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This article addresses the problems of solving quartic and cubic equations in computer graphics. These equations are interesting as they can be solved by analytic algorithms, and in principle need no iterative techniques.

Quartic equations need to be solved when ray tracing 4th degree surfaces e.g., a torus. Quartics also need to be solved in a number of problems involving quadric surfaces. Quadric surfaces (i.e. ellipsoids, paraboloids, hyperboloids, cones) are useful in computer graphics for generating objects with curved surfaces (Badler, 1979). Fewer primitives are required than with planar surfaces to approximate a curved surface to a given accuracy (Herbison-Evans, 1982).

Bicubic surfaces may also be used for the composition of curved objects. They have the advantage of being able to incorporate recurves: lines of inflection. There is a problem, however, when drawing the outlines of bicubics in the calculation of hidden arcs. The visibility of an outline can change where its projection intersects that of another outline. The intersection can be found as the simultaneous solution of the two projected outlines. For bicubic surfaces, these outlines are cubics, and the simultaneous solution of two of these is a sextic which can only be solved by iterative techniques. For quadric surfaces, the projected outlines are quadratic. The simultaneous solution of two of these leads to a quartic equation.

The need to solve cubic equations in computer graphics arises in the solution of the quartic equations mentioned above. Also, a number of problems which involve the use of cubic splines require the solution of cubic equations.

One simplifying feature of the computer graphics problem is that often, only the real roots (if there are any) are required. The full solution of the quartic in the complex domain (Nonweiler, 1967) is then an unnecessary use of computing resources.

Another simplification in the graphics problem is that displays have only a limited resolution, so that only a limited number of accurate digits in the solution of a cubic or quartic may be required. A resolution of 1 in 1,000,000 should in principle be achievable using single precision floating point (32 bit) arithmetic, which would be more than adequate for most current displays.

 

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