_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 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;
}
_ccd_inline int portalReachTolerance(const ccd_simplex_t *portal,
const ccd_support_t *v4,
const ccd_vec3_t *dir,
const ccd_t *ccd)
{
ccd_real_t dv1, dv2, dv3, dv4;
ccd_real_t dot1, dot2, dot3;
/* find the smallest dot product of dir and {v1-v4, v2-v4, v3-v4}*/
dv1 = ccdVec3Dot(&ccdSimplexPoint(portal, 1)->v, dir);
dv2 = ccdVec3Dot(&ccdSimplexPoint(portal, 2)->v, dir);
dv3 = ccdVec3Dot(&ccdSimplexPoint(portal, 3)->v, dir);
dv4 = ccdVec3Dot(&v4->v, dir);
dot1 = dv4 - dv1;
dot2 = dv4 - dv2;
dot3 = dv4 - dv3;
dot1 = CCD_FMIN(dot1, dot2);
dot1 = CCD_FMIN(dot1, dot3);
return ccdEq(dot1, ccd->mpr_tolerance) || dot1 < ccd->mpr_tolerance;
}
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;
}