[Tex/LaTex] plot 1d curves in 3d space with nice looking intersections

3ddiagrams

I am trying to draw multiple colored one-dimensional parametric
curves in three dimensional euclidean space. The curves
themselves do not intersect in 3d, but sometimes their
projections intersect when projected into 2d for visualization.
In these cases, I would like the color at the intersection to
be the color of the curve 'in front.'

The problem is that the curves are not strictly ordered (they
could be knot-like) so I have some variant of the painter's
problem, except a bit simpler because all my objects are
one-dimensional and they do not intersect in 3d.

Is there a nice LaTeX or related thing that could help me with
this? I've looked at examples for asymptote and some other ps
and tikz packages but I can't find a nice example where this is
done automatically.

Currently my work-around is to break all of the curves into
tiny segments and then depth sort. This works as long as the
segments are small relative to the minimum distance between
curves in 3d, I think. But it is a hack and is ugly.

Best Answer

Doing these computations in tex is usually not a pretty sight, so Z-ordering is mostly absent in these plotting tools. I believe you need to look into an external tool, like sketch which can generate the tikz code for you.

Related Solutions

[Tex/LaTex] Drawing 3d diagrams

Asymptote is definitely a tool to look at, it can do nut just parametric surfaces, but graphs of functions, and I believe even isosurfaces, although I am not sure.

Take a look at sketch. It does not seem to be great at surfaces, though.

Metapost can do pretty interesting things, take a look at these examples.

[Tex/LaTex] Stereographic Projections

Something like this?

enter image description here

size(250,250);
import solids;

projection Prj=orthographic(25,25,6,center=false);
currentprojection=Prj;

void Draw3(guide3 g,pen p=currentpen){
  draw(g,p);
}

void FillDraw3(guide3 g,pen fillPen=currentpen, pen linePen=nullpen){
  filldraw(project(g),fillPen,linePen);
}

real R=1;

triple O=(0,0,0);

guide3 circ(triple sh=O, real r){
  guide3 uc=(r,0,0)..(0,r,0)..(-r,0,0)..(0,-r,0)..cycle;
  return shift(sh)*uc;
}

guide3 semicirc(triple sh=O, real r){
  guide3 uc=(r,0,0)..(0,r,0)..(-r,0,0);
  return shift(sh)*uc;
}

guide3 Parallel(real h, real R){
  return circ((0,0,h),sqrt(R^2-h^2));
}

pen dashed=linetype(new real[] {4,3}); // set up dashed pattern

pen dashCircPen=gray(0.4)+dashed+opacity(1)+0.32pt;
pen boldLine=darkblue+1.2pt;
pen floorCircPen=olive+opacity(0.75);
pen sphereSpotPen=blue+opacity(0.382);


Draw3(O--(0,0,R),boldLine);
Draw3(O--(R,0,0),boldLine);
Draw3(O--(0,R,0),boldLine);
Draw3(O--(-R,0,0),boldLine);
Draw3(O--(0,-R,0),boldLine);
Draw3(circ(R),boldLine);

real h=0;
real dphi=10;
real phi=dphi;
for(int i=1;i<9;++i){
  h=R*Sin(phi);
  Draw3(Parallel(h,R),dashCircPen);
  phi+=dphi;
}

for(int i=0;i<6;++i){
  Draw3(rotate(30*i,(0,0,1))*rotate(90,(1,0,0))*semicirc(R),dashCircPen);
}

triple P=(0,0,R);
triple A=(2R,0,0);
triple B=(0,1.5R,0);
triple sA,sB;

surface hs=scale(R,R,R)*unithemisphere;

guide3 lin;

lin=subpath(A--P,0.0,0.999);
triple[] p;
p=intersectionpoints(lin,hs);
Draw3(A--p[0],red);
Draw3(p[0]--P,blue+dashed);
sA=p[0];
dot(A);

lin=subpath(B--P,0.0,0.999);
p=intersectionpoints(lin,hs);

Draw3(B--p[0],red);
Draw3(p[0]--P,blue+dashed);
sB=p[0];

real r=0.2;
guide3 g=circ((A.x,A.y+r,0), r);

draw(g);

guide3 sphCirc(guide3 g, int n=100){
  guide3 sphg;
  triple[] p;
  triple v;
  guide3 lin;
  for(int i=0;i<n;++i){
    v=point(g,length(g)*i/(n-1));
    lin=subpath(v--P,0.0,0.999);
    p=intersectionpoints(lin,hs);
    sphg=sphg--p[0];
  }
  sphg=sphg--cycle;
  return (sphg);
}

draw(sphCirc(g),red+dashed);

g=B--(B.x+0.7R,B.y+0.3R,0)--(B.x-0.5R,B.y-0.5R,0)--cycle;

draw(g,orange);
draw(sphCirc(g),red);

dot(new triple[]{(0,0,0),A,sA,B,sB,P});

label("$P$",P,N);
label("$A$",A,NW);
label("$B$",B,SE);
label("$A'$",sA,NW);
label("$B'$",sB,NE);

Compile this s.asy file with asy -f pdf -noprc -render=0 s.asy to get a standalone s.pdf.

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