Geometric Methods in Engineering Applications

Summary

In this work, we introduce two sets of algorithms inspired by the ideas from modern geometry. One is computational conformal geometry method, including harmonic maps, holomorphic 1-forms and Ricci flow. The other one is optimization method using affine normals.

In the first part, we focus on conformal geometry. Conformal structure is a natural structure of metric surfaces. The concepts and methods from conformal geometry play important roles for real applications in scientific computing, computer graphics, computer vision and medical imaging fields.

This work systematically introduces the concepts, methods for numerically computing conformal structures inspired by conformal geometry. The algorithms are theoretically rigorous and practically efficient.

We demonstrate the algorithms by real applications, such as surface matching, global conformal parameterization, conformal brain mapping etc.

In the second part, we consider minimization of a real-valued function ƒ over ℝⁿ⁺¹ and study the choice of the affine normal of the level set hypersurfaces of ƒ as a direction for minimization. The affine normal vector arises in affine differential geometry when answering the question of what hypersurfaces are invariant under unimodular affine transformations. It can be computed at points of a hypersurface from local geometry or, in an alternative description, centers of gravity of slices. In the case where ƒ is quadratic, the line passing through any chosen point parallel to its affine normal will pass through the critical point of ƒ. We study numerical techniques for calculating affine normal directions of level set surfaces of convex ƒ for minimization algorithms.