Transcript 3DCartesianSysComps

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