Torus

torus
The torus is one of few surfaces which can be parametrized with one patch. It is the easiest example surface for the study of geodesics (see anaglyphs below).
torus rolled stretched 001
The torus can be thought of as a rectangular piece of rubber, rolled and stretched around.
torus 5 2 geodesic
The geodesic surrounds the hole twice and swings up and down 5 times.
torus 4 1 geodesic
The geodesic surrounds the hole ones and circles the torus 4 times.
torus 7 2 knotted geodesic
The geodesic is a 7-2-knot on the torus.
torus 11 2 knotted geodesic
The geodesic is a 11-2-knot on the torus.
torus near asymptotic oneside
The geodesic is not quite asymptotic to the inner equator. it approaches the equator from above and leaves in the same direction.
torus near asymptotic round
The geodesic is not quite asymptotic to the inner equator. it approaches the equator, crosses it and leaves to the other side.
torus st
torus st
torus sw
torus sw
      Torus Surface Parametric Equations

  halftwists := Round(ff);
		x = aa * (cos(v) + u * cos(halftwists * v / 2) * cos(v));
		y := aa * (sin(v) + u * cos(halftwists * v / 2) * sin(v));
		z := aa * u * sin(v / 2);

MonkeyTorusCyclide.pdf