Horn Torus
Shape · (x, y, z) = f(u, v, t)
((R + a · cos(v)) · cos(u), a · sin(v), (R + a · cos(v)) · sin(u))
Open in the app The dials and keys named below are the app's.
What it draws
The parameter v goes round the tube and u carries the tube round the ring. Read R + a · cos(v) first: it is the distance from the vertical axis, R at the top and bottom of the tube, R + a on the outside and R − a on the inside. The width and the depth are that distance turned through u as cos(u) and sin(u), so the ring lies flat on the ground plane. The second expression is the height, a · sin(v), so the tube reaches ±a.
So R is the radius of the centre circle and a is the radius of the tube swept round it. Both are sliders, and the whole character of the surface is the ratio between them.
Ring, horn and spindle
While a is smaller than R the hole through the middle has radius exactly R − a, and this is the ordinary ring torus. Dragging a up closes that hole steadily: 1 at a = 1, 0.2 at a = 1.8.
At a = R exactly, and the preset opens there with both at 2, the hole is gone. The inner edge of every tube circle has arrived at the axis, so the whole surface meets itself at one single point on the axis. That is the horn torus, the boundary case between the two other kinds.
Push a past R and R + a · cos(v) turns negative over part of the tube. Those points swing across the axis to the far side, and the surface passes through itself: a spindle torus, with a lemon shaped inner shell sitting inside the outer one. A fourth case hides at the other end of the R slider. With R = 0 the distance from the axis is just a · cos(v) and the height a · sin(v), which is an exact sphere of radius a, covered twice as v goes all the way round.
Try
- Drag a down to 1 and R up to 3 for the shape of the preset Torus: a ring torus with a hole of radius 2 and an outer radius of 4.
- Drag a to 3 with R left at 2: the spindle torus, its inner shell turned inside out and buried in the outer one.
- Drag R to 0: a sphere of radius 2, and the mesh lines double up because every point of it is reached twice.
- Set Span of v (×π) to 1: v stops half way round the tube, leaving the upper half of the surface as an open gutter.
- Drag a to 0: the tube has no thickness left and only the centre circle is drawn.