that describes the effect of rotating the entire coordinate system about an axis through a specified angle.
• Rotation matrices are square, normal orthogonal matrices.
• Transpose of an normal orthogonal matrix is equivalent to its inverse.
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Euler Angles• D, C, B rotations retain the values for Z, Y
and X respectively.
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Rotations
• The order in which single axis rotations are performed is critical.
• Algebraic sign of the rotation angle is considered positive if the rotation is viewed as a counterclockwise rotation from the positive end of the axis when looking toward the origin.
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Transformation Parameters
Note: milli-arcseconds (mas) = (1/3600)*(pi/180)
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Local <-> Geocentric• We cannot transform directly from local
geodetic coordinates to geodetic coordinates.
– We must use geocentric coordinates.
• Perform two rotations to align local system with geocentric.
– Make U-axis parallel with Z-axis geocentric
– Then align all corresponding axes with one another.
• Rotation matrix consists of geodetic coordinates of standpoint (origin).
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Geocentric to Local
The transpose of this rotation matrix can be used to transpose from local to geocentric. Result [ X;Y;Z] input [ E;N;U]
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Matlab code