Project an ellipse onto a plane, orthogonally.
Copyright (1995), California Institute of Technology. U.S. Government sponsorship acknowledged.
ELLIPSES PLANES
Variable I/O Description -------- --- -------------------------------------------------- elin I A CSPICE ellipse to be projected. plane I A plane onto which elin is to be projected. elout O A CSPICE ellipse resulting from the projection.
elin,
plane are, respectively, a cspice ellipse and a
cspice plane. The geometric ellipse represented
by elin is to be orthogonally projected onto the
geometric plane represented by plane.
elout is a cspice ellipse that represents the geometric
ellipse resulting from orthogonally projecting the
ellipse represented by inel onto the plane
represented by plane.
None.
Projecting an ellipse orthogonally onto a plane can be thought of finding the points on the plane that are `under' or `over' the ellipse, with the `up' direction considered to be perpendicular to the plane. More mathematically, the orthogonal projection is the set of points Y in the plane such that for some point X in the ellipse, the vector Y - X is perpendicular to the plane. The orthogonal projection of an ellipse onto a plane yields another ellipse.
1) With center = { 1., 1., 1. },
vect1 = { 2., 0., 0. },
vect2 = { 0., 1., 1. },
normal = { 0., 0., 1. }
the code fragment
nvc2pl_c ( normal, 0., plane );
cgv2el_c ( center, vect1, vect2, elin );
pjelpl_c ( elin, plane, elout );
el2cgv_c ( elout, prjctr, prjmaj, prjmin );
returns
prjctr = { 1., 1., 0. },
prjmaj = { 2., 0., 0. },
prjmin = { 0., 1., 0. }
2) With vect1 = { 2., 0., 0. },
vect2 = { 1., 1., 1. },
center = { 0., 0., 0. },
normal = { 0., 0., 1. },
the code fragment
nvc2pl_c ( normal, 0., plane );
cgv2el_c ( center, vect1, vect2, elin );
pjelpl_c ( elin, plane, elout );
el2cgv_c ( elout, prjctr, prjmaj, prjmin );
returns
prjctr = { 0., 0., 0. };
prjmaj = { -2.227032728823213,
-5.257311121191336e-1,
0. };
prjmin = { 2.008114158862273e-1,
-8.506508083520399e-1,
0. };
3) An example of actual use: Suppose we wish to compute the
distance from an ellipsoid to a line. Let the line be
defined by a point P and a direction vector DIRECT; the
line is the set of points
P + t * DIRECT,
where t is any real number. Let the ellipsoid have semi-
axis lengths A, B, and C.
We can reduce the problem to that of finding the distance
between the line and an ellipse on the ellipsoid surface by
considering the fact that the surface normal at the nearest
point to the line will be orthogonal to DIRECT; the set of
surface points where this condition holds lies in a plane,
and hence is an ellipse on the surface. The problem can be
further simplified by projecting the ellipse orthogonally
onto the plane defined by
< X, DIRECT > = 0.
The problem is then a two dimensional one: find the
distance of the projected ellipse from the intersection of
the line and this plane (which is necessarily one point).
A `paraphrase' of the relevant code is:
#include "SpiceUsr.h"
.
.
.
/.
Step 1. Find the candidate ellipse cand.
normal is a normal vector to the plane
containing the candidate ellipse. The
ellipse must exist, since it's the
intersection of an ellipsoid centered at
the origin and a plane containing the
origin. For this reason, we don't check
inedpl_c's "found flag" found below.
./
normal[0] = direct[0] / (a*a);
normal[1] = direct[1] / (b*b);
normal[2] = direct[2] / (c*c);
nvc2pl_c ( normal, 0., &candpl );
inedpl_c ( a, b, c, &candpl, cand, &found );
/.
Step 2. Project the candidate ellipse onto a
plane orthogonal to the line. We'll
call the plane prjpl and the
projected ellipse prjel.
./
nvc2pl_c ( direct, 0., &prjpl );
pjelpl_c ( &cand, &prjpl, &prjel );
/.
Step 3. Find the point on the line lying in the
projection plane, and then find the
near point pjnear on the projected
ellipse. Here prjpt is the point on the
input line that lies in the projection
plane. The distance between prjpt and
pjnear is dist.
./
vprjp_c ( linept, &prjpl, prjpt );
npelpt_c ( &prjel, prjpt, pjnear, &dist );
/.
Step 4. Find the near point pnear on the
ellipsoid by taking the inverse
orthogonal projection of PJNEAR; this is
the point on the candidate ellipse that
projects to pjnear. Note that the output
dist was computed in step 3.
The inverse projection of pjnear is
guaranteed to exist, so we don't have to
check found.
./
vprjpi_c ( pjnear, &prjpl, &candpl, pnear, &found );
/.
The value of dist returned is the distance we're looking
for.
The procedure described here is carried out in the routine
npedln_c.
./
None.
1) If the input plane is invalid, the error will be diagnosed
by routines called by this routine.
2) The input ellipse may be degenerate--its semi-axes may be
linearly dependent. Such ellipses are allowed as inputs.
3) The ellipse resulting from orthogonally projecting the input
ellipse onto a plane may be degenerate, even if the input
ellipse is not.
None.
N.J. Bachman (JPL)
None.
-CSPICE Version 1.0.0, 02-SEP-1999 (NJB)
project ellipse onto plane