void npedln_c ( SpiceDouble a, SpiceDouble b, SpiceDouble c, ConstSpiceDouble linept[3], ConstSpiceDouble linedr[3], SpiceDouble pnear[3], SpiceDouble * dist ) /* -Brief_I/O Variable I/O Description -------- --- -------------------------------------------------- a I Length of ellipsoid's semi-axis in the x direction b I Length of ellipsoid's semi-axis in the y direction c I Length of ellipsoid's semi-axis in the z direction linept I Point on line linedr I Direction vector of line pnear O Nearest point on ellipsoid to line dist O Distance of ellipsoid from line -Detailed_Input a, b, c are the lengths of the semi-axes of a triaxial ellipsoid which is centered at the origin and oriented so that its axes lie on the x-, y- and z- coordinate axes. a, b, and c are the lengths of the semi-axes that point in the x, y, and z directions respectively. linept linedr are, respectively, a point and a direction vector that define a line. The line is the set of vectors linept + t * linedr where t is any real number. -Detailed_Output pnear is the point on the ellipsoid that is closest to the line, if the line doesn't intersect the ellipsoid. If the line intersects the ellipsoid, pnear will be a point of intersection. If linept is outside of the ellipsoid, pnear will be the closest point of intersection. If linept is inside the ellipsoid, pnear will not necessarily be the closest point of intersection. dist is the distance of the line from the ellipsoid. This is the minimum distance between any point on the line and any point on the ellipsoid. If the line intersects the ellipsoid, dist is zero. -Parameters None. -Exceptions If this routine detects an error, the output arguments nearp and dist are not modified. 1) If the length of any semi-axis of the ellipsoid is non-positive, the error SPICE(INVALIDAXISLENGTH) is signaled. 2) If the line's direction vector is the zero vector, the error SPICE(ZEROVECTOR) is signaled. 3) If the length of any semi-axis of the ellipsoid is zero after the semi-axis lengths are scaled by the reciprocal of the magnitude of the longest semi-axis and then squared, the error SPICE(DEGENERATECASE) is signaled. 4) If the input ellipsoid is extremely flat or needle-shaped and has its shortest axis close to perpendicular to the input line, numerical problems could cause this routine's algorithm to fail, in which case the error SPICE(DEGENERATECASE) is signaled. -Files None. -Particulars For any ellipsoid and line, if the line does not intersect the ellipsoid, there is a unique point on the ellipsoid that is closest to the line. Therefore, the distance dist between ellipsoid and line is well-defined. The unique line segment of length dist that connects the line and ellipsoid is normal to both of these objects at its endpoints. If the line intersects the ellipsoid, the distance between the line and ellipsoid is zero. -Examples 1) We can find the distance between an instrument optic axis ray and the surface of a body modelled as a tri-axial ellipsoid using this routine. If the instrument position and pointing unit vector in body-fixed coordinates are linept = ( 1.0e6, 2.0e6, 3.0e6 ) and linedr = ( -4.472091234e-1 -8.944182469e-1, -4.472091234e-3 ) and the body semi-axes lengths are a = 7.0e5 b = 7.0e5 c = 6.0e5, then the call to npedln_c npedln_c ( a, b, c, linept, linedr, pnear, &dist ); yields a value for pnear, the nearest point on the body to the optic axis ray, of ( -.16333110792340931E+04, -.32666222157820771E+04, .59999183350006724E+06 ) and a value for dist, the distance to the ray, of .23899679338299707E+06 (These results were obtained on a PC-Linux system under gcc.) In some cases, it may not be clear that the closest point on the line containing an instrument boresight ray is on the boresight ray itself; the point may lie on the ray having the same vertex as the boresight ray and pointing in the opposite direction. To rule out this possibility, we can make the following test: /. Find the difference vector between the closest point on the ellipsoid to the line containing the boresight ray and the boresight ray's vertex. Find the angular separation between this difference vector and the boresight ray. If the angular separation does not exceed pi/2, we have the nominal geometry. Otherwise, we have an error. ./ vsub_c ( pnear, linept, diff ); sep = vsep_c ( diff, linedr ); if ( sep <= halfpi_c() ) { [ perform normal processing ] } else { [ handle error case ] } -Restrictions None. -Literature_References None. -Author_and_Institution N.J. Bachman (JPL) -Version -CSPICE Version 1.1.0, 01-JUN-2010 (NJB) Added touchd_ calls to tests for squared, scaled axis length underflow. This forces rounding to zero in certain cases where it otherwise might not occur due to use of extended registers. -CSPICE Version 1.0.1, 06-DEC-2002 (NJB) Outputs shown in header example have been corrected to be consistent with those produced by this routine. -CSPICE Version 1.0.0, 03-SEP-1999 (NJB) -Index_Entries distance between line and ellipsoid distance between line of sight and body nearest point on ellipsoid to line -& */ { /* Begin npedln_c */ /* Local variables */ SpiceBoolean found [2]; SpiceBoolean ifound; SpiceBoolean xfound; SpiceDouble oppdir [3]; SpiceDouble mag; SpiceDouble normal [3]; SpiceDouble prjpt [3]; SpiceDouble prjnpt [3]; SpiceDouble pt [2][3]; SpiceDouble scale; SpiceDouble scla; SpiceDouble scla2; SpiceDouble sclb; SpiceDouble sclb2; SpiceDouble sclc; SpiceDouble sclc2; SpiceDouble sclpt [3]; SpiceDouble udir [3]; SpiceEllipse cand; SpiceEllipse prjel; SpiceInt i; SpicePlane candpl; SpicePlane prjpl; /* Static variables */ /* Participate in error tracing. */ chkin_c ( "npedln_c" ); /* The algorithm used in this routine has two parts. The first part handles the case where the input line and ellipsoid intersect. Our procedure is simple in that case; we just call surfpt_c twice, passing it first one ray determined by the input line, then a ray pointing in the opposite direction. The second part of the algorithm handles the case where surfpt_c doesn't find an intersection. Finding the nearest point on the ellipsoid to the line, when the two do not intersect, is a matter of following four steps: 1) Find the points on the ellipsoid where the surface normal is normal to the line's direction. This set of points is an ellipse centered at the origin. The point we seek MUST lie on this `candidate' ellipse. 2) Project the candidate ellipse onto a plane that is normal to the line's direction. This projection preserves distance from the line; the nearest point to the line on this new ellipse is the projection of the nearest point to the line on the candidate ellipse, and these two points are exactly the same distance from the line. If computed using infinite-precision arithmetic, this projection would be guaranteed to be non-degenerate as long as the input ellipsoid were non-degenerate. This can be verified by taking the inner product of the scaled normal to the candidate ellipse plane and the line's unitized direction vector (these vectors are called normal and udir in the code below); the inner product is strictly greater than 1 if the ellipsoid is non-degenerate. 3) The nearest point on the line to the projected ellipse will be contained in the plane onto which the projection is done; we find this point and then find the nearest point to it on the projected ellipse. The distance between these two points is the distance between the line and the ellipsoid. 4) Finally, we find the point on the candidate ellipse that was projected to the nearest point to the line on the projected ellipse that was found in step 3. This is the nearest point on the ellipsoid to the line. Glossary of Geometric Variables a, b, c Input ellipsoid's semi-axis lengths. point Point of intersection of line and ellipsoid if the intersection is non-empty. candpl Plane containing candidate ellipse. normal Normal vector to the candidate plane candpl. cand Candidate ellipse. linept, linedr, Point and direction vector on input line. udir Unitized line direction vector. prjpl Projection plane; the candidate ellipse is projected onto this plane to yield prjel. prjel Projection of the candidate ellipse cand onto the projection plane prjel. prjpt Projection of line point. prjnpt Nearest point on projected ellipse to projection of line point. pnear Nearest point on ellipsoid to line. */ /* We need a valid normal vector. */ unorm_c ( linedr, udir, &mag ); if ( mag == 0. ) { setmsg_c( "Line direction vector is the zero vector. " ); sigerr_c( "SPICE(ZEROVECTOR)" ); chkout_c( "npedln_c" ); return; } if ( ( a <= 0. ) || ( b <= 0. ) || ( c <= 0. ) ) { setmsg_c ( "Semi-axis lengths: a = #, b = #, c = #." ); errdp_c ( "#", a ); errdp_c ( "#", b ); errdp_c ( "#", c ); sigerr_c ( "SPICE(INVALIDAXISLENGTH)" ); chkout_c ( "npedln_c" ); return; } /* Scale the semi-axes lengths for better numerical behavior. If squaring any one of the scaled lengths causes it to underflow to zero, we cannot continue the computation. Otherwise, scale the viewing point too. */ scale = maxd_c ( 3, a, b, c ); scla = a / scale; sclb = b / scale; sclc = c / scale; scla2 = scla*scla; sclb2 = sclb*sclb; sclc2 = sclc*sclc; if ( ( (SpiceDouble)touchd_(&scla2) == 0. ) || ( (SpiceDouble)touchd_(&sclb2) == 0. ) || ( (SpiceDouble)touchd_(&sclc2) == 0. ) ) { setmsg_c ( "Semi-axis too small: a = #, b = #, c = #. " ); errdp_c ( "#", a ); errdp_c ( "#", b ); errdp_c ( "#", c ); sigerr_c ( "SPICE(DEGENERATECASE)" ); chkout_c ( "npedln_c" ); return; } /* Scale linept. */ sclpt[0] = linept[0] / scale; sclpt[1] = linept[1] / scale; sclpt[2] = linept[2] / scale; /* Hand off the intersection case to surfpt_c. surfpt_c determines whether rays intersect a body, so we treat the line as a pair of rays. */ vminus_c ( udir, oppdir ); surfpt_c ( sclpt, udir, scla, sclb, sclc, pt[0], &(found[0]) ); surfpt_c ( sclpt, oppdir, scla, sclb, sclc, pt[1], &(found[1]) ); for ( i = 0; i < 2; i++ ) { if ( found[i] ) { *dist = 0.0; vequ_c ( pt[i], pnear ); vscl_c ( scale, pnear, pnear ); chkout_c ( "npedln_c" ); return; } } /* Getting here means the line doesn't intersect the ellipsoid. Find the candidate ellipse CAND. NORMAL is a normal vector to the plane containing the candidate ellipse. Mathematically the ellipse must exist, since it's the intersection of an ellipsoid centered at the origin and a plane containing the origin. Only numerical problems can prevent the intersection from being found. */ normal[0] = udir[0] / scla2; normal[1] = udir[1] / sclb2; normal[2] = udir[2] / sclc2; nvc2pl_c ( normal, 0., &candpl ); inedpl_c ( scla, sclb, sclc, &candpl, &cand, &xfound ); if ( !xfound ) { setmsg_c ( "Candidate ellipse could not be found." ); sigerr_c ( "SPICE(DEGENERATECASE)" ); chkout_c ( "npedln_c" ); return; } /* Project the candidate ellipse onto a plane orthogonal to the line. We'll call the plane prjpl and the projected ellipse prjel. */ nvc2pl_c ( udir, 0., &prjpl ); pjelpl_c ( &cand, &prjpl, &prjel ); /* Find the point on the line lying in the projection plane, and then find the near point PRJNPT on the projected ellipse. Here PRJPT is the point on the line lying in the projection plane. The distance between PRJPT and PRJNPT is DIST. */ vprjp_c ( sclpt, &prjpl, prjpt ); npelpt_c ( prjpt, &prjel, prjnpt, dist ); /* Find the near point pnear on the ellipsoid by taking the inverse orthogonal projection of prjnpt; this is the point on the candidate ellipse that projects to prjnpt. Note that the output dist was computed in step 3 and needs only to be re-scaled. The inverse projection of pnear ought to exist, but may not be calculable due to numerical problems (this can only happen when the input ellipsoid is extremely flat or needle-shaped). */ vprjpi_c ( prjnpt, &prjpl, &candpl, pnear, &ifound ); if ( !ifound ) { setmsg_c ( "Inverse projection could not be found." ); sigerr_c ( "SPICE(DEGENERATECASE)" ); chkout_c ( "npedln_c" ); return; } /* Undo the scaling. */ vscl_c ( scale, pnear, pnear ); *dist *= scale; chkout_c ( "npedln_c" ); } /* End npedln_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 */