void vprjpi_c ( ConstSpiceDouble vin [3],
ConstSpicePlane * projpl,
ConstSpicePlane * invpl,
SpiceDouble vout [3],
SpiceBoolean * found )
/*
-Brief_I/O
Variable I/O Description
-------- --- --------------------------------------------------
vin I The projected vector.
projpl I Plane containing vin.
invpl I Plane containing inverse image of vin.
vout O Inverse projection of vin.
found O Flag indicating whether vout could be calculated.
-Detailed_Input
vin,
projpl,
invpl are, respectively, a 3-vector, a CSPICE plane
containing the vector, and a CSPICE plane
containing the inverse image of the vector under
orthogonal projection onto projpl.
-Detailed_Output
vout is the inverse orthogonal projection of vin. This
is the vector lying in the plane invpl whose
orthogonal projection onto the plane projpl is
vin. vout is valid only when found (defined below)
is SPICETRUE. Otherwise, vout is undefined.
found indicates whether the inverse orthogonal projection
of vin could be computed. found is SPICETRUE if so,
SPICEFALSE otherwise.
-Parameters
None.
-Exceptions
1) If the geometric planes defined by projpl and invpl are
orthogonal, or nearly so, the inverse orthogonal projection
of vin may be undefined or have magnitude too large to
represent with double precision numbers. In either such
case, found will be set to SPICEFALSE.
2) Even when found is SPICETRUE, vout may be a vector of extremely
large magnitude, perhaps so large that it is impractical to
compute with it. It's up to you to make sure that this
situation does not occur in your application of this routine.
-Files
None.
-Particulars
Projecting a vector orthogonally onto a plane can be thought of
as finding the closest vector in the plane to the original vector.
This `closest vector' always exists; it may be coincident with the
original vector. Inverting an orthogonal projection means finding
the vector in a specified plane whose orthogonal projection onto
a second specified plane is a specified vector. The vector whose
projection is the specified vector is the inverse projection of
the specified vector, also called the `inverse image under
orthogonal projection' of the specified vector. This routine
finds the inverse orthogonal projection of a vector onto a plane.
Related routines are vprjp_c, which projects a vector onto a plane
orthogonally, and vproj_c, which projects a vector onto another
vector orthogonally.
-Examples
1) Suppose
vin = ( 0.0, 1.0, 0.0 ),
and that projpl has normal vector
projn = ( 0.0, 0.0, 1.0 ).
Also, let's suppose that invpl has normal vector and constant
invn = ( 0.0, 2.0, 2.0 )
invc = 4.0.
Then vin lies on the y-axis in the x-y plane, and we want to
find the vector vout lying in invpl such that the orthogonal
projection of vout the x-y plane is vin. Let the notation
< a, b > indicate the inner product of vectors a and b.
Since every point x in invpl satisfies the equation
< x, (0.0, 2.0, 2.0) > = 4.0,
we can verify by inspection that the vector
( 0.0, 1.0, 1.0 )
is in invpl and differs from vin by a multiple of projn. So
( 0.0, 1.0, 1.0 )
must be vout.
To find this result using CSPICE, we can create the
CSPICE planes projpl and invpl using the code fragment
nvp2pl_c ( projn, vin, &projpl );
nvc2pl_c ( invn, invc, &invpl );
and then perform the inverse projection using the call
vprjpi_c ( vin, &projpl, &invpl, vout );
vprjpi_c will return the value
vout = ( 0.0, 1.0, 1.0 );
-Restrictions
None.
-Literature_References
[1] `Calculus and Analytic Geometry', Thomas and Finney.
-Author_and_Institution
N.J. Bachman (JPL)
-Version
-CSPICE Version 1.1.0, 05-APR-2004 (NJB)
Computation of LIMIT was re-structured to avoid
run-time underflow warnings on some platforms.
-CSPICE Version 1.0.0, 05-MAR-1999 (NJB)
-Index_Entries
vector projection onto plane inverted
-&
*/
/*
-Revisions
-CSPICE Version 1.1.0, 05-APR-2004 (NJB)
Computation of LIMIT was re-structured to avoid run-time
underflow warnings on some platforms. In the revised code,
BOUND/dpmax_c() is never scaled by a number having absolute value
< 1.
-&
*/
{ /* Begin vprjpi_c */
/*
Local constants
*/
/*
BOUND is used to bound the magnitudes of the numbers that we
try to take the reciprocal of, since we can't necessarily invert
any non-zero number. We won't try to invert any numbers with
magnitude less than
BOUND / dpmax_c()
BOUND is chosen somewhat arbitrarily....
*/
#define BOUND 10.0
/*
Local variables
*/
SpiceDouble denom;
SpiceDouble invc;
SpiceDouble invn [3];
SpiceDouble limit;
SpiceDouble mult;
SpiceDouble numer;
SpiceDouble projc;
SpiceDouble projn [3];
/*
Participate in error tracing.
*/
if ( return_c() )
{
return;
}
chkin_c ( "vprjpi_c" );
/*
Unpack the planes.
*/
pl2nvc_c ( projpl, projn, &projc );
pl2nvc_c ( invpl, invn, &invc );
/*
We'll first discuss the computation of VOUT in the nominal case,
and then deal with the exceptional cases.
When projpl and invpl are not orthogonal to each other, the
inverse projection of vin will differ from vin by a multiple of
projn, the unit normal vector to projpl. We find this multiple
by using the fact that the inverse projection vout satisfies the
plane equation for the inverse projection plane invpl.
We have
vout = vin + mult * projn; (1)
since vout satisfies
< vout, invn > = invc
we must have
< vin + mult * projn, invn > = invc
which in turn implies
invc - < vin, invn >
mult = ------------------------. (2)
< projn, invn >
Having mult, we can compute vout according to equation (1).
Now, if the denominator in the above expression for mult is zero
or just too small, performing the division would cause a
divide-by-zero error or an overflow of mult. In either case, we
will avoid carrying out the division, and we'll set found to
SPICEFALSE.
Compute the numerator and denominator of the right side of (2).
*/
numer = invc - vdot_c ( vin, invn );
denom = vdot_c ( projn, invn );
/*
If the magnitude of the denominator is greater than
BOUND
limit = abs ( ---------- * numer ),
dpmax_c()
we can safely divide the numerator by the denominator, and the
magnitude of the result will be no greater than
dpmax_c()
----------- .
BOUND
Note that we have ruled out the case where numer and denom are
both zero by insisting on strict inequality in the comparison of
denom and limit:
*/
if ( fabs(numer) < 1.0 )
{
limit = fabs ( BOUND / dpmax_c() );
}
else
{
limit = fabs ( ( BOUND / dpmax_c() ) * numer );
}
*found = ( fabs (denom) > limit );
if ( *found )
{
/*
We'll compute vout after all.
*/
mult = numer / denom;
vlcom_c ( 1.0, vin, mult, projn, vout );
}
chkout_c ( "vprjpi_c" );
} /* End vprjpi_c */
void vprjp_c ( ConstSpiceDouble vin [3],
ConstSpicePlane * plane,
SpiceDouble vout [3] )
/*
-Brief_I/O
Variable I/O Description
-------- --- --------------------------------------------------
vin I Vector to be projected.
plane I A CSPICE plane onto which vin is projected.
vout O Vector resulting from projection.
-Detailed_Input
vin is a 3-vector that is to be orthogonally projected
onto a specified plane.
plane is a CSPICE plane that represents the geometric
plane onto which vin is to be projected.
-Detailed_Output
vout is the vector resulting from the orthogonal
projection of vin onto plane. vout is the closest
point in the specified plane to vin.
-Parameters
None.
-Exceptions
1) Invalid input planes are diagnosed by the routine pl2nvc_c,
which is called by this routine.
-Files
None.
-Particulars
Projecting a vector v orthogonally onto a plane can be thought of
as finding the closest vector in the plane to v. This `closest
vector' always exists; it may be coincident with the original
vector.
Two related routines are vprjpi_c, which inverts an orthogonal
projection of a vector onto a plane, and vproj_c, which projects
a vector orthogonally onto another vector.
-Examples
1) Find the closest point in the ring plane of a planet to a
spacecraft located at positn (in body-fixed coordinates).
Suppose the vector normal is normal to the ring plane, and
that origin, which represents the body center, is in the
ring plane. Then we can make a `plane' with the code
pnv2pl_c ( origin, normal, &plane );
can find the projection by making the call
vprjp_c ( positn, &plane, proj );
-Restrictions
None.
-Literature_References
[1] `Calculus and Analytic Geometry', Thomas and Finney.
-Author_and_Institution
N.J. Bachman (JPL)
-Version
-CSPICE Version 1.0.0, 05-MAR-1999 (NJB)
-Index_Entries
vector projection onto plane
-&
*/
{ /* Begin vprjp_c */
/*
Local variables
*/
SpiceDouble constant;
SpiceDouble normal [3];
/*
Participate in error tracing.
*/
if ( return_c() )
{
return;
}
chkin_c ( "vprjp_c" );
/*
Obtain a unit vector normal to the input plane, and a constant
for the plane.
*/
pl2nvc_c ( plane, normal, &constant );
/*
Let the notation < a, b > indicate the inner product of vectors
a and b.
vin differs from its projection onto plane by some multiple of
normal. That multiple is
< vin - vout, normal > * normal
= ( < vin, normal > - < vout, normal > ) * normal
= ( < vin, normal > - const ) * normal
Subtracting this multiple of normal from vin yields vout.
*/
vlcom_c ( 1.0,
vin,
constant - vdot_c ( vin, normal ),
normal,
vout );
chkout_c ( "vprjp_c" );
} /* End vprjp_c */
void pjelpl_c ( ConstSpiceEllipse * elin,
ConstSpicePlane * plane,
SpiceEllipse * elout )
/*
-Brief_I/O
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.
-Detailed_Input
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.
-Detailed_Output
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.
-Parameters
None.
-Exceptions
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.
-Files
None.
-Particulars
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.
-Examples
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.
./
-Restrictions
None.
-Literature_References
None.
-Author_and_Institution
N.J. Bachman (JPL)
-Version
-CSPICE Version 1.0.0, 02-SEP-1999 (NJB)
-Index_Entries
project ellipse onto plane
-&
*/
{ /* Begin pjelpl_c */
/*
Local variables
*/
SpiceDouble center[3];
SpiceDouble cnst;
SpiceDouble normal[3];
SpiceDouble prjctr[3];
SpiceDouble prjvc1[3];
SpiceDouble prjvc2[3];
SpiceDouble smajor[3];
SpiceDouble sminor[3];
/*
Participate in error tracing.
*/
chkin_c ( "pjelpl_c" );
/*
Find generating vectors of the input ellipse.
*/
el2cgv_c ( elin, center, smajor, sminor );
/*
Find a normal vector for the input plane.
*/
pl2nvc_c ( plane, normal, &cnst );
/*
Find the components of the semi-axes that are orthogonal to the
input plane's normal vector. The components are generating
vectors for the projected plane.
*/
vperp_c ( smajor, normal, prjvc1 );
vperp_c ( sminor, normal, prjvc2 );
/*
Find the projection of the ellipse's center onto the input plane.
This is the center of the projected ellipse.
In case the last assertion is non-obvious, note that the
projection we're carrying out is the composition of a linear
mapping (projection to a plane containing the origin and parallel
to PLANE) and a translation mapping (adding the closest point to
the origin in PLANE to every point), and both linear mappings and
translations carry the center of an ellipse to the center of the
ellipse's image. Let's state this using mathematical symbols.
Let L be a linear mapping and let T be a translation mapping,
say
T(x) = x + A.
Then
T ( L ( center + cos(theta)smajor + sin(theta)sminor ) )
= A + L ( center + cos(theta)smajor + sin(theta)sminor )
= A + L (center)
+ cos(theta) L(smajor)
+ sin(theta) L(sminor)
From the form of this last expression, we see that we have an
ellipse centered at
A + L (center)
= T ( L (center) )
This last term is the image of the center of the original ellipse,
as we wished to demonstrate.
Now in the case of orthogonal projection onto a plane PL, L can be
taken as the orthogonal projection onto a parallel plane PL'
containing the origin. Then L is a linear mapping. Let M be
the multiple of the normal vector of PL such that M is contained
in PL (M is the closest point in PL to the origin). Then the
orthogonal projection mapping onto PL, which we will name PRJ,
can be defined by
PRJ (x) = L (x) + M.
So PRJ is the composition of a translation and a linear mapping,
as claimed.
*/
vprjp_c ( center, plane, prjctr );
/*
Put together the projected ellipse.
*/
cgv2el_c ( prjctr, prjvc1, prjvc2, elout );
chkout_c ( "pjelpl_c" );
} /* End pjelpl_c */
