TriDist.cpp

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                       University of N. Carolina
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  EMail:               geom@cs.unc.edu


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//--------------------------------------------------------------------------
// File:   TriDist.cpp
// Author: Eric Larsen
// Description:
// contains SegPoints() for finding closest points on a pair of line
// segments and TriDist() for finding closest points on a pair of triangles
//--------------------------------------------------------------------------

#include "MatVec.h"
#ifdef _WIN32
#include <float.h>
#define isnan _isnan
#endif

//--------------------------------------------------------------------------
// SegPoints() 
//
// Returns closest points between an segment pair.
// Implemented from an algorithm described in
//
// Vladimir J. Lumelsky,
// On fast computation of distance between line segments.
// In Information Processing Letters, no. 21, pages 55-61, 1985.   
//--------------------------------------------------------------------------

void
SegPoints(PQP_REAL VEC[3], 
          PQP_REAL X[3], PQP_REAL Y[3],             // closest points
          const PQP_REAL P[3], const PQP_REAL A[3], // seg 1 origin, vector
          const PQP_REAL Q[3], const PQP_REAL B[3]) // seg 2 origin, vector
{
  PQP_REAL T[3], A_dot_A, B_dot_B, A_dot_B, A_dot_T, B_dot_T;
  PQP_REAL TMP[3];

  VmV(T,Q,P);
  A_dot_A = VdotV(A,A);
  B_dot_B = VdotV(B,B);
  A_dot_B = VdotV(A,B);
  A_dot_T = VdotV(A,T);
  B_dot_T = VdotV(B,T);

  // t parameterizes ray P,A 
  // u parameterizes ray Q,B 

  PQP_REAL t,u;

  // compute t for the closest point on ray P,A to
  // ray Q,B

  PQP_REAL denom = A_dot_A*B_dot_B - A_dot_B*A_dot_B;

  t = (A_dot_T*B_dot_B - B_dot_T*A_dot_B) / denom;

  // clamp result so t is on the segment P,A

  if ((t < 0) || isnan(t)) t = 0; else if (t > 1) t = 1;

  // find u for point on ray Q,B closest to point at t

  u = (t*A_dot_B - B_dot_T) / B_dot_B;

  // if u is on segment Q,B, t and u correspond to 
  // closest points, otherwise, clamp u, recompute and
  // clamp t 

  if ((u <= 0) || isnan(u)) {

    VcV(Y, Q);

    t = A_dot_T / A_dot_A;

    if ((t <= 0) || isnan(t)) {
      VcV(X, P);
      VmV(VEC, Q, P);
    }
    else if (t >= 1) {
      VpV(X, P, A);
      VmV(VEC, Q, X);
    }
    else {
      VpVxS(X, P, A, t);
      VcrossV(TMP, T, A);
      VcrossV(VEC, A, TMP);
    }
  }
  else if (u >= 1) {

    VpV(Y, Q, B);

    t = (A_dot_B + A_dot_T) / A_dot_A;

    if ((t <= 0) || isnan(t)) {
      VcV(X, P);
      VmV(VEC, Y, P);
    }
    else if (t >= 1) {
      VpV(X, P, A);
      VmV(VEC, Y, X);
    }
    else {
      VpVxS(X, P, A, t);
      VmV(T, Y, P);
      VcrossV(TMP, T, A);
      VcrossV(VEC, A, TMP);
    }
  }
  else {

    VpVxS(Y, Q, B, u);

    if ((t <= 0) || isnan(t)) {
      VcV(X, P);
      VcrossV(TMP, T, B);
      VcrossV(VEC, B, TMP);
    }
    else if (t >= 1) {
      VpV(X, P, A);
      VmV(T, Q, X);
      VcrossV(TMP, T, B);
      VcrossV(VEC, B, TMP);
    }
    else {
      VpVxS(X, P, A, t);
      VcrossV(VEC, A, B);
      if (VdotV(VEC, T) < 0) {
        VxS(VEC, VEC, -1);
      }
    }
  }
}

//--------------------------------------------------------------------------
// TriDist() 
//
// Computes the closest points on two triangles, and returns the 
// distance between them.
// 
// S and T are the triangles, stored tri[point][dimension].
//
// If the triangles are disjoint, P and Q give the closest points of 
// S and T respectively. However, if the triangles overlap, P and Q 
// are basically a random pair of points from the triangles, not 
// coincident points on the intersection of the triangles, as might 
// be expected.
//--------------------------------------------------------------------------

PQP_REAL 
TriDist(PQP_REAL P[3], PQP_REAL Q[3],
        const PQP_REAL S[3][3], const PQP_REAL T[3][3])  
{
  // Compute vectors along the 6 sides

  PQP_REAL Sv[3][3], Tv[3][3];
  PQP_REAL VEC[3];

  VmV(Sv[0],S[1],S[0]);
  VmV(Sv[1],S[2],S[1]);
  VmV(Sv[2],S[0],S[2]);

  VmV(Tv[0],T[1],T[0]);
  VmV(Tv[1],T[2],T[1]);
  VmV(Tv[2],T[0],T[2]);

  // For each edge pair, the vector connecting the closest points 
  // of the edges defines a slab (parallel planes at head and tail
  // enclose the slab). If we can show that the off-edge vertex of 
  // each triangle is outside of the slab, then the closest points
  // of the edges are the closest points for the triangles.
  // Even if these tests fail, it may be helpful to know the closest
  // points found, and whether the triangles were shown disjoint

  PQP_REAL V[3];
  PQP_REAL Z[3];
  PQP_REAL minP[3], minQ[3], mindd;
  int shown_disjoint = 0;

  mindd = VdistV2(S[0],T[0]) + 1;  // Set first minimum safely high

  for (int i = 0; i < 3; i++)
  {
    for (int j = 0; j < 3; j++)
    {
      // Find closest points on edges i & j, plus the 
      // vector (and distance squared) between these points

      SegPoints(VEC,P,Q,S[i],Sv[i],T[j],Tv[j]);
      
      VmV(V,Q,P);
      PQP_REAL dd = VdotV(V,V);

      // Verify this closest point pair only if the distance 
      // squared is less than the minimum found thus far.

      if (dd <= mindd)
      {
        VcV(minP,P);
        VcV(minQ,Q);
        mindd = dd;

        VmV(Z,S[(i+2)%3],P);
        PQP_REAL a = VdotV(Z,VEC);
        VmV(Z,T[(j+2)%3],Q);
        PQP_REAL b = VdotV(Z,VEC);

        if ((a <= 0) && (b >= 0)) return sqrt(dd);

        PQP_REAL p = VdotV(V, VEC);

        if (a < 0) a = 0;
        if (b > 0) b = 0;
        if ((p - a + b) > 0) shown_disjoint = 1;        
      }
    }
  }

  // No edge pairs contained the closest points.  
  // either:
  // 1. one of the closest points is a vertex, and the
  //    other point is interior to a face.
  // 2. the triangles are overlapping.
  // 3. an edge of one triangle is parallel to the other's face. If
  //    cases 1 and 2 are not true, then the closest points from the 9
  //    edge pairs checks above can be taken as closest points for the
  //    triangles.
  // 4. possibly, the triangles were degenerate.  When the 
  //    triangle points are nearly colinear or coincident, one 
  //    of above tests might fail even though the edges tested
  //    contain the closest points.

  // First check for case 1

  PQP_REAL Sn[3], Snl;       
  VcrossV(Sn,Sv[0],Sv[1]); // Compute normal to S triangle
  Snl = VdotV(Sn,Sn);      // Compute square of length of normal
  
  // If cross product is long enough,

  if (Snl > 1e-15)  
  {
    // Get projection lengths of T points

    PQP_REAL Tp[3]; 

    VmV(V,S[0],T[0]);
    Tp[0] = VdotV(V,Sn);

    VmV(V,S[0],T[1]);
    Tp[1] = VdotV(V,Sn);

    VmV(V,S[0],T[2]);
    Tp[2] = VdotV(V,Sn);

    // If Sn is a separating direction,
    // find point with smallest projection

    int point = -1;
    if ((Tp[0] > 0) && (Tp[1] > 0) && (Tp[2] > 0))
    {
      if (Tp[0] < Tp[1]) point = 0; else point = 1;
      if (Tp[2] < Tp[point]) point = 2;
    }
    else if ((Tp[0] < 0) && (Tp[1] < 0) && (Tp[2] < 0))
    {
      if (Tp[0] > Tp[1]) point = 0; else point = 1;
      if (Tp[2] > Tp[point]) point = 2;
    }

    // If Sn is a separating direction, 

    if (point >= 0) 
    {
      shown_disjoint = 1;

      // Test whether the point found, when projected onto the 
      // other triangle, lies within the face.
    
      VmV(V,T[point],S[0]);
      VcrossV(Z,Sn,Sv[0]);
      if (VdotV(V,Z) > 0)
      {
        VmV(V,T[point],S[1]);
        VcrossV(Z,Sn,Sv[1]);
        if (VdotV(V,Z) > 0)
        {
          VmV(V,T[point],S[2]);
          VcrossV(Z,Sn,Sv[2]);
          if (VdotV(V,Z) > 0)
          {
            // T[point] passed the test - it's a closest point for 
            // the T triangle; the other point is on the face of S

            VpVxS(P,T[point],Sn,Tp[point]/Snl);
            VcV(Q,T[point]);
            return sqrt(VdistV2(P,Q));
          }
        }
      }
    }
  }

  PQP_REAL Tn[3], Tnl;       
  VcrossV(Tn,Tv[0],Tv[1]); 
  Tnl = VdotV(Tn,Tn);      
  
  if (Tnl > 1e-15)  
  {
    PQP_REAL Sp[3]; 

    VmV(V,T[0],S[0]);
    Sp[0] = VdotV(V,Tn);

    VmV(V,T[0],S[1]);
    Sp[1] = VdotV(V,Tn);

    VmV(V,T[0],S[2]);
    Sp[2] = VdotV(V,Tn);

    int point = -1;
    if ((Sp[0] > 0) && (Sp[1] > 0) && (Sp[2] > 0))
    {
      if (Sp[0] < Sp[1]) point = 0; else point = 1;
      if (Sp[2] < Sp[point]) point = 2;
    }
    else if ((Sp[0] < 0) && (Sp[1] < 0) && (Sp[2] < 0))
    {
      if (Sp[0] > Sp[1]) point = 0; else point = 1;
      if (Sp[2] > Sp[point]) point = 2;
    }

    if (point >= 0) 
    { 
      shown_disjoint = 1;

      VmV(V,S[point],T[0]);
      VcrossV(Z,Tn,Tv[0]);
      if (VdotV(V,Z) > 0)
      {
        VmV(V,S[point],T[1]);
        VcrossV(Z,Tn,Tv[1]);
        if (VdotV(V,Z) > 0)
        {
          VmV(V,S[point],T[2]);
          VcrossV(Z,Tn,Tv[2]);
          if (VdotV(V,Z) > 0)
          {
            VcV(P,S[point]);
            VpVxS(Q,S[point],Tn,Sp[point]/Tnl);
            return sqrt(VdistV2(P,Q));
          }
        }
      }
    }
  }

  // Case 1 can't be shown.
  // If one of these tests showed the triangles disjoint,
  // we assume case 3 or 4, otherwise we conclude case 2, 
  // that the triangles overlap.
  
  if (shown_disjoint)
  {
    VcV(P,minP);
    VcV(Q,minQ);
    return sqrt(mindd);
  }
  else return 0;
}