,r,~'lSll'dl x omputcr The mathematics of computer graphics Rae A. Earnshaw University of Leeds, LEEDS LS2 9JT, United Kingdom Until relatively recently, researchers in computer graphics paid scant attention to the numerics of their computations. Com- putation was used as a simple tool to eval- uate algorithms or transform data into some appropriate pictoral representation. Thus standard computer graphics texts have little to say about numerical meth- ods, just as earlier numerical analysis text- books had little to say about computer graphics. This is now changing, for the im- portant reasons outlined in this paper. Key words: Numerical methods - Euclide- an geometry - Picture generation - Com- puter graphics - Image processing - Inter- polation - Fractal mathematics The Visual Computer (1987) 3:115-124 9 Springer-Verlag 1987 1 Computer graphics and numerical methods 1.1 Introduction To date there has been relatively little interaction between computer graphics and numerical meth- ods, except in standard 'mathematical' areas. A number of important considerations are currently reversing this trend. Geometric computations must be performed more accurately (in some sense) since the power of rendering and presentation now en- ables model construction to be more clearly seen. Computer-aided animation and image synthesis can consume large amounts of compute power - such computations need to be more carefully con- structed. Computer graphics and image processing techniques are drawing closer together as a result of the developments in parallel processing architec- tures, silicon-compilation and execution, trans- puters, and other engines. Thus traditional aspects of image processing such as sampling, aliasing, Fourier transforms, convolution and basic systems theory are developing their correlates in the com- puter graphics field. And finally, further areas of overlap and intersection are in the areas of differ- ential and algebraic geometry, fractal mathematics, curve definition, dynamics, and shape deforma- tion, for reasons to do with the exploitation of these techniques (often originating in more 'classi- cal' fields) in specialised areas of computer graph- ics. This synergism will produce an added rigour to the definition and execution of graphics processes - whether in hardware or software - and enable the coupling of numerics and pictures to take place to their mutual benefit. This in turn will enable the next generation of graphics processors to be designed on a rigorous and consistent basis. When this is coupled with the application of formal meth- ods to algorithm specification and execution, pipe- line transformations, graphics language and inter- face design, and parallelism, our understanding of the processes of picture production will be greatly enhanced. This paper reviews the aspects of mathematics and numerical analysis of relevance to computer graph- ics and the anticipated developments in the future. 1.2 Survey and background Historically there has always been an overlap be- tween computer graphics and numerical methods 115