Find the vector in a specified plane that maps to a specified vector in another plane under orthogonal projection.
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.
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 );
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.
N.J. Bachman (JPL)
 `Calculus and Analytic Geometry', Thomas and Finney.
-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
-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.