Interval methods for computer graphics and geometric modeling

This research focuses on robust and adaptive methods for the solution of problems in computer graphics and geometric modeling. Robustness means that we are interested in using computers to prove properties of curves and surfaces. This usually takes the form of solving equations in several variables. The main tools for achieving robustness are interval computation methods using interval arithmetic and affine arithmetic. Interval methods provide guaranteed numerical results that are not affected by rounding errors in floating-point computations. More importantly, interval methods allow us to analyse the global behaviour of functions over whole regions of the space without sampling it. Adaptiveness means that we want to concentrate the computational effort near interesting regions of the space, such as near a solution curve or in regions where the surface curvature is high. Global analysis with interval methods leads naturally to adaptive methods.

adaptive meshes for implicit surfaces

implicit curves on triangulations

beam casting implicit surfaces

bounds for Julia sets

bounds for strange attractors

distance fields for parametric curves

strip trees for parametric curves

offsets of parametric curves