Sign up ×
TeX - LaTeX Stack Exchange is a question and answer site for users of TeX, LaTeX, ConTeXt, and related typesetting systems. It's 100% free, no registration required.

Is there an easy way to draw a contour image of torus below with tikz? Or for that matter with any other graphics package.

enter image description here

share|improve this question

8 Answers 8

up vote 22 down vote accepted

One fairly easy, but a bit rough-and-ready, would be to load that picture as the background in Inkscape, then draw over the top an SVG version of it, and finally export it to TikZ using the export-tikz plugin.

Actually, for a simple picture like this one you could do it "by hand" in TikZ: use TikZ to draw on top of the picture, adjust the parameters until it looks right, then remove the background.

Other than that, work out the equation of what you're seeing and code that into TikZ. I thought about doing this when I was trying to draw a torus (see my other answer) and decided that I couldn't be bothered to work out the details so would draw a torus "as it was meant to be" (namely, a product of circles).

Edit: Here's the result, a little tweaked afterwards:

\draw (-3.5,0) .. controls (-3.5,2) and (-1.5,2.5) .. (0,2.5);
\draw[xscale=-1] (-3.5,0) .. controls (-3.5,2) and (-1.5,2.5) .. (0,2.5);
\draw[rotate=180] (-3.5,0) .. controls (-3.5,2) and (-1.5,2.5) .. (0,2.5);
\draw[yscale=-1] (-3.5,0) .. controls (-3.5,2) and (-1.5,2.5) .. (0,2.5);

\draw (-2,.2) .. controls (-1.5,-0.3) and (-1,-0.5) .. (0,-.5) .. controls (1,-0.5) and (1.5,-0.3) .. (2,0.2);

\draw (-1.75,0) .. controls (-1.5,0.3) and (-1,0.5) .. (0,.5) .. controls (1,0.5) and (1.5,0.3) .. (1.75,0);


Produced the following:

torus via tikz

share|improve this answer
I feared that this would be the answer. Thank you for your work! – Caramdir Jul 28 '10 at 14:08
@Caramdir: In reaction to the word "feared", I'd point out that the alternative is to work out the exact equations giving the points on the torus tangent to the angle of incidence (solving a tedious set of equations). Given that it's only as good as the medium it'll be displayed on, I think that the simplicity outweighs the inaccuracy. Looking at it now, I'd pull out the edges a little. – Loop Space Jul 28 '10 at 14:18
I hoped someone has already calculated that and put it in a nice tikz extension. But you are right, pictures like this are always intended for illustration anyway, so they do not need to be completely accurate. – Caramdir Jul 28 '10 at 14:24
@Caramdir: For some unknown reason, this one's been bugging me recently. I've found a reasonable set of beziers that draw the outside of the torus when seen from a reasonable range of angles, but the inside has so far eluded me. Any ideas on how to define the inner curve (conceptually, I mean: what properties should it satisfy)? – Loop Space Jun 13 '11 at 21:19
@Andrew: It should be where the direction of projection is tangent to the torus. I finally sat down and did the math: If the torus is parametrized by (φ,ψ) (i.e. given by (R.cos φ + r.cos(φ)*cos(ψ), R.sin(φ) + r.sin(φ).cos(ψ), r.sin(ψ))), and the projection is along the vector (a,b,c), then the lines are the (visible parts of the) solutions to c.sin ψ + (a.cos φ + b.sin φ).cos ψ = 0. Solving for one variable and plotting gives a neat way to draw the torus, except that it also draws the hidden parts of the inner line. – Caramdir Jun 13 '11 at 23:35

I'll see your torus and raise you two more (both SVGs, for PDFs go here and here).

The sources can be found on this page and this page.

(In general, I'd recommend the page for seeing what's capable with TikZ)

share|improve this answer
Thank you, but I am looking for something more like the picture I linked. Shaded images are often not so useful for black-and-white printing. – Caramdir Jul 27 '10 at 13:14

Anthony Phan wrote a 3d extension of Metapost, m3D, which is well suited to such things. As an example, he wrote some code to draw a graph on a Torus (last example):

torus with graph

The downside is that this fork doesn't support nice things like the mptosvg SVG converter, &c, nor the nice Metapost 2 extensions. I seem to recall some discussion of adding 3d support to the mainstream (i.e. Taco Hoekwater stream) Metapost, but I guess that didn't come to anything. But there is some fairly well established 3d drawing support for the regular Metapost language by Dennis Riegel.

share|improve this answer
Looks good, but not quite what I was looking for. It is however always good to know useful tools, so thank you for the links! – Caramdir Jul 28 '10 at 14:15

I traced the original image to get the critical points. By setting showgrid to top and commenting out %\rput(0,0){\usebox\IBox}, you can edit the critical points to get a better result that suits your preferences.




    \temp\psscalebox{-1 1}{\temp}

The following is the output:

enter image description here

And the original one:

enter image description here

share|improve this answer
Is my answer the most similar to the sample in question? – kiss my armpit Sep 8 '12 at 13:37

You could parametrize the surface as (for example)

x(t,s) = (2+cos(t))*cos(s+pi/2) 
y(t,s) = (2+cos(t))*sin(s+pi/2) 
z(t,s) = sin(t)

where both t and s take values on [0,2pi] and then use the pgfplots package.

Admittedly, I'm not sure if this package was available at the time when the question was written :)




       domain=0:2*pi,y domain=0:2*pi,
       z buffer=sort]


Or else with PSTricks


\psset{viewpoint=20 40 40 rtp2xyz,Decran=30,lightsrc=20 10 10}
         base=-10 10 0 6.28,fillcolor=black!70,incolor=orange,
         function=torus,ngrid=60 0.4,



share|improve this answer
Can I insert autmatically ticks by at -pi 0 pi etc.? – lazyboy Sep 5 '12 at 22:59
@lazyboy yep, have a look at axis-with-trigonometric-labels-in-pgfplots – cmhughes Sep 6 '12 at 0:15
The OP asked for a silhouette, and not a plot of the torus. – Dror Sep 6 '12 at 7:03
@Herbert thanks for the feedback. Could you clarify- who can I use Pi without pst-math? I also don't understand how I've defined viewpoint twice – cmhughes Sep 6 '12 at 16:11
Very nice diagram! I added it to the PGFplots example gallery. If you would like to show further plots made by you, let me know - this would be great! I noticed that you are experienced in plotting. Also, if you sometimes might think about a guest blog post oh, to share some pgfplots tricks, I would be glad. – Stefan Kottwitz Mar 17 '14 at 11:19

without a grid


\psset{viewpoint=30 0 15 rtp2xyz,Decran=30,lightsrc=viewpoint}
\psSolid[object=tore,r1=5,r0=1,ngrid=36 72,fillcolor=blue!30,grid=false]%


enter image description here

with a grid a colors


\psset{Decran=30,viewpoint=20 40 30 rtp2xyz,lightsrc=viewpoint}
 \psSolid[object=tore,r1=2.5,r0=1.5,ngrid=18 36,fillcolor=green!30]%

\psset{Decran=30,viewpoint=20 40 30 rtp2xyz,lightsrc=viewpoint}
 \psSolid[object=tore,r1=2.5,r0=1.5,ngrid=18 36,
  tablez=0 0.3 1.5 { } for, zcolor=1 0 0 0 1 1]%


enter image description here

enter image description here

share|improve this answer

Along the line of @AndrewStacey, I tried something slightly simpler. Using one ellipse and an two elliptical arcs, translated, I get the (almost) right visual effect, which is not at all accurate:

enter image description here

The code is rather simple and easy to tweak in case one wants to get a better/different visual effect:

  \draw[domain=0:2*\PI] plot ({\a*cos(\x r)},{\b*sin(\x r)});
  \draw[domain=\PI/4:3*\PI/4] plot ({\a*cos(\x r)},{\b*sin(\x r) -1});
  \draw[domain=-0.1+5*\PI/4:0.1+7*\PI/4] plot ({\a*cos(\x r)},{\b*sin(\x r) +1.1});
share|improve this answer

Here's a solution using an Asymptote module I am writing (which is still in its very early stages).

The images:

A vector image of the contour:

enter image description here

or, "just for fun," in a gif animation (my first ever):

Note that, by design, this animation pauses momentarily when it is the same image (up to resolution) as the one above.

The code:

First, save the following code in a file 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);

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; };
  else eyedir = new triple(triple pt) { return - 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())));
  } 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];

  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) {
    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);
  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);
    triple startingPoint = p + fuzz*directionToCamera;
    return startingPoint --;
  else {
    triple directionToCamera = unit(;
    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) {
  } 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]) {

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) {
  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);
  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) {

    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];
    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) {
    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) {
    current = prospective;
    return toReturn;

  void drawSurfacePath(path parampath, pen p=currentpen, bool onTop=true) {
    path[] toDraw;
    real[] crossingTimes = splitTimes(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);

SmoothSurface operator *(transform3 t, SmoothSurface s) {
  return SmoothSurface(t*s.s);

To get the clean image, compile the following tex file as described in the comments. (The tex file should be in the same directory as surfacepaths.asy.)

%usage (if file is named foo.tex):
%> pdflatex foo.tex
%> asy foo-*.asy
%> pdflatex foo.tex
import surfacepaths;
int niceangle = 70;
currentprojection = orthographic(camera=10Z + .1Y, up=Y);
surface torus = surface(Circle(c=2Y,normal=X,r=0.5,n=32), c=O, axis=Z, n=32);
SmoothSurface Torus = SmoothSurface(rotate(angle=-niceangle, v=X) * torus);

To get the animated version (as a gif file), run the following Asymptote code. (For instance, save it in the file foo.asy, and then enter asy foo at the command line.)

import surfacepaths;
import animation;

size(50cm,0);    // Increased size and line width for better resolution

int niceangle = 70;

currentprojection = orthographic(camera=10Z + .1Y, up=Y);
surface torus = surface(Circle(c=2Y,normal=X,r=0.5,n=100), c=O, axis=Z, n=32);
SmoothSurface Torus = SmoothSurface(rotate(angle=-niceangle, v=X) * torus);

animation A;

for (int angle = 0; angle <= 180; angle += 5) {
  (rotate(angle=-angle, v=X) * Torus).drawSilhouette(linewidth(2pt));    // Increase size and line width for better resolution
  write("computed angle " + (string)angle);  //output some progress indicator
share|improve this answer
@Stiff Jokes: I was able to eliminate most of the flicker by increasing the number of Bezier patches making up the surfaces. Because of the way the algorithms are implemented here, this actually does not hugely affect the compilation time. – Charles Staats Jan 5 '14 at 20:10
Extremely excellent. Thanks. – kiss my armpit Jan 5 '14 at 20:11

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.