void vprjpi_c ( ConstSpiceDouble vin [3],
ConstSpicePlane * projpl,
ConstSpicePlane * invpl,
SpiceDouble vout [3],
SpiceBoolean * found )
Find the vector in a specified plane that maps to a specified
vector in another plane under orthogonal projection.
PLANES
GEOMETRY
MATH
PLANE
VECTOR
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.
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.
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.
None.
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.
None.
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.
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 );
None.
[1] `Calculus and Analytic Geometry', Thomas and Finney.
N.J. Bachman (JPL)
-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)
vector projection onto plane inverted
Link to routine vprjpi_c source file vprjpi_c.c
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