The height function on a Torus

Question

See the torus as a surface of revolution around the x (or y) axis. Define the height function on it.

How to shade only a part of the torus based on the height function?

There are illustrations of homotopy type and level curves on the torus on wikipedia; illustrations of gradient flows on the torus here.

[wikipedia]

Answer 1

Here's a starting point.

It may be compiled as follows.

First, save the following code, inspired by splitpatch.asy, in a file called crop3D.asy:

import three;

/**********************************************/
/* Code for splitting surfaces: */

struct possibleInt {
  int value;
  bool holds;
}

int operator cast(possibleInt i) { return i.value; }

restricted int maxdepth = 20;
restricted void maxdepth(int n) { maxdepth = n; }

surface[] divide(surface s, possibleInt region(patch), int numregions,
         bool keepregion(int) = null) {

  if (keepregion == null) keepregion = new bool(int region) {
      return (0 <= region && region < numregions);
    };

  surface[] toreturn = new surface[numregions];
  for (int i = 0; i < numregions; ++i)
    toreturn[i] = new surface;

  void addPatch(patch P, int region) {
    if (keepregion(region)) toreturn[region].push(P);
  }

  void divide(patch P, int depth) {
    if (depth == 0) {
      addPatch(P, region(P));
      return;
    }

    possibleInt region = region(P);
    if (region.holds) {
      addPatch(P, region);
      return;
    }

    // Choose the splitting function based on the parity of the recursion depth.
    triple[][][] Split(triple[][] P) = (depth % 2 == 0 ? hsplit : vsplit);

    patch[] Split(patch P) {
      triple[][][] patches = Split(P.P);
      return sequence(new patch(int i) {return patch(patches[i]);}, patches.length);
    }

    patch[] patches = Split(P);
    for (patch PP : patches)
      divide(PP, depth-1);
  }

  for (patch P : s.s)
    divide(P, maxdepth);

  return toreturn;
}


surface[] divide(surface s, int region(triple), int numregions,
         bool keepregion(int) = null) {
  possibleInt patchregion(patch P) {
    triple[][] controlpoints = P.P;
    possibleInt theRegion;
    theRegion.value = region(controlpoints[0][0]);
    theRegion.holds = true;
    for (triple[] ta : controlpoints) {
      for (triple t : ta) {
    if (region(t) != theRegion.value) {
      theRegion.holds = false;
      break;
    }
      }
      if (!theRegion.holds) break;
    }
    return theRegion;
  }

  return divide(s, patchregion, numregions, keepregion);
}


/**************************************************/
/* Code for cropping surfaces */

// Return 0 iff the point lies in box(a,b).
int cropregion(triple pt, triple a=O, triple b=(1,1,1)) {
  real x=pt.x, y=pt.y, z=pt.z;
  int toreturn=0;
  real xmin=a.x, xmax=b.x, ymin = a.y, ymax=b.y, zmin=a.z, zmax=b.z;
  if (xmin > xmax) { xmin = b.x; xmax = a.x; }
  if (ymin > ymax) { ymin = b.y; ymax = a.y; }
  if (zmin > zmax) { zmin = b.z; zmax = a.z; }
  if (x < xmin) --toreturn;
  else if (x > xmax) ++toreturn;
  toreturn *= 2;
  if (y < ymin) --toreturn;
  else if (y > ymax) ++toreturn;
  toreturn *= 2;
  if (z < zmin) --toreturn;
  else if (z > zmax) ++toreturn;
  return toreturn;
}

//bool keepregion(int region) { return (region == 0); }

// Crop the surface to box(a,b).
surface crop(surface s, triple a, triple b) {
  int region(triple pt) {
    return cropregion(pt, a, b);
  }
  return divide(s, region=region, numregions=1)[0];
}

// A rectangular solid with opposite vertices a, b:
surface surfacebox(triple a, triple b) {
  return shift(a)*scale((b-a).x,(b-a).y,(b-a).z)*unitcube;
}

bool containedInBox(triple pt, triple a, triple b) {
  return cropregion(pt, a, b) == 0;
}

// Crop a path3 to box(a,b).
path3[] crop(path3 g, triple a, triple b) {
  surface thebox = surfacebox(a,b);
  path3[] toreturn;
  real[] times = new real[] {0};
  real[][] alltimes = intersections(g, thebox);
  for (real[] threetimes : alltimes)
    times.push(threetimes[0]);
  times.push(length(g));
  for (int i = 1; i < times.length; ++i) {
    real mintime = times[i-1];
    real maxtime = times[i];
    triple midpoint = point(g, (mintime+maxtime)/2);
    if (containedInBox(midpoint, a, b))
      toreturn.push(subpath(g, mintime, maxtime));
  }
  return toreturn;
}

path3[] crop(path3[] g, triple a, triple b) {
  path3[] toreturn;
  for (path3 gi : g)
    toreturn.append(crop(gi, a, b));
  return toreturn;
}

/***************************************/
/* Code to return only the portion of the surface facing the camera */

bool facingCamera(triple vec, triple pt=O, projection P = currentprojection, bool towardsCamera = true) {
  triple normal = P.camera;
  if (!P.infinity) {
    normal = P.camera - pt;
  }
  if (towardsCamera) return (dot(vec, normal) >= 0);
  else return (dot(vec, normal) <= 0);
}

surface facingCamera(surface s, bool towardsCamera = true) {

  possibleInt facingregion(patch P) {
    int n = 2;
    possibleInt toreturn;
    unravel toreturn;
    bool facingcamera = facingCamera(P.normal(1/2, 1/2), pt=P.point(1/2,1/2), towardsCamera);
    value = facingcamera ? 0 : 1;
    holds = true;
    for (int i = 0; i <= n; ++i) {
      real u = i/n;
      for (int j = 0; j <= n; ++j) {
    real v = j/n;
    if (facingCamera(P.normal(u,v), P.point(u,v), towardsCamera) != facingcamera) {
      holds = false;
      break;
    }
      }
      if (!holds) break;
    }
    return toreturn;
  }

  return divide(s, facingregion, numregions=1)[0];
}

Then, save the following code in a file (in the same directory) called surfacepaths.asy:

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);
  }
}

triple[] derivative(surface s, pair pt) {
  patchWithCoords thepatch = patchWithCoords(s, pt.x, pt.y);
  return new triple[] {thepatch.partialu(), thepatch.partialv()};
}

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;

}

path3[] onSurface(surface s, path[] p) {
  return sequence(new path3(int i) {return onSurface(s,p[i]);}, p.length);
}

/*
 * 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));
  };
}


path3[] levelcurve(function3 f, surface s, real value=0) {
  function2 fparam = pullback(f, s);
  path[] paramcurve = contour(fparam, (0,0), uvmax(s), new real[] {value})[0];
  return onSurface(s, paramcurve);
}

/*
 * 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);
    }
  }
  layeredPaths.delete();
}

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;
}

Finally, here's the code that actually creates the image (using the previous two modules). Save it in a file called (e.g.) morse_theory.asy in the same directory as the other two files, and then run asy morse_theory at the command line.

settings.outformat="png";
settings.render=16;
settings.prc=false;
usepackage("lmodern");
usepackage("amssymb");
defaultpen(fontsize(10pt));
unitsize(1cm);

import graph3;

/**********************************************/
/* Code for splitting surfaces: */

import crop3D;

/************************************************/
/* Code for drawing a level curve on a surface: */

import surfacepaths;

/************************************************/
/* The actual drawing: */


currentprojection = orthographic(X+.2Z);

real r = 1.4;
real R = 2.6;
real rc = (r+R)/2;
real rtube = (R-r)/2;
transform3 yscale = scale(1,.8,1);

surface torus = yscale * surface(rotate(angle=90,rc*Y,rc*Y+Z)*Circle(c=rc*Y, r=rtube, normal=Z, n=8), c=O, axis=X, n=8);

maxdepth(10);
surface torusFront = facingCamera(torus, towardsCamera=false);
maxdepth(17);

real[] dividerHeights = sort(new real[] {-rc+.1, -r, -rc+2rtube-.1});

int region(triple t) {
  real z = t.z;
  for (int i = 0; i < dividerHeights.length; ++i) {
    if (z < dividerHeights[i]) return i;
  }
  return dividerHeights.length;
}

int numregions = dividerHeights.length + 1;

surface[] torusCells = divide(torusFront, region, numregions);

pen transparency = opacity(0.8);

material[] materials = new material[] {material(black+transparency),
                       material(gray(1/3)+transparency, ambientpen=gray(1/3)),
                       material(gray(2/3)+transparency, ambientpen=gray(2/3)),
                       material(white+transparency, ambientpen=white)};

draw(torusCells[0], surfacepen=materials[0]);
draw(torusCells[1], surfacepen=materials[1]);
draw(torusCells[2], surfacepen=materials[2]);
draw(torusCells[3], surfacepen=materials[3]);

real height(triple pt) { return pt.z; }

for (real h : dividerHeights) {
  draw(levelcurve(height, torus, h));
}

path3[] circularBands = levelcurve(planeEqn(pt=O,normal=Y+.3X), torus, 0);

real maxdivide = dividerHeights[dividerHeights.length - 1];
real mindivide = dividerHeights[0];

path3[] bandsToDraw = crop(circularBands, (-5,-5, mindivide), (5, 5, maxdivide));

// In our case, as it happens, there is only one band--but let's make sure of that.
assert(bandsToDraw.length == 1);
path3 bandToDraw = bandsToDraw[0];

draw(bandToDraw);

triple bandPoint = relpoint(bandToDraw, 0.25);
//dot(bandPoint);
draw(maxdivide*Z {(-1,-.4,-.2)} .. {(-.4,.6,-1)} bandPoint, arrow=ArcArrow3(DefaultHead2,emissive(black)), L=Label("$e^k$", position=BeginPoint, align=ENE));

real y = R;
draw((0,y,-R) -- (0,y,R), L=Label("$\mathbb{R}$",position=EndPoint,align=NE), arrow=Arrow3(TeXHead2,emissive(black)));

//draw bars with size specified in postscript coordinates
draw(shift(0,y,dividerHeights[0]) * ((0,-.1,0) -- (0,.1,0)), L=Label("$c-\varepsilon$",position=EndPoint,align=E));
draw(shift(0,y,dividerHeights[1]) * ((0,-.1,0) -- (0,.1,0)), L=Label("$c$",position=EndPoint,align=E));
draw(shift(0,y,dividerHeights[2]) * ((0,-.1,0) -- (0,.1,0)), L=Label("$c+\varepsilon$",position=EndPoint,align=E));


transform3 shiftright = shift(0,6,0);

real perturbdist = abs(min(torus).y)/2;
path perturbpath = (-2*perturbdist,0) --- (-perturbdist,0) .. (0,.08) .. (perturbdist,0) --- (2*perturbdist,0);

real perturbedHeight(triple pt) {
  real z = pt.z, y=pt.y;
  if (abs(y) > perturbdist) return z;
  else return z + point(perturbpath,times(perturbpath,y)[0]).y;
}

real perturbedDivide = -r + .01;

region = new int(triple t) {
  if (perturbedHeight(t) < perturbedDivide) return 0;
  else if (height(t) < perturbedDivide) return 1;
  else if (height(t) < dividerHeights[2]) return 2;
  else return 3;
};

surface[] newTorusCells = divide(torusFront, region, numregions=3);
newTorusCells.push(torusCells[3]);
for (int i = 0; i < newTorusCells.length; ++i)
  draw(shiftright*newTorusCells[i], surfacepen=materials[i]);

draw(shiftright * crop(levelcurve(perturbedHeight, torus,perturbedDivide), (0,-5,-5), (5,5,5)));
draw(shiftright * crop(levelcurve(height, torus, perturbedDivide), (0,-5,-5), (5,5,5)));
draw(shiftright * levelcurve(height, torus, maxdivide));

draw(shiftright * crop(Circle(c=-rc*Z, r=rtube, normal=Y), (5,5,perturbedDivide-perturbedHeight(O)), (0,-5, 5)));

draw(shiftright * ((0,R,-R) -- (0,R,maxdivide)), Bars3, arrow=Arrows3(DefaultHead2), L=Label("$M \leq c + \varepsilon$", position=MidPoint, align=E));

Answer 2

Please find attached an inkscape code (download the code and save it with .svg extension). You can freely modify it to your needs. The resulting image is shown below.

<?xml version="1.0" encoding="UTF-8" standalone="no"?>
<!-- Created with Inkscape (http://www.inkscape.org/) -->

<svg
   xmlns:osb="http://www.openswatchbook.org/uri/2009/osb"
   xmlns:dc="http://purl.org/dc/elements/1.1/"
   xmlns:cc="http://creativecommons.org/ns#"
   xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
   xmlns:svg="http://www.w3.org/2000/svg"
   xmlns="http://www.w3.org/2000/svg"
   xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
   xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
   width="800"
   height="800"
   id="svg2"
   version="1.1"
   inkscape:version="0.48.4 r9939"
   sodipodi:docname="torus.svg"
   inkscape:export-filename="torus.png"
   inkscape:export-xdpi="600"
   inkscape:export-ydpi="600">
  <defs
     id="defs4">
    <marker
       inkscape:stockid="Arrow2Mstart"
       orient="auto"
       refY="0.0"
       refX="0.0"
       id="Arrow2Mstart"
       style="overflow:visible">
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