Introduction to Geometric Processing through Optimization

As an introduction to the field, this article shows how to formulate several geometry-processing operations to solve systems of equations in the“least-squares” sense. The equations are derived from local geometric relations using elementary concepts from analytic geometry, such as points, lines, planes, vectors, and polygons. Simple and useful tools for interactive polygon mesh editing result from the most basic descent strategies to solve these optimization problems. Throughout the article, the author develops the mathematical formulations incrementally, keeping in mind that the objective is to implement simple software for interactive editing applications that works well in practice. Readers can implement higher-performance versions of these algorithms by replacing the simple solvers proposed here with more advanced ones.