Surface-surface intersection with validated error bounds

Abstract

This thesis presents a robust method for tracing intersection curve segments between continuous rational parametric surfaces, typically rational polynomial parametric surface patches. Using a validated ordinary differential equation (ODE) system solver based on interval arithmetic, we obtain a continuous, validated upper bound for the intersection curve segment in the parametric space of each surface. Application of the validated ODE solver in the context of eliminating the pathological phenomena of straying and looping is discussed. We develop a method to achieve a continuous gap-free boundary with a definite numerically verified upper bound for the intersection curve error in parameter space. This bound in parametric space is further mapped to an upper bound for the intersection curve error in 3D model space, denoted as model space error, which assists in defining robust boundary representation models of complex three-dimensional solids. In addition, we also discuss a method for controlling this model space error so that it takes values below a predefined threshold (tolerance). Application of the above method to various examples is further demonstrated.