_ccd_inline ccd_real_t __ccdVec3PointSegmentDist2(const ccd_vec3_t *P,
const ccd_vec3_t *x0,
const ccd_vec3_t *b,
ccd_vec3_t *witness)
{
// The computation comes from solving equation of segment:
// S(t) = x0 + t.d
// where - x0 is initial point of segment
// - d is direction of segment from x0 (|d| > 0)
// - t belongs to <0, 1> interval
//
// Than, distance from a segment to some point P can be expressed:
// D(t) = |x0 + t.d - P|^2
// which is distance from any point on segment. Minimization
// of this function brings distance from P to segment.
// Minimization of D(t) leads to simple quadratic equation that's
// solving is straightforward.
//
// Bonus of this method is witness point for free.
ccd_real_t dist, t;
ccd_vec3_t d, a;
// direction of segment
ccdVec3Sub2(&d, b, x0);
// precompute vector from P to x0
ccdVec3Sub2(&a, x0, P);
t = -CCD_REAL(1.) * ccdVec3Dot(&a, &d);
t /= ccdVec3Len2(&d);
if (t < CCD_ZERO || ccdIsZero(t)){
dist = ccdVec3Dist2(x0, P);
if (witness)
ccdVec3Copy(witness, x0);
}else if (t > CCD_ONE || ccdEq(t, CCD_ONE)){
dist = ccdVec3Dist2(b, P);
if (witness)
ccdVec3Copy(witness, b);
}else{
if (witness){
ccdVec3Copy(witness, &d);
ccdVec3Scale(witness, t);
ccdVec3Add(witness, x0);
dist = ccdVec3Dist2(witness, P);
}else{
// recycling variables
ccdVec3Scale(&d, t);
ccdVec3Add(&d, &a);
dist = ccdVec3Len2(&d);
}
}
return dist;
}
_ccd_inline void portalDir(const ccd_simplex_t *portal, ccd_vec3_t *dir)
{
ccd_vec3_t v2v1, v3v1;
ccdVec3Sub2(&v2v1, &ccdSimplexPoint(portal, 2)->v,
&ccdSimplexPoint(portal, 1)->v);
ccdVec3Sub2(&v3v1, &ccdSimplexPoint(portal, 3)->v,
&ccdSimplexPoint(portal, 1)->v);
ccdVec3Cross(dir, &v2v1, &v3v1);
ccdVec3Normalize(dir);
}
_ccd_inline void findOrigin(const void *obj1, const void *obj2, const ccd_t *ccd,
ccd_support_t *center)
{
ccd->center1(obj1, ¢er->v1);
ccd->center2(obj2, ¢er->v2);
ccdVec3Sub2(¢er->v, ¢er->v1, ¢er->v2);
}
_ccd_inline int portalReachTolerance(const ccd_simplex_t *portal,
const ccd_support_t *v4,
const ccd_vec3_t *dir,
const ccd_t *ccd)
{
ccd_vec3_t vec;
ccd_real_t dot;
ccdVec3Sub2(&vec, &v4->v, &ccdSimplexPoint(portal, 3)->v);
dot = ccdVec3Dot(&vec, dir);
return ccdEq(dot, ccd->mpr_tolerance) || dot < ccd->mpr_tolerance;
}
static int discoverPortal(const void *obj1, const void *obj2,
const ccd_t *ccd, ccd_simplex_t *portal)
{
ccd_vec3_t dir, va, vb;
ccd_real_t dot;
int cont;
/* vertex 0 is center of portal*/
findOrigin(obj1, obj2, ccd, ccdSimplexPointW(portal, 0));
ccdSimplexSetSize(portal, 1);
if (ccdVec3Eq(&ccdSimplexPoint(portal, 0)->v, ccd_vec3_origin)) {
/* Portal's center lies on origin (0,0,0) => we know that objects*/
/* intersect but we would need to know penetration info.*/
/* So move center little bit...*/
ccdVec3Set(&va, CCD_EPS * CCD_REAL(10.), CCD_ZERO, CCD_ZERO);
ccdVec3Add(&ccdSimplexPointW(portal, 0)->v, &va);
}
/* vertex 1 = support in direction of origin*/
ccdVec3Copy(&dir, &ccdSimplexPoint(portal, 0)->v);
ccdVec3Scale(&dir, CCD_REAL(-1.));
ccdVec3Normalize(&dir);
__ccdSupport(obj1, obj2, &dir, ccd, ccdSimplexPointW(portal, 1));
ccdSimplexSetSize(portal, 2);
/* test if origin isn't outside of v1*/
dot = ccdVec3Dot(&ccdSimplexPoint(portal, 1)->v, &dir);
if (ccdIsZero(dot) || dot < CCD_ZERO)
return -1;
/* vertex 2*/
ccdVec3Cross(&dir, &ccdSimplexPoint(portal, 0)->v,
&ccdSimplexPoint(portal, 1)->v);
if (ccdIsZero(ccdVec3Len2(&dir))) {
if (ccdVec3Eq(&ccdSimplexPoint(portal, 1)->v, ccd_vec3_origin)) {
/* origin lies on v1*/
return 1;
} else {
/* origin lies on v0-v1 segment*/
return 2;
}
}
ccdVec3Normalize(&dir);
__ccdSupport(obj1, obj2, &dir, ccd, ccdSimplexPointW(portal, 2));
dot = ccdVec3Dot(&ccdSimplexPoint(portal, 2)->v, &dir);
if (ccdIsZero(dot) || dot < CCD_ZERO)
return -1;
ccdSimplexSetSize(portal, 3);
/* vertex 3 direction*/
ccdVec3Sub2(&va, &ccdSimplexPoint(portal, 1)->v,
&ccdSimplexPoint(portal, 0)->v);
ccdVec3Sub2(&vb, &ccdSimplexPoint(portal, 2)->v,
&ccdSimplexPoint(portal, 0)->v);
ccdVec3Cross(&dir, &va, &vb);
ccdVec3Normalize(&dir);
/* it is better to form portal faces to be oriented "outside" origin*/
dot = ccdVec3Dot(&dir, &ccdSimplexPoint(portal, 0)->v);
if (dot > CCD_ZERO) {
ccdSimplexSwap(portal, 1, 2);
ccdVec3Scale(&dir, CCD_REAL(-1.));
}
while (ccdSimplexSize(portal) < 4) {
__ccdSupport(obj1, obj2, &dir, ccd, ccdSimplexPointW(portal, 3));
dot = ccdVec3Dot(&ccdSimplexPoint(portal, 3)->v, &dir);
if (ccdIsZero(dot) || dot < CCD_ZERO)
return -1;
cont = 0;
/* test if origin is outside (v1, v0, v3) - set v2 as v3 and*/
/* continue*/
ccdVec3Cross(&va, &ccdSimplexPoint(portal, 1)->v,
&ccdSimplexPoint(portal, 3)->v);
dot = ccdVec3Dot(&va, &ccdSimplexPoint(portal, 0)->v);
if (dot < CCD_ZERO && !ccdIsZero(dot)) {
ccdSimplexSet(portal, 2, ccdSimplexPoint(portal, 3));
cont = 1;
}
if (!cont) {
/* test if origin is outside (v3, v0, v2) - set v1 as v3 and*/
/* continue*/
ccdVec3Cross(&va, &ccdSimplexPoint(portal, 3)->v,
&ccdSimplexPoint(portal, 2)->v);
dot = ccdVec3Dot(&va, &ccdSimplexPoint(portal, 0)->v);
if (dot < CCD_ZERO && !ccdIsZero(dot)) {
ccdSimplexSet(portal, 1, ccdSimplexPoint(portal, 3));
cont = 1;
}
}
if (cont) {
ccdVec3Sub2(&va, &ccdSimplexPoint(portal, 1)->v,
&ccdSimplexPoint(portal, 0)->v);
ccdVec3Sub2(&vb, &ccdSimplexPoint(portal, 2)->v,
&ccdSimplexPoint(portal, 0)->v);
ccdVec3Cross(&dir, &va, &vb);
ccdVec3Normalize(&dir);
} else {
ccdSimplexSetSize(portal, 4);
}
}
return 0;
}
ccd_real_t ccdVec3PointTriDist2(const ccd_vec3_t *P,
const ccd_vec3_t *x0, const ccd_vec3_t *B,
const ccd_vec3_t *C,
ccd_vec3_t *witness)
{
// Computation comes from analytic expression for triangle (x0, B, C)
// T(s, t) = x0 + s.d1 + t.d2, where d1 = B - x0 and d2 = C - x0 and
// Then equation for distance is:
// D(s, t) = | T(s, t) - P |^2
// This leads to minimization of quadratic function of two variables.
// The solution from is taken only if s is between 0 and 1, t is
// between 0 and 1 and t + s < 1, otherwise distance from segment is
// computed.
ccd_vec3_t d1, d2, a;
ccd_real_t u, v, w, p, q, r, d;
ccd_real_t s, t, dist, dist2;
ccd_vec3_t witness2;
ccdVec3Sub2(&d1, B, x0);
ccdVec3Sub2(&d2, C, x0);
ccdVec3Sub2(&a, x0, P);
u = ccdVec3Dot(&a, &a);
v = ccdVec3Dot(&d1, &d1);
w = ccdVec3Dot(&d2, &d2);
p = ccdVec3Dot(&a, &d1);
q = ccdVec3Dot(&a, &d2);
r = ccdVec3Dot(&d1, &d2);
d = w * v - r * r;
if (ccdIsZero(d)){
// To avoid division by zero for zero (or near zero) area triangles
s = t = -1.;
}else{
s = (q * r - w * p) / d;
t = (-s * r - q) / w;
}
if ((ccdIsZero(s) || s > CCD_ZERO)
&& (ccdEq(s, CCD_ONE) || s < CCD_ONE)
&& (ccdIsZero(t) || t > CCD_ZERO)
&& (ccdEq(t, CCD_ONE) || t < CCD_ONE)
&& (ccdEq(t + s, CCD_ONE) || t + s < CCD_ONE)){
if (witness){
ccdVec3Scale(&d1, s);
ccdVec3Scale(&d2, t);
ccdVec3Copy(witness, x0);
ccdVec3Add(witness, &d1);
ccdVec3Add(witness, &d2);
dist = ccdVec3Dist2(witness, P);
}else{
dist = s * s * v;
dist += t * t * w;
dist += CCD_REAL(2.) * s * t * r;
dist += CCD_REAL(2.) * s * p;
dist += CCD_REAL(2.) * t * q;
dist += u;
}
}else{
dist = __ccdVec3PointSegmentDist2(P, x0, B, witness);
dist2 = __ccdVec3PointSegmentDist2(P, x0, C, &witness2);
if (dist2 < dist){
dist = dist2;
if (witness)
ccdVec3Copy(witness, &witness2);
}
dist2 = __ccdVec3PointSegmentDist2(P, B, C, &witness2);
if (dist2 < dist){
dist = dist2;
if (witness)
ccdVec3Copy(witness, &witness2);
}
}
return dist;
}