SAELGV_C: Semi-axes of ellipse from generating vectors
Find semi-axis vectors of an ellipse generated by two arbitrary
three-dimensional vectors.
ELLIPSES
ELLIPSE
GEOMETRY
MATH
Variable I/O Description
-------- --- --------------------------------------------------
vec1,
vec2 I Two vectors used to generate an ellipse.
smajor O Semi-major axis of ellipse.
sminor O Semi-minor axis of ellipse.
vec1,
vec2 are two vectors that define an ellipse.
The ellipse is the set of points in 3-space
center + cos(theta) vec1 + sin(theta) vec2
where theta is in the interval ( -pi, pi ] and
center is an arbitrary point at which the ellipse
is centered. An ellipse's semi-axes are
independent of its center, so the vector center
shown above is not an input to this routine.
vec2 and vec1 need not be linearly independent;
degenerate input ellipses are allowed.
smajor
sminor are semi-major and semi-minor axes of the ellipse,
respectively. smajor and sminor may overwrite
either of vec1 or vec2.
None.
We note here that two linearly independent but not necessarily
orthogonal vectors vec1 and vec2 can define an ellipse
centered at the origin: the ellipse is the set of points in
3-space
center + cos(theta) vec1 + sin(theta) vec2
where theta is in the interval (-pi, pi] and center is an
arbitrary point at which the ellipse is centered.
This routine finds vectors that constitute semi-axes of an
ellipse that is defined, except for the location of its center,
by vec1 and vec2. The semi-major axis is a vector of largest
possible magnitude in the set
cos(theta) vec1 + sin(theta) vec2
There are two such vectors; they are additive inverses of each
other. The semi-minor axis is an analogous vector of smallest
possible magnitude. The semi-major and semi-minor axes are
orthogonal to each other. If smajor and sminor are choices of
semi-major and semi-minor axes, then the input ellipse can also
be represented as the set of points
center + cos(theta) smajor + sin(theta) sminor
where theta is in the interval (-pi, pi].
The capability of finding the axes of an ellipse is useful in
finding the image of an ellipse under a linear transformation.
Finding this image is useful for determining the orthogonal and
gnomonic projections of an ellipse, and also for finding the limb
and terminator of an ellipsoidal body.
1) An example using inputs that can be readily checked by
hand calculation.
Let
vec1 = ( 1., 1., 1. )
vec2 = ( 1., -1., 1. )
The function call
saelgv_c ( vec1, vec2, smajor, sminor );
returns
smajor = ( -1.414213562373095,
0.0,
-1.414213562373095 )
and
sminor = ( -2.4037033579794549D-17
1.414213562373095,
-2.4037033579794549D-17 )
2) This example is taken from the code of the CSPICE routine
pjelpl_c, which finds the orthogonal projection of an ellipse
onto a plane. The code listed below is the portion used to
find the semi-axes of the projected ellipse.
#include "SpiceUsr.h"
.
.
.
/.
Project vectors defining axes of ellipse onto plane.
./
vperp_c ( vec1, normal, proj1 );
vperp_c ( vec2, normal, proj2 );
.
.
.
saelgv_c ( proj1, proj2, smajor, sminor );
The call to saelgv_c determines the required semi-axes.
None.
1) If one or more semi-axes of the ellipse is found to be the
zero vector, the input ellipse is degenerate. This case is
not treated as an error; the calling program must determine
whether the semi-axes are suitable for the program's intended
use.
None.
N.J. Bachman (JPL)
W.L. Taber (JPL)
[1] Calculus, Vol. II. Tom Apostol. John Wiley & Sons, 1969.
See Chapter 5, `Eigenvalues of Operators Acting on Euclidean
Spaces'.
-CSPICE Version 1.0.0, 12-JUN-1999 (NJB)
semi-axes of ellipse from generating vectors
Link to routine saelgv_c source file saelgv_c.c
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