Dr Neil Dodgson, University of Cambridge Computer Laboratory
Part II course, 1998

The ray tracing primitives, described in Lecture 1C, 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.

These are an extension of quadrics, where the power on the coordinate does not have to be 2. The general form of a superquadric centred at the origin and aligned along the coordinate axes is:

Super-ellipsoids tend to be the only members of this family that are actually used, and even they are only used in very limited areas. The effect of n on a super-ellipsoid is roughly as follows: n=2 is a standard ellipsoid; n<2 is a more pointy version, the "points" being along the main axes; n>2 becomes closer to a box as n increases; n=1 is allegedly an octahedral shape; and n<1 is truly pointy along the main axes.

The interested student may like to have a quick look at Alan Barr's two papers on superquadrics. The papers can be found in the Computer Laboratory library in IEEE Transactions on Computer Graphics and Applications volume 1, number 1 (January 1981), pages 11-23, and volume 1, number 3 (July 1981), pages 41-47.

Brian Wyvill describes a use of super-ellipsoids on pages 264 and 265 of "A Computer Animation Tutorial" in Computer Graphics Techniques: Theory and Practice, Rogers and Earnshaw (editors), Springer-Verlag, 1990, ISBN 0-387-97237-4.

Neil Dodgson | Advanced Graphics | Computer Laboratory

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Page last updated on Mon Sep 7 12:47:49 BST 1998
by Neil Dodgson (nad@cl.cam.ac.uk)