void dvhat_c ( ConstSpiceDouble s1 [6],
SpiceDouble sout[6] )
/*
-Brief_I/O
VARIABLE I/O DESCRIPTION
-------- --- --------------------------------------------------
s1 I State to be normalized.
sout O Unit vector s1 / |s1|, and its time derivative.
-Detailed_Input
s1 This is any double precision state. If the position
component of the state is the zero vector, this routine
will detect it and will not attempt to divide by zero.
-Detailed_Output
sout sout is a state containing the unit vector pointing in
the direction of position component of s1 and the
derivative of the unit vector with respect to time.
sout may overwrite s1.
-Parameters
None.
-Exceptions
Error free.
1) If s1 represents the zero vector, then the position
component of sout will also be the zero vector. The
velocity component will be the velocity component
of s1.
-Files
None.
-Particulars
Let s1 be a state vector with position and velocity components p
and v respectively. From these components one can compute the
unit vector parallel to p, call it u and the derivative of u
with respect to time, du. This pair (u,du) is the state returned
by this routine in sout.
-Examples
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.
Suppose that 'state' gives the apparent state of a body with
respect to an observer. This routine can be used to compute the
instantaneous angular rate of the object across the sky as seen
from the observers vantage.
#include "SpiceUsr.h"
#include <stdio.h>
#include <math.h>
int main()
{
SpiceDouble et;
SpiceDouble ltime;
SpiceDouble omega;
SpiceDouble state [6];
SpiceDouble ustate [6];
SpiceChar * epoch = "Jan 1 2009";
SpiceChar * target = "MOON";
SpiceChar * frame = "J2000";
SpiceChar * abcorr = "LT+S";
SpiceChar * obsrvr = "EARTH BARYCENTER";
/.
Load SPK, PCK, and LSK kernels, use a meta kernel for convenience.
./
furnsh_c ( "standard.tm" );
/.
Define an arbitrary epoch, convert the epoch to ephemeris time.
./
str2et_c ( epoch, &et );
/.
Calculate the state of the moon with respect to the earth-moon
barycenter in J2000, corrected for light time and stellar aberration
at ET.
./
spkezr_c ( target, et, frame, abcorr, obsrvr, state, <ime );
/.
Calculate the unit vector of STATE and the derivative of the
unit vector.
./
dvhat_c ( state, ustate );
/.
Calculate the instantaneous angular velocity from the magnitude of the
derivative of the unit vector.
v = r x omega
||omega|| = ||v|| for r . v = 0
-----
||r||
||omega|| = ||v|| for ||r|| = 1
./
omega = vnorm_c( &ustate[3] );
printf( "Instantaneous angular velocity, rad/sec %.10g\n", omega );
return 0;
}
The program outputs:
Instantaneous angular velocity, rad/sec 2.48106658e-06
-Restrictions
None.
-Literature_References
None.
-Author_and_Institution
W.L. Taber (JPL)
E.D. Wright (JPL)
-Version
-CSPICE Version 1.0.1, 06-MAY-2010 (EDW)
Reordered header sections to proper NAIF convention.
Minor edit to code comments eliminating typo.
-CSPICE Version 1.0.0, 07-JUL-1999 (EDW)
-Index_Entries
State of a unit vector parallel to a state vector
-&
*/
{ /* Begin dvhat_c */
/*
Local variables
*/
SpiceDouble length;
SpiceDouble posin [3];
SpiceDouble posout[3];
SpiceDouble velin [3];
SpiceDouble velout[3];
/*
We'll do this the obvious way for now. Unpack the input vector
into two working vectors.
*/
posin[0] = s1[0];
posin[1] = s1[1];
posin[2] = s1[2];
velin[0] = s1[3];
velin[1] = s1[4];
velin[2] = s1[5];
/*
Get the position portion of the output state and the length of
the input position.
*/
unorm_c ( posin, posout, &length );
if ( length == 0. )
{
/*
If the length of the input position is zero, just copy
the input velocity to the output velocity.
*/
vequ_c ( velin, velout );
}
else
{
/*
Otherwise the derivative of the unit vector is just the
component of the input velocity perpendicular to the input
position, scaled by the reciprocal of the length of the
input position.
*/
vperp_c ( velin , posout, velout );
vscl_c ( 1./length, velout, velout );
}
/*
Pack everything and return. Hazar!
*/
sout[0] = posout[0];
sout[1] = posout[1];
sout[2] = posout[2];
sout[3] = velout[0];
sout[4] = velout[1];
sout[5] = velout[2];
} /* End dvhat_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 */
