## Resumen de Improving the interval ray tracing of implicit surfaces based on interval arithmetic

Jorge Eliécer Flórez Díaz

• Implicit surfaces are useful in many areas related to computer graphics, One of their main advantages over other representations is that they can be easily used as primitives for modeling (using Constructive Solid Geometry or blending). However, they are not widely used for this purpose because the models created with implicit surfaces take a long time to be rendered.

When a precise visualization of an implicit surface is required, the best option is to use ray tracing. If simple surfaces are rendered, then the results are suitable, however, thin features can be missed in models that have thin parts, and are not rendered. These problems are caused by the truncation performed in the floating-point representation in the computers: some bits are lost in the mathematical operations during the intersection tests between the rays and the surfaces.

Many authors have used Interval Arithmetic in the intersection test to solve the problems related with point sampling, that cause the loss of intersection points, however, there are still two open problems in the ray tracing of implicit surfaces based on Interval Arithmetic: - Ray tracing is slow, and Interval Arithmetic is slow too. Many floating-point operations are required for every Interval Arithmetic operation (besides rounding). For that reason, the ray tracing becomes even slower.

- Although Interval Arithmetic has been applied to replace point sampling during the intersection tests, the application of Interval Arithmetic to replace point sampling in the distribution of the rays inside the pixel, has not been studied. This also causes loss of thin parts of the surface in the final image.

In this work algorithms to deal with those problems are presented. The research is based on Modal Interval Analysis, which is a logical completion of classic interval analysis that includes tools for solving quantified uncertainty. Modal interval Analysis gives the mathematical foundations used in the development of these algorithms.

The efficiency of the ray tracing of implicit surfaces is improved by means of the evaluation of groups of rays instead of individual ones, which permits thus saving computational time in the entire ray tracing process. The quality of the visualization is improved by means of the creation of an adaptive anti aliasing algorithm. Using this strategy, it is possible to evaluate areas of the pixels instead of points. The efficiency of animated scenes composed by implicit surfaces is also improved. The improvement is achieved by means of an algorithm that exploits the coherence between frames.

This research also shows how it is possible to reduce the rendering time by means of the use of cluster of computers and also by means of graphic processing cards, where the improvement in efficiency is between two and three orders of magnitude.