Reference:Torus

A torus is a 4th order quartic polynomial shape that looks like a donut or inner tube. Because this shape is so useful and quartics are difficult to define, POV-Ray lets you take a short-cut and define a torus by:

TORUS:
  torus {
    Major, Minor [SPINDLE_MODE]
    [TORUS_MODIFIER...]
    }
TORUS_MODIFIER:
  sturm | OBJECT_MODIFIER
SPINDLE_MODE:
  difference | intersection | merge | union

Torus default values:

union
sturm : off

where Major is a float value giving the major radius and Minor is a float specifying the minor radius. The major radius extends from the center of the hole to the mid-line of the rim while the minor radius is the radius of the cross-section of the rim. The torus is centered at the origin and lies in the x-z-plane with the y-axis sticking through the hole.

Major and minor radius of a torus.

The torus is internally bounded by two cylinders and two rings forming a thick cylinder. With this bounding cylinder the performance of the torus intersection test is vastly increased. The test for a valid torus intersection, i.e. solving a 4th order polynomial, is only performed if the bounding cylinder is hit. Thus a lot of slow root solving calculations are avoided.

Spindle Torus

Template:New.3.7.1 A torus with a minor radius greater than the major radius (aka spindle torus) will self-intersect in a spindle-shaped region. The behaviour with respect to the spindle can be controlled by specifying either of the difference, intersection, merge or union keywords, which act similar to the corresponding CSG operations:

• Using the difference keyword, the self-intersecting portion is cut away from the torus, so that the spindle volume is considered outside the primitive; the spindle surface is visible (provided you cut open the torus, or make it semi-transparent).
• Using the intersection keyword, the resulting shape consists of only the self-intersecting portion, so that only the spindle volume is considered inside the primitive, and only the spindle surface is visible.
• Using the merge keyword, the surface within the self-intersecting portion is hidden, so that the spindle surface is not visible; the spindle volume is considered inside the primitive.
• Using the union keyword, the entire torus surface remains visible and the spindle volume is considered inside the primitive (this is the default).

In a non-spindle torus, the intersection keyword will cause a "possible parse error" warning and make the torus invisible, while the other spindle mode keywords will have no effect whatsoever.

Note: The difference spindle mode does not affect the behaviour with respect to the interior_texture keyword. An interior_texture will always be applied to the side of the spindle surface facing the spindle volume.

cutaway view of spindle torus using the difference, intersection, merge and union mode, respectively

Sturm

Calculations for all higher order polynomials must be very accurate. If the torus renders improperly you may add the keyword sturm to use POV-Ray's slower-yet-more-accurate Sturmian root solver.