Table of contents
CSPICE_EDNMPT returns the unique point on an ellipsoid's surface where
the outward normal direction is a given vector.
Given:
a the length of the semi-axis of the ellipsoid that is parallel
to the X-axis of the body-fixed coordinate system.
[1,1] = size(a); double = class(a)
b the length of the semi-axis of the ellipsoid that is parallel
to the Y-axis of the body-fixed coordinate system.
[1,1] = size(b); double = class(b)
c the length of the semi-axis of the ellipsoid that is parallel
to the Z-axis of the body-fixed coordinate system.
[1,1] = size(c); double = class(c)
normal a non-zero vector.
[3,1] = size(normal); double = class(normal)
The unique point on the ellipsoid at which `normal' is an
outward normal vector is sought.
the call:
[point] = cspice_ednmpt( a, b, c, normal )
returns:
point the unique point on the ellipsoid at which `normal' is an
outward normal vector.
[3,1] = size(point); double = class(point)
`point' is a 3-vector giving the body-fixed coordinates
of a point on the ellipsoid. In body-fixed coordinates,
the semi-axes of the ellipsoid are aligned with the X,
Y, and Z-axes of the coordinate system.
None.
Any numerical results shown for this example may differ between
platforms as the results depend on the SPICE kernels used as input
and the machine specific arithmetic implementation.
1) Choose a triaxial ellipsoid with three unequal semi-axis
lengths. Pick several vectors; find the points on the
ellipsoid where the respective outward normals are parallel to
those vectors.
Check the results: at each point, a computed outward normal
vector should have very small angular separation from the
input vector. Also, the point should be on the surface of the
ellipsoid. The ellipsoid can be thought of as a level surface
of the function
2 2 2
f(x, y, z) = (x/a) + (y/b) + (z/c)
where `a', `b', `c' are the semi-axis lengths of the ellipsoid.
Specifically, the ellipsoid is the set
{ (x, y, z) : f(x, y, z) = 1 }
We can evaluate F at a point to determine whether that point
is close to the ellipsoid's surface.
Example code begins here.
function ednmpt_ex1()
%
% Initialize the ellipsoid semi-axes.
%
a = 10.0;
b = 5.0;
c = 2.0;
%
% Pick several vectors; find the points
% on the ellipsoid where the respective
% outward normals are parallel to those
% vectors; check the results.
%
xnorml = [ 0.0, 0.0, 3.0 ]';
[point] = cspice_ednmpt( a, b, c, xnorml );
[normal] = cspice_surfnm( a, b, c, point );
fprintf( ' \n' )
fprintf( 'Semi-axis lengths: %13.8f %13.8f %13.8f\n', a, b, c )
fprintf( 'Input vector: %13.8f %13.8f %13.8f\n', ...
xnorml(1), xnorml(2), xnorml(3) )
fprintf( 'Point: %13.8f %13.8f %13.8f\n', ...
point(1), point(2), point(3) )
fprintf( 'Outward normal: %13.8f %13.8f %13.8f\n', ...
normal(1), normal(2), normal(3) )
fprintf( 'Angular error (rad): %13.8f\n', ...
cspice_vsep( normal, xnorml ) )
fprintf( 'Off-surface error: %13.8f\n', ...
(point(1)/a) ^ 2 + (point(2)/b) ^ 2 + ...
(point(3)/c) ^ 2 - 1 )
fprintf( ' \n' )
xnorml = [ 15.0, -7.0, 3.0 ]';
[point] = cspice_ednmpt( a, b, c, xnorml );
[normal] = cspice_surfnm( a, b, c, point );
fprintf( 'Semi-axis lengths: %13.8f %13.8f %13.8f\n', a, b, c )
fprintf( 'Input vector: %13.8f %13.8f %13.8f\n', ...
xnorml(1), xnorml(2), xnorml(3) )
fprintf( 'Point: %13.8f %13.8f %13.8f\n', ...
point(1), point(2), point(3) )
fprintf( 'Outward normal: %13.8f %13.8f %13.8f\n', ...
normal(1), normal(2), normal(3) )
fprintf( 'Angular error (rad): %13.8f\n', ...
cspice_vsep( normal, xnorml ) )
fprintf( 'Off-surface error: %13.8f\n', ...
(point(1)/a) ^ 2 + (point(2)/b) ^ 2 + ...
(point(3)/c) ^ 2 - 1 )
fprintf( ' \n' )
xnorml = [ 15.0, -7.0, 3.0 ]';
[point] = cspice_ednmpt( a, b, c, xnorml );
[normal] = cspice_surfnm( a, b, c, point );
fprintf( 'Semi-axis lengths: %13.8f %13.8f %13.8f\n', a, b, c )
fprintf( 'Input vector: %13.8f %13.8f %13.8f\n', ...
xnorml(1), xnorml(2), xnorml(3) )
fprintf( 'Point: %13.8f %13.8f %13.8f\n', ...
point(1), point(2), point(3) )
fprintf( 'Outward normal: %13.8f %13.8f %13.8f\n', ...
normal(1), normal(2), normal(3) )
fprintf( 'Angular error (rad): %13.8f\n', ...
cspice_vsep( normal, xnorml ) )
fprintf( 'Off-surface error: %13.8f\n', ...
(point(1)/a) ^ 2 + (point(2)/b) ^ 2 + ...
(point(3)/c) ^ 2 - 1 )
fprintf( ' \n' )
xnorml = [ a/2, b/2, c/2 ]';
[point] = cspice_ednmpt( a, b, c, xnorml );
[normal] = cspice_surfnm( a, b, c, point );
fprintf( 'Semi-axis lengths: %13.8f %13.8f %13.8f\n', a, b, c )
fprintf( 'Input vector: %13.8f %13.8f %13.8f\n', ...
xnorml(1), xnorml(2), xnorml(3) )
fprintf( 'Point: %13.8f %13.8f %13.8f\n', ...
point(1), point(2), point(3) )
fprintf( 'Outward normal: %13.8f %13.8f %13.8f\n', ...
normal(1), normal(2), normal(3) )
fprintf( 'Angular error (rad): %13.8f\n', ...
cspice_vsep( normal, xnorml ) )
fprintf( 'Off-surface error: %13.8f\n', ...
(point(1)/a) ^ 2 + (point(2)/b) ^ 2 + ...
(point(3)/c) ^ 2 - 1 )
fprintf( ' \n' )
When this program was executed on a Mac/Intel/Octave5.x/64-bit
platform, the output was:
Semi-axis lengths: 10.00000000 5.00000000 2.00000000
Input vector: 0.00000000 0.00000000 3.00000000
Point: 0.00000000 0.00000000 2.00000000
Outward normal: 0.00000000 0.00000000 1.00000000
Angular error (rad): 0.00000000
Off-surface error: 0.00000000
Semi-axis lengths: 10.00000000 5.00000000 2.00000000
Input vector: 15.00000000 -7.00000000 3.00000000
Point: 9.73103203 -1.13528707 0.07784826
Outward normal: 0.89165745 -0.41610681 0.17833149
Angular error (rad): 0.00000000
Off-surface error: 0.00000000
Semi-axis lengths: 10.00000000 5.00000000 2.00000000
Input vector: 15.00000000 -7.00000000 3.00000000
Point: 9.73103203 -1.13528707 0.07784826
Outward normal: 0.89165745 -0.41610681 0.17833149
Angular error (rad): 0.00000000
Off-surface error: 0.00000000
Semi-axis lengths: 10.00000000 5.00000000 2.00000000
Input vector: 5.00000000 2.50000000 1.00000000
Point: 9.69412864 1.21176608 0.07755303
Outward normal: 0.88045091 0.44022545 0.17609018
Angular error (rad): 0.00000000
Off-surface error: 0.00000000
This routine can be used to determine the distance between an
ellipsoid and a non-intersecting plane. This distance computation
supports computation of terminator points on an ellipsoid.
1) If any of the semi-axis lengths is non-positive, the error
SPICE(BADAXISLENGTH) is signaled by a routine in the call tree
of this routine.
2) If any of the semi-axis lengths underflows to zero when
divided by the largest semi-axis length, the error
SPICE(AXISUNDERFLOW) is signaled by a routine in the call tree
of this routine.
3) If `normal' is the zero vector, the error SPICE(ZEROVECTOR)
is signaled by a routine in the call tree of this routine.
4) If the input pass the above checks but lead to a
divide-by-zero error or to a computing an invalid argument of
a fractional exponential expression, the error
SPICE(DEGENERATECASE) is signaled by a routine in the call
tree of this routine.
5) If any of the input arguments, `a', `b', `c' or `normal', is
undefined, an error is signaled by the Matlab error handling
system.
6) If any of the input arguments, `a', `b', `c' or `normal', is
not of the expected type, or it does not have the expected
dimensions and size, an error is signaled by the Mice
interface.
None.
None.
MICE.REQ
None.
J. Diaz del Rio (ODC Space)
-Mice Version 1.0.0, 09-AUG-2021 (JDR)
point on an ellipsoid having given surface normal
|