latest changes from Charles Staats for torus with three paths

asymptote/torus-three-paths/Makefile

@@ -12,7 +12,7 @@ make:
 	make clean
 
 clean:
-	rm -rf  $(TARGET) *.class *.html *.log *.aux *.data *.gnuplot *.out *.prc *.pre *.asy $(SOURCE)-1_0.pdf $(SOURCE)1+0_0.pdf $(SOURCE)-1.tex
+	rm -rf  $(TARGET) *.class *.html *.log *.aux *.data *.gnuplot *.out *.prc *.pre $(SOURCE)-1_0.pdf $(SOURCE)1+0_0.pdf $(SOURCE)-1.tex
 
 gif:
 	pdfcrop $(SOURCE).pdf

asymptote/torus-three-paths/next.asy

@@ -0,0 +1,540 @@
+settings.outformat="pdf";
+
+import graph3;
+import contour;
+
+// A bunch of auxiliary functions.
+
+real fuzz = .001;
+
+real umin(surface s) { return 0; }
+real vmin(surface s) { return 0; }
+pair uvmin(surface s) { return (umin(s), vmin(s)); }
+real umax(surface s, real fuzz=fuzz) {
+  if (s.ucyclic()) return s.index.length;
+  else return s.index.length - fuzz;
+}
+real vmax(surface s, real fuzz=fuzz) {
+  if (s.vcyclic()) return s.index[0].length;
+  return s.index[0].length - fuzz;
+}
+pair uvmax(surface s, real fuzz=fuzz) { return (umax(s,fuzz), vmax(s,fuzz)); }
+
+typedef real function(real, real);
+
+function normalDot(surface s, triple eyedir(triple)) {
+  real toreturn(real u, real v) {
+    return dot(s.normal(u, v), eyedir(s.point(u,v)));
+  }
+  return toreturn;
+}
+
+struct patchWithCoords {
+  patch p;
+  real u;
+  real v;
+  void operator init(patch p, real u, real v) {
+    this.p = p;
+    this.u = u;
+    this.v = v;
+  }
+  void operator init(surface s, real u, real v) {
+    int U=floor(u);
+    int V=floor(v);
+    int index = (s.index.length == 0 ? U+V : s.index[U][V]);
+
+    this.p = s.s[index];
+    this.u = u-U;
+    this.v = v-V;
+  }
+  triple partialu() {
+    return p.partialu(u,v);
+  }
+  triple partialv() {
+    return p.partialv(u,v);
+  }
+}
+
+typedef triple paramsurface(pair);
+
+paramsurface tangentplane(surface s, pair pt) {
+  patchWithCoords thepatch = patchWithCoords(s, pt.x, pt.y);
+  triple partialu = thepatch.partialu();
+  triple partialv = thepatch.partialv();
+  return new triple(pair tangentvector) {
+    return s.point(pt.x, pt.y) + (tangentvector.x * partialu) + (tangentvector.y * partialv);
+  };
+}
+
+guide[] normalpathuv(surface s, projection P = currentprojection, int n = ngraph) {
+  triple eyedir(triple a);
+  if (P.infinity) eyedir = new triple(triple) { return P.camera; };
+  else eyedir = new triple(triple pt) { return P.camera - pt; };
+  return contour(normalDot(s, eyedir), uvmin(s), uvmax(s), new real[] {0}, nx=n)[0];
+}
+
+path3 onSurface(surface s, path p) {
+  triple f(int t) {
+    pair point = point(p,t);
+    return s.point(point.x, point.y);
+  }
+
+  guide3 toreturn = f(0);
+  paramsurface thetangentplane = tangentplane(s, point(p,0));
+  triple oldcontrol, newcontrol;
+  int size = length(p);
+  for (int i = 1; i <= size; ++i) {
+    oldcontrol = thetangentplane(postcontrol(p,i-1) - point(p,i-1));
+    thetangentplane = tangentplane(s, point(p,i));
+    newcontrol = thetangentplane(precontrol(p, i) - point(p,i));
+    toreturn = toreturn .. controls oldcontrol and newcontrol .. f(i);
+  }
+
+  if (cyclic(p)) toreturn = toreturn & cycle;
+
+  return toreturn;
+
+}
+
+/*
+ * This method returns an array of paths that trace out all the
+ * points on s at which s is parallel to eyedir.
+ */
+
+path[] paramSilhouetteNoEdges(surface s, projection P = currentprojection, int n = ngraph) {
+   guide[] uvpaths = normalpathuv(s, P, n);
+  //Reduce the number of segments to conserve memory
+  for (int i = 0; i < uvpaths.length; ++i) {
+    real len = length(uvpaths[i]);
+    uvpaths[i] = graph(new pair(real t) {return point(uvpaths[i],t);}, 0, len, n=n);
+  }
+  return uvpaths;
+}   
+
+private typedef real function2(real, real);
+private typedef real function3(triple);
+
+triple[] normalVectors(triple dir, triple surfacen) {
+  dir = unit(dir);
+  surfacen = unit(surfacen);
+  triple v1, v2;
+  int i = 0;
+  do {
+    v1 = unit(cross(dir, (unitrand(), unitrand(), unitrand())));
+    v2 = unit(cross(dir, (unitrand(), unitrand(), unitrand())));
+    ++i;
+  } while ((abs(dot(v1,v2)) > Cos(10) || abs(dot(v1,surfacen)) > Cos(5) || abs(dot(v2,surfacen)) > Cos(5)) && i < 1000);
+  if (i >= 1000) {
+    write("problem: Unable to comply.");
+    write(" dir = " + (string)dir);
+    write(" surface normal = " + (string)surfacen);
+  }
+  return new triple[] {v1, v2};
+}
+
+function3 planeEqn(triple pt, triple normal) {
+  return new real(triple r) {
+    return dot(normal, r - pt);
+  };
+}
+
+function2 pullback(function3 eqn, surface s) {
+  return new real(real u, real v) {
+    return eqn(s.point(u,v));
+  };
+}
+
+/*
+ * returns the distinct points in which the surface intersects
+ * the line through the point pt in the direction dir
+ */
+
+triple[] intersectionPoints(surface s, pair parampt, triple dir) {
+  triple pt = s.point(parampt.x, parampt.y);
+  triple[] lineNormals = normalVectors(dir, s.normal(parampt.x, parampt.y));
+  path[][] curves;
+  for (triple n : lineNormals) {
+    function3 planeEn = planeEqn(pt, n);
+    function2 pullback = pullback(planeEn, s);
+    guide[] contour = contour(pullback, uvmin(s), uvmax(s), new real[]{0})[0];
+
+    curves.push(contour);
+  }
+  pair[] intersectionPoints;
+  for (path c1 : curves[0])
+    for (path c2 : curves[1])
+      intersectionPoints.append(intersectionpoints(c1, c2));
+  triple[] toreturn;
+  for (pair P : intersectionPoints)
+    toreturn.push(s.point(P.x, P.y));
+  return toreturn;
+}
+
+
+
+/*
+ * Returns those intersection points for which the vector from pt forms an
+ * acute angle with dir.
+ */
+ int numPointsInDirection(surface s, pair parampt, triple dir, real fuzz=.05) {
+  triple pt = s.point(parampt.x, parampt.y);
+  dir = unit(dir);
+  triple[] intersections = intersectionPoints(s, parampt, dir);
+  int num = 0;
+  for (triple isection: intersections)
+    if (dot(isection - pt, dir) > fuzz) ++num;
+  return num;
+}
+
+bool3 increasing(real t0, real t1) {
+  if (t0 < t1) return true;
+  if (t0 > t1) return false;
+  return default;
+}
+
+int[] extremes(real[] f, bool cyclic = f.cyclic) {
+  bool3 lastIncreasing;
+  bool3 nextIncreasing;
+  int max;
+  if (cyclic) {
+    lastIncreasing = increasing(f[-1], f[0]);
+    max = f.length - 1;
+  } else {
+    max = f.length - 2;
+    if (increasing(f[0], f[1])) lastIncreasing = false;
+    else lastIncreasing = true;
+  }
+  int[] toreturn;
+  for (int i = 0; i <= max; ++i) {
+    nextIncreasing = increasing(f[i], f[i+1]);
+    if (lastIncreasing != nextIncreasing) {
+      toreturn.push(i);
+    }
+    lastIncreasing = nextIncreasing;
+  }
+  if (!cyclic) toreturn.push(f.length - 1);
+  toreturn.cyclic = cyclic;
+  return toreturn;
+}
+
+int[] extremes(path path, real f(pair) = new real(pair P) {return P.x;})
+{
+  real[] fvalues = new real[size(path)];
+  for (int i = 0; i < fvalues.length; ++i) {
+    fvalues[i] = f(point(path, i));
+  }
+  fvalues.cyclic = cyclic(path);
+  int[] toreturn = extremes(fvalues);
+  fvalues.delete();
+  return toreturn;
+}
+
+path[] splitAtExtremes(path path, real f(pair) = new real(pair P) {return P.x;})
+{
+  int[] splittingTimes = extremes(path, f);
+  path[] toreturn;
+  if (cyclic(path)) toreturn.push(subpath(path, splittingTimes[-1], splittingTimes[0]));
+  for (int i = 0; i+1 < splittingTimes.length; ++i) {
+    toreturn.push(subpath(path, splittingTimes[i], splittingTimes[i+1]));
+  }
+  return toreturn;
+}
+
+path[] splitAtExtremes(path[] paths, real f(pair P) = new real(pair P) {return P.x;})
+{
+  path[] toreturn;
+  for (path path : paths) {
+    toreturn.append(splitAtExtremes(path, f));
+  }
+  return toreturn;
+}
+
+path3 toCamera(triple p, projection P=currentprojection, real fuzz = .01, real upperLimit = 100) {
+  if (!P.infinity) {
+    triple directionToCamera = unit(P.camera - p);
+    triple startingPoint = p + fuzz*directionToCamera;
+    return startingPoint -- P.camera;
+  }
+  else {
+    triple directionToCamera = unit(P.camera);
+    triple startingPoint = p + fuzz*directionToCamera;
+
+    return startingPoint -- (p + upperLimit*directionToCamera);
+  }
+}
+
+int numSheetsHiding(surface s, pair parampt, projection P = currentprojection) {
+  triple p = s.point(parampt.x, parampt.y);
+  path3 tocamera = toCamera(p, P);
+  triple pt = beginpoint(tocamera);
+  triple dir = endpoint(tocamera) - pt;
+  return numPointsInDirection(s, parampt, dir);
+}
+
+struct coloredPath {
+  path path;
+  pen pen;
+  void operator init(path path, pen p=currentpen) {
+    this.path = path;
+    this.pen = p;
+  }
+  /* draws the path with the pen having the specified weight (using colors)*/
+  void draw(real weight) {
+    draw(path, p=weight*pen + (1-weight)*white);
+  }
+}
+coloredPath[][] layeredPaths;
+// onTop indicates whether the path should be added at the top or bottom of the specified layer
+void addPath(path path, pen p=currentpen, int layer, bool onTop=true) {
+  coloredPath toAdd = coloredPath(path, p);
+  if (layer >= layeredPaths.length) {
+    layeredPaths[layer] = new coloredPath[] {toAdd};
+  } else if (onTop) {
+    layeredPaths[layer].push(toAdd);
+  } else layeredPaths[layer].insert(0, toAdd);
+}
+
+void drawLayeredPaths() {
+  for (int layer = layeredPaths.length - 1; layer >= 0; --layer) {
+    real layerfactor = (1/3)^layer;
+    for (coloredPath toDraw : layeredPaths[layer]) {
+      toDraw.draw(layerfactor);
+    }
+  }
+}
+
+real[] cutTimes(path tocut, path[] knives) {
+  real[] intersectionTimes = new real[] {0, length(tocut)};
+  for (path knife : knives) {
+    real[][] complexIntersections = intersections(tocut, knife);
+    for (real[] times : complexIntersections) {
+      intersectionTimes.push(times[0]);
+    }
+  }
+  return sort(intersectionTimes);
+}
+
+path[] cut(path tocut, path[] knives) {
+  real[] cutTimes = cutTimes(tocut, knives);
+  path[] toreturn;
+  for (int i = 0; i + 1 < cutTimes.length; ++i) {
+    toreturn.push(subpath(tocut,cutTimes[i], cutTimes[i+1]));
+  }
+  return toreturn;
+}
+
+real[] condense(real[] values, real fuzz=.001) {
+  values = sort(values);
+  values.push(infinity);
+  real previous = -infinity;
+  real lastMin;
+  real[] toReturn;
+  for (real t : values) {
+    if (t - fuzz > previous) {
+      if (previous > -infinity) toReturn.push((lastMin + previous) / 2);
+      lastMin = t;
+    }
+    previous = t;
+  }
+  return toReturn;
+}
+
+/*
+ * A smooth surface parametrized by the domain [0,1] x [0,1]
+ */
+struct SmoothSurface {
+  surface s;
+  private real sumax;
+  private real svmax;
+  path[] paramSilhouette;
+  path[] projectedSilhouette;
+  projection theProjection;
+
+  path3 onSurface(path paramPath) {
+    return onSurface(s, scale(sumax,svmax)*paramPath);
+  }
+
+  triple point(real u, real v) { return s.point(sumax*u, svmax*v); }
+  triple point(pair uv) { return point(uv.x, uv.y); }
+  triple normal(real u, real v) { return s.normal(sumax*u, svmax*v); }
+  triple normal(pair uv) { return normal(uv.x, uv.y); }
+
+  void operator init(surface s, projection P=currentprojection) {
+    this.s = s;
+    this.sumax = umax(s);
+    this.svmax = vmax(s);
+    this.theProjection = P;
+    this.paramSilhouette = scale(1/sumax, 1/svmax) * paramSilhouetteNoEdges(s,P);
+    this.projectedSilhouette = sequence(new path(int i) {
+    path3 truePath = onSurface(paramSilhouette[i]);
+    path projectedPath = project(truePath, theProjection, ninterpolate=1);
+    return projectedPath;
+      }, paramSilhouette.length);
+  }
+
+  int numSheetsHiding(pair parampt) {
+    return numSheetsHiding(s, scale(sumax,svmax)*parampt);
+  }
+
+  void drawSilhouette(pen p=currentpen, bool includePathsBehind=false, bool onTop = true) {
+    int[][] extremes;
+    for (path path : projectedSilhouette) {
+      extremes.push(extremes(path));
+    }
+
+    path[] splitSilhouette;
+    path[] paramSplitSilhouette;
+
+    /*
+     * First, split at extremes to ensure that there are no
+     * self-intersections of any one subpath in the projected silhouette.
+     */
+
+    for (int j = 0; j < paramSilhouette.length; ++j) {
+      path current = projectedSilhouette[j];
+
+      path currentParam = paramSilhouette[j];
+
+      int[] dividers = extremes[j];
+      for (int i = 0; i + 1 < dividers.length; ++i) {
+    int start = dividers[i];
+    int end = dividers[i+1];
+    splitSilhouette.push(subpath(current,start,end));
+    paramSplitSilhouette.push(subpath(currentParam, start, end));
+      }
+    }
+
+    /*
+     * Now, split at intersections of distinct subpaths.
+     */
+
+    for (int j = 0; j < splitSilhouette.length; ++j) {
+      path current = splitSilhouette[j];
+      path currentParam = paramSplitSilhouette[j];
+
+      real[] splittingTimes = new real[] {0,length(current)};
+      for (int k = 0; k < splitSilhouette.length; ++k) {
+    if (j == k) continue;
+    real[][] times = intersections(current, splitSilhouette[k]);
+    for (real[] time : times) {
+      real relevantTime = time[0];
+      if (.01 < relevantTime && relevantTime < length(current) - .01) splittingTimes.push(relevantTime);
+    }
+      }
+      splittingTimes = sort(splittingTimes);
+      for (int i = 0; i + 1 < splittingTimes.length; ++i) {
+    real start = splittingTimes[i];
+    real end = splittingTimes[i+1];
+    real mid = start + ((end-start) / (2+0.1*unitrand()));
+    pair theparampoint = point(currentParam, mid);
+    int sheets = numSheetsHiding(theparampoint);
+
+    if (sheets == 0 || includePathsBehind) {
+      path currentSubpath = subpath(current, start, end);
+      addPath(currentSubpath, p=p, onTop=onTop, layer=sheets);
+    }
+
+      }
+    }
+  }
+
+  /*
+    Splits a parametrized path along the parametrized silhouette,
+    taking [0,1] x [0,1] as the
+    fundamental domain.  Could be implemented more efficiently.
+  */
+  private real[] splitTimes(path thepath) {
+    pair min = min(thepath);
+    pair max = max(thepath);
+    path[] baseknives = paramSilhouette;
+    path[] knives;
+    for (int u = floor(min.x); u < max.x + .001; ++u) {
+      for (int v = floor(min.y); v < max.y + .001; ++v) {
+    knives.append(shift(u,v)*baseknives);
+      }
+    }
+    return cutTimes(thepath, knives);
+  }
+
+  /*
+    Returns the times at which the projection of the given path3 intersects
+    the projection of the surface silhouette. This may miss unstable
+    intersections that can be detected by the previous method.
+  */
+  private real[] silhouetteCrossingTimes(path3 thepath, real fuzz = .01) {
+    path projectedpath = project(thepath, theProjection, ninterpolate=1);
+    real[] crossingTimes = cutTimes(projectedpath, projectedSilhouette);
+    if (crossingTimes.length == 0) return crossingTimes;
+    real current = 0;
+    real[] toReturn = new real[] {0};
+    for (real prospective : crossingTimes) {
+      if (prospective > current + fuzz
+      && prospective < length(thepath) - fuzz) {
+    toReturn.push(prospective);
+    current = prospective;
+      }
+    }
+    toReturn.push(length(thepath));
+    return toReturn;
+  }
+
+  void drawSurfacePath(path parampath, pen p=currentpen, bool onTop=true) {
+    path[] toDraw;
+    real[] crossingTimes = splitTimes(parampath);
+    crossingTimes.append(silhouetteCrossingTimes(onSurface(parampath)));
+    crossingTimes = condense(crossingTimes);
+    for (int i = 0; i+1 < crossingTimes.length; ++i) {
+      toDraw.push(subpath(parampath, crossingTimes[i], crossingTimes[i+1]));
+    }
+    for (path thepath : toDraw) {
+      pair midpoint = point(thepath, length(thepath) / (2+.1*unitrand()));
+      int sheets = numSheetsHiding(midpoint);
+      path path3d = project(onSurface(thepath), theProjection, ninterpolate = 1);
+      addPath(path3d, p=p, onTop=onTop, layer=sheets);
+    }
+  }
+}
+
+real unit = 4cm;
+unitsize(unit);
+triple eye = (10,1,4);
+//currentprojection=perspective(2*eye);
+currentprojection=orthographic(eye);
+
+surface torus = surface(Circle(c=2Y, r=0.6, normal=X, n=32), c=O, axis=Z, n=32);
+torus.ucyclic(true);
+torus.vcyclic(true);
+
+SmoothSurface Torus = SmoothSurface(torus);
+
+Torus.drawSilhouette(p=black, includePathsBehind=true);
+
+pair a = (22/40, 3/40);
+pair b = (5/40, .5);
+
+path abpathparam(int ucycles, int vcycles) {
+  pair bshift = (ucycles, vcycles);
+  pair f(real t) {
+    return (1-t)*a + t*(b+bshift);
+  }
+  return graph(f, 0, 1);
+}
+
+real linewidth = 0.8pt;
+
+Torus.drawSurfacePath(abpathparam(0,0), p=linewidth + orange);
+Torus.drawSurfacePath(abpathparam(1,0), p=linewidth + red);
+Torus.drawSurfacePath(abpathparam(1,-1), p=linewidth + (darkgreen + 0.2blue));
+
+pen meshpen = gray(0.6);
+for (real u = 0; u < 1; u += 1/10) {
+  Torus.drawSurfacePath(graph(new pair(real v) {return (u,v);}, 0,1,n=10), p=meshpen, onTop=false);
+}
+for (real v = 0; v < 1; v += 1/10) {
+  Torus.drawSurfacePath(graph(new pair(real u) {return (u,v);}, 0, 1, n=10), p=meshpen, onTop=false);
+}
+
+drawLayeredPaths();
+
+dot(project(Torus.point(a.x,a.y)), L="$a$", align=W);
+dot(project(Torus.point(b.x,b.y)), L="$b$", align=N);