Nurbs (#473)

* Limit u,v range between 0 and 1 in Newton. Fixes issue #471
* Change the math for projecting a point onto a plane to work better with non-orthogonal U,V derivatives in several places. Fixes #472.

src/srf/ratpoly.cpp

@@ -439,9 +439,14 @@ void SSurface::ClosestPointTo(Vector p, double *u, double *v, bool mustConverge)
                bu   = (ctrl[1][0]).Minus(orig),
                bv   = (ctrl[0][1]).Minus(orig);
         if((ctrl[1][1]).Equals(orig.Plus(bu).Plus(bv))) {
+
+            Vector n = bu.Cross(bv);
+            Vector ty = n.Cross(bu).ScaledBy(1.0/bu.MagSquared());
+            Vector tx = bv.Cross(n).ScaledBy(1.0/bv.MagSquared());
+
             Vector dp = p.Minus(orig);
-            *u = dp.Dot(bu) / bu.MagSquared();
-            *v = dp.Dot(bv) / bv.MagSquared();
+            *u = dp.Dot(bu) / tx.MagSquared();
+            *v = dp.Dot(bv) / ty.MagSquared();
             return;
         }
     }
@@ -501,16 +506,28 @@ bool SSurface::ClosestPointNewton(Vector p, double *u, double *v, bool mustConve
             }
         }
 
-        Vector tu, tv;
+        Vector tu, tv, tx, ty;
         TangentsAt(*u, *v, &tu, &tv);
+        Vector n = tu.Cross(tv);
+        // since tu and tv may not be orthogonal, use y in place of v.
+        // |y| = |v|sin(theta) where theta is the angle between tu and tv.
+        ty = n.Cross(tu).ScaledBy(1.0/tu.MagSquared());
+        tx = tv.Cross(n).ScaledBy(1.0/tv.MagSquared());
 
         // Project the point into a plane through p0, with basis tu, tv; a
         // second-order thing would converge faster but needs second
         // derivatives.
         Vector dp = p.Minus(p0);
-        double du = dp.Dot(tu), dv = dp.Dot(tv);
-        *u += du / (tu.MagSquared());
-        *v += dv / (tv.MagSquared());
+        double du = dp.Dot(tx),
+               dv = dp.Dot(ty);
+        *u += du / (tx.MagSquared());
+        *v += dv / (ty.MagSquared());
+
+        if (*u < 0.0) *u = 0.0;
+        else if (*u > 1.0) *u = 1.0;
+        if (*v < 0.0) *v = 0.0;
+        else if (*v > 1.0) *v = 1.0;
+
     }
 
     if(mustConverge) {
@@ -540,13 +557,17 @@ bool SSurface::PointIntersectingLine(Vector p0, Vector p1, double *u, double *v)
         // Check for convergence
         if(pi.Equals(p, RATPOLY_EPS)) return true;
 
+        n = tu.Cross(tv);
+        Vector ty = n.Cross(tu).ScaledBy(1.0/tu.MagSquared());
+        Vector tx = tv.Cross(n).ScaledBy(1.0/tv.MagSquared());
+
         // Adjust our guess and iterate
         Vector dp = pi.Minus(p);
-        double du = dp.Dot(tu), dv = dp.Dot(tv);
-        *u += du / (tu.MagSquared());
-        *v += dv / (tv.MagSquared());
+        double du = dp.Dot(tx), dv = dp.Dot(ty);
+        *u += du / tx.MagSquared();
+        *v += dv / ty.MagSquared();
     }
-//    dbp("didn't converge (surface intersecting line)");
+    dbp("didn't converge (surface intersecting line)");
     return false;
 }
 
@@ -582,10 +603,14 @@ Vector SSurface::ClosestPointOnThisAndSurface(SSurface *srf2, Vector p) {
 
         // Adjust our guess and iterate
         for(j = 0; j < 2; j++) {
+            Vector n = tu[j].Cross(tv[j]);
+            Vector ty = n.Cross(tu[j]).ScaledBy(1.0/tu[j].MagSquared());
+            Vector tx = tv[j].Cross(n).ScaledBy(1.0/tv[j].MagSquared());
+
             Vector dc = pc.Minus(cp[j]);
-            double du = dc.Dot(tu[j]), dv = dc.Dot(tv[j]);
-            puv[j].x += du / ((tu[j]).MagSquared());
-            puv[j].y += dv / ((tv[j]).MagSquared());
+            double du = dc.Dot(tx), dv = dc.Dot(ty);
+            puv[j].x += du / tx.MagSquared();
+            puv[j].y += dv / ty.MagSquared();
         }
     }
     if(i >= 10) {
@@ -637,10 +662,15 @@ void SSurface::PointOnSurfaces(SSurface *s1, SSurface *s2, double *up, double *v
         if(parallel) break;
 
         for(j = 0; j < 3; j++) {
+            Vector n = tu[j].Cross(tv[j]);
+            Vector ty = n.Cross(tu[j]).ScaledBy(1.0/tu[j].MagSquared());
+            Vector tx = tv[j].Cross(n).ScaledBy(1.0/tv[j].MagSquared());
+
             Vector dp = pi.Minus(p[j]);
-            double du = dp.Dot(tu[j]), dv = dp.Dot(tv[j]);
-            u[j] += du / (tu[j]).MagSquared();
-            v[j] += dv / (tv[j]).MagSquared();
+            double du = dp.Dot(tx), dv = dp.Dot(ty);
+
+            u[j] += du / tx.MagSquared();
+            v[j] += dv / ty.MagSquared();
         }
     }
     dbp("didn't converge (three surfaces intersecting)");