/
intersect.h
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/
intersect.h
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/*
* FUNCTION:
* This file contains a number of utilities useful to 3D graphics in
* general, and to the generation of tubing and extrusions in particular
*
* HISTORY:
* Written by Linas Vepstas, August 1991
* Updated to correctly handle degenerate cases, Linas, February 1993
*/
#include <math.h>
#include "port.h"
#include "vvector.h"
#define BACKWARDS_INTERSECT (2)
/* ========================================================== */
/*
* the Degenerate_Tolerance token represents the greatest amount by
* which different scales in a graphics environment can differ before
* they should be considered "degenerate". That is, when one vector is
* a million times longer than another, changces are that the second will
* be less than a pixel int, and therefore was probably meant to be
* degenerate (by the CAD package, etc.) But what should this tolerance
* be? At least 1 in onethousand (since screen sizes are 1K pixels), but
* les than 1 in 4 million (since this is the limit of single-precision
* floating point accuracy). Of course, if double precision were used,
* then the tolerance could be increased.
*
* Potentially, this naive assumption could cause problems if the CAD
* package attempts to zoom in on small details, and turns out, certain
* points should not hvae been degenerate. The problem presented here
* is that the tolerance could run out before single-precision ran
* out, and so the CAD packages would perceive this as a "bug".
* One alternative is to fiddle around & try to tighten the tolerance.
* However, the right alternative is to code the graphics pipeline in
* double-precision (and tighten the tolerance).
*
* By the way, note that Degernate Tolerance is a "dimensionless"
* quantitiy -- it has no units -- it does not measure feet, inches,
* millimeters or pixels. It is used only in the computations of ratios
* and relative lengths.
*/
/*
* Right now, the tolerance is set to 2 parts in a million, which
* corresponds to a 19-bit distinction of mantissas. Note that
* single-precsion numbers have 24 bit mantissas.
*/
#define DEGENERATE_TOLERANCE (0.000002)
/* ========================================================== */
/*
* The macro and subroutine INTERSECT are designed to compute the
* intersection of a line (defined by the points v1 and v2) and a plane
* (defined as plane which is normal to the vector n, and contains the
* point p). Both return the point sect, which is the point of
* interesection.
*
* This MACRO attemps to be fairly robust by checking for a divide by
* zero.
*/
/* ========================================================== */
/*
* HACK ALERT
* The intersection parameter t has the nice property that if t>1,
* then the intersection is "in front of" p1, and if t<0, then the
* intersection is "behind" p2. Unfortunately, as the intersecting plane
* and the line become parallel, t wraps through infinity -- i.e. t can
* become so large that t becomes "greater than infinity" and comes back
* as a negative number (i.e. winding number hopped by one unit). We
* have no way of detecting this situation without adding gazzillions
* of lines of code of topological algebra to detect the winding number;
* and this would be incredibly difficult, and ruin performance.
*
* Thus, we've installed a cheap hack for use by the "cut style" drawing
* routines. If t proves to be a large negative number (more negative
* than -5), then we assume that t was positive and wound through
* infinity. This makes most cuts look good, without introducing bogus
* cuts at infinity.
*/
/* ========================================================== */
#define INTERSECT(sect,p,n,v1,v2) \
{ \
gleDouble deno, numer, t, omt; \
\
deno = (v1[0] - v2[0]) * n[0]; \
deno += (v1[1] - v2[1]) * n[1]; \
deno += (v1[2] - v2[2]) * n[2]; \
\
if (deno == 0.0) { \
VEC_COPY (n, v1); \
/* printf ("Intersect: Warning: line is coplanar with plane \n"); */ \
} else { \
\
numer = (p[0] - v2[0]) * n[0]; \
numer += (p[1] - v2[1]) * n[1]; \
numer += (p[2] - v2[2]) * n[2]; \
\
t = numer / deno; \
omt = 1.0 - t; \
\
sect[0] = t * v1[0] + omt * v2[0]; \
sect[1] = t * v1[1] + omt * v2[1]; \
sect[2] = t * v1[2] + omt * v2[2]; \
} \
}
/* ========================================================== */
/*
* The macro and subroutine BISECTING_PLANE compute a normal vector that
* describes the bisecting plane between three points (v1, v2 and v3).
* This bisecting plane has the following properties:
* 1) it contains the point v2
* 2) the angle it makes with v21 == v2 - v1 is equal to the angle it
* makes with v32 == v3 - v2
* 3) it is perpendicular to the plane defined by v1, v2, v3.
*
* Having input v1, v2, and v3, it returns a unit vector n.
*
* In some cases, the user may specify degenerate points, and still
* expect "reasonable" or "obvious" behaviour. The "expected"
* behaviour for these degenerate cases is:
*
* 1) if v1 == v2 == v3, then return n=0
* 2) if v1 == v2, then return v32 (normalized).
* 3) if v2 == v3, then return v21 (normalized).
* 4) if v1, v2 and v3 co-linear, then return v21 (normalized).
*
* Mathematically, these special cases "make sense" -- we just have to
* code around potential divide-by-zero's in the code below.
*/
/* ========================================================== */
#define BISECTING_PLANE(valid,n,v1,v2,v3) \
{ \
double v21[3], v32[3]; \
double len21, len32; \
double vdot; \
\
VEC_DIFF (v21, v2, v1); \
VEC_DIFF (v32, v3, v2); \
\
VEC_LENGTH (len21, v21); \
VEC_LENGTH (len32, v32); \
\
if (len21 <= DEGENERATE_TOLERANCE * len32) { \
\
if (len32 == 0.0) { \
/* all three points lie ontop of one-another */ \
VEC_ZERO (n); \
valid = FALSE; \
} else { \
/* return a normalized copy of v32 as bisector */ \
len32 = 1.0 / len32; \
VEC_SCALE (n, len32, v32); \
valid = TRUE; \
} \
\
} else { \
\
valid = TRUE; \
\
if (len32 <= DEGENERATE_TOLERANCE * len21) { \
/* return a normalized copy of v21 as bisector */ \
len21 = 1.0 / len21; \
VEC_SCALE (n, len21, v21); \
\
} else { \
\
/* normalize v21 to be of unit length */ \
len21 = 1.0 / len21; \
VEC_SCALE (v21, len21, v21); \
\
/* normalize v32 to be of unit length */ \
len32 = 1.0 / len32; \
VEC_SCALE (v32, len32, v32); \
\
VEC_DOT_PRODUCT (vdot, v32, v21); \
\
/* if vdot == 1 or -1, then points are colinear */ \
if ((vdot >= (1.0-DEGENERATE_TOLERANCE)) || \
(vdot <= (-1.0+DEGENERATE_TOLERANCE))) { \
VEC_COPY (n, v21); \
} else { \
\
/* go do the full computation */ \
n[0] = vdot * (v32[0] + v21[0]) - v32[0] - v21[0]; \
n[1] = vdot * (v32[1] + v21[1]) - v32[1] - v21[1]; \
n[2] = vdot * (v32[2] + v21[2]) - v32[2] - v21[2]; \
\
/* if above if-test's passed, \
* n should NEVER be of zero length */ \
VEC_NORMALIZE (n); \
} \
} \
} \
}
/* ========================================================== */
/*
* The block of code below is ifdef'd out, and is here for reference
* purposes only. It performs the "mathematically right thing" for
* computing a bisecting plane, but is, unfortunately, subject ot noise
* in the presence of near degenerate points. Since computer graphics,
* due to sloppy coding, laziness, or correctness, is filled with
* degenerate points, we can't really use this version. The code above
* is far more appropriate for graphics.
*/
#ifdef MATHEMATICALLY_EXACT_GRAPHICALLY_A_KILLER
#define BISECTING_PLANE(n,v1,v2,v3) \
{ \
double v21[3], v32[3]; \
double len21, len32; \
double vdot; \
\
VEC_DIFF (v21, v2, v1); \
VEC_DIFF (v32, v3, v2); \
\
VEC_LENGTH (len21, v21); \
VEC_LENGTH (len32, v32); \
\
if (len21 == 0.0) { \
\
if (len32 == 0.0) { \
/* all three points lie ontop of one-another */ \
VEC_ZERO (n); \
valid = FALSE; \
} else { \
/* return a normalized copy of v32 as bisector */ \
len32 = 1.0 / len32; \
VEC_SCALE (n, len32, v32); \
} \
\
} else { \
\
/* normalize v21 to be of unit length */ \
len21 = 1.0 / len21; \
VEC_SCALE (v21, len21, v21); \
\
if (len32 == 0.0) { \
/* return a normalized copy of v21 as bisector */ \
VEC_COPY (n, v21); \
} else { \
\
/* normalize v32 to be of unit length */ \
len32 = 1.0 / len32; \
VEC_SCALE (v32, len32, v32); \
\
VEC_DOT_PRODUCT (vdot, v32, v21); \
\
/* if vdot == 1 or -1, then points are colinear */ \
if ((vdot == 1.0) || (vdot == -1.0)) { \
VEC_COPY (n, v21); \
} else { \
\
/* go do the full computation */ \
n[0] = vdot * (v32[0] + v21[0]) - v32[0] - v21[0]; \
n[1] = vdot * (v32[1] + v21[1]) - v32[1] - v21[1]; \
n[2] = vdot * (v32[2] + v21[2]) - v32[2] - v21[2]; \
\
/* if above if-test's passed, \
* n should NEVER be of zero length */ \
VEC_NORMALIZE (n); \
} \
} \
} \
}
#endif
/* ========================================================== */
/*
* This macro computes the plane perpendicular to the the plane
* defined by three points, and whose normal vector is givven as the
* difference between the two vectors ...
*
* (See way below for the "math" model if you want to understand this.
* The comments about relative errors above apply here.)
*/
#define CUTTING_PLANE(valid,n,v1,v2,v3) \
{ \
double v21[3], v32[3]; \
double len21, len32; \
double lendiff; \
\
VEC_DIFF (v21, v2, v1); \
VEC_DIFF (v32, v3, v2); \
\
VEC_LENGTH (len21, v21); \
VEC_LENGTH (len32, v32); \
\
if (len21 <= DEGENERATE_TOLERANCE * len32) { \
\
if (len32 == 0.0) { \
/* all three points lie ontop of one-another */ \
VEC_ZERO (n); \
valid = FALSE; \
} else { \
/* return a normalized copy of v32 as cut-vector */ \
len32 = 1.0 / len32; \
VEC_SCALE (n, len32, v32); \
valid = TRUE; \
} \
\
} else { \
\
valid = TRUE; \
\
if (len32 <= DEGENERATE_TOLERANCE * len21) { \
/* return a normalized copy of v21 as cut vector */ \
len21 = 1.0 / len21; \
VEC_SCALE (n, len21, v21); \
} else { \
\
/* normalize v21 to be of unit length */ \
len21 = 1.0 / len21; \
VEC_SCALE (v21, len21, v21); \
\
/* normalize v32 to be of unit length */ \
len32 = 1.0 / len32; \
VEC_SCALE (v32, len32, v32); \
\
VEC_DIFF (n, v21, v32); \
VEC_LENGTH (lendiff, n); \
\
/* if the perp vector is very small, then the two \
* vectors are darn near collinear, and the cut \
* vector is probably poorly defined. */ \
if (lendiff < DEGENERATE_TOLERANCE) { \
VEC_ZERO (n); \
valid = FALSE; \
} else { \
lendiff = 1.0 / lendiff; \
VEC_SCALE (n, lendiff, n); \
} \
} \
} \
}
/* ========================================================== */
#ifdef MATHEMATICALLY_EXACT_GRAPHICALLY_A_KILLER
#define CUTTING_PLANE(n,v1,v2,v3) \
{ \
double v21[3], v32[3]; \
\
VEC_DIFF (v21, v2, v1); \
VEC_DIFF (v32, v3, v2); \
\
VEC_NORMALIZE (v21); \
VEC_NORMALIZE (v32); \
\
VEC_DIFF (n, v21, v32); \
VEC_NORMALIZE (n); \
}
#endif
/* ============================================================ */
/* This macro is used in several places to cycle through a series of
* points to find the next non-degenerate point in a series */
#define FIND_NON_DEGENERATE_POINT(inext,npoints,len,diff,point_array) \
{ \
gleDouble slen; \
gleDouble summa[3]; \
\
do { \
/* get distance to next point */ \
VEC_DIFF (diff, point_array[inext+1], point_array[inext]); \
VEC_LENGTH (len, diff); \
VEC_SUM (summa, point_array[inext+1], point_array[inext]); \
VEC_LENGTH (slen, summa); \
slen *= DEGENERATE_TOLERANCE; \
inext ++; \
} while ((len <= slen) && (inext < npoints-1)); \
}
/* ========================================================== */
extern int bisecting_plane (gleDouble n[3], /* returned */
gleDouble v1[3], /* input */
gleDouble v2[3], /* input */
gleDouble v3[3]); /* input */