A noniterative method for robustly computing the intersections between a line and a curve or surface

Abstract

SummaryThe need to compute the intersections between a line and a high‐order curve or surface arises in a large number of finite element applications. Such intersection problems are easy to formulate but hard to solve robustly. We introduce a noniterative method for computing intersections by solving a matrix singular value decomposition and an eigenvalue problem. That is, all intersection points and their parametric coordinates are determined in one‐shot using only standard linear algebra techniques available in most software libraries. As a result, the introduced technique is far more robust than the widely used Newton‐Raphson iteration or its variants. The maximum size of the considered matrices depends on the polynomial degree q of the shape functions and is 2q × 3q for curves and 6q2 × 8q2 for surfaces. The method has its origin in algebraic geometry and has here been considerably simplified with a view to widely used high‐order finite elements. In addition, the method is derived from a purely linear algebra perspective without resorting to algebraic geometry terminology. A complete implementation is available from http://bitbucket.org/nitro-project/.