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#### 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 [TORUS_MODIFIER...] `}`
TORUS_MODIFIER:
`
  sturm
` | OBJECT_MODIFIER

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.

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.

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