Dr Neil Dodgson, University of Cambridge Computer Laboratory

The ray tracing primitives have relatively simple mathematical definitions. This is what makes them attractive: the simple mathematical definition allows for simple ray-object intersection code. Following from this, it would seem logical to investigate other shapes with simple mathematical definitions. Spheres, cones and cylinders are part of a more general family of parametric surfaces called quadrics (N.B. tori are not quadrics). Quadrics are the 3D analogue of 2D conics. We describe these general families below, but it turns out that they are of little practical use. It would seem that the general quadrics are a "dead end" in graphics research.

## Conics

A conic is a two dimensional curve desribed by the general equation:

This general form can be rotated, scaled, and translated so that it is aligned along the axes of the coordinate system. It will then have the simpler equation:

The useful conics are the ellipse (of which the circle is a special case), the hyperbola, and the parabola. For more details see R&A section 4-10, especially Table 4-8 on page 242.

The quadrics are the three dimensional analogue of the conics. The general equation is:

This general form can be rotated, scaled, and translated so that it is aligned along the axes of the coordinate system. It will then have the simpler equation:

The useful conics are the ellipsoid (of which the sphere is a special case), the infinite cylinder, and the infinite cone. Various hyperboloids, and paraboloids are also defined by these equations, but these have little real use unless one is designing satellite dishes (paraboloid), headlamp reflectors (also paraboloid), or power station cooling towers (hyperboloid). For more details see R&A section 6-4, especially Figure 6-18 on page 403.