Lie Group Approach to Envelope Surfaces

TL;DR

This paper introduces a Lie group-based method for efficiently computing envelope surfaces by interpreting surface systems as curves in homogeneous Lie group spaces, enabling precomputation and rational parameterization.

Contribution

The paper presents a novel Lie group and Lie algebra framework for envelope surface computation, allowing precomputation of characteristic curves and explicit rational parameterizations.

Findings

Characteristic curves can be precomputed as intersections in Lie group spaces.

Envelopes of cones under rational motions are rational surfaces.

Explicit rational parameterizations facilitate solving the trimming problem.

Abstract

In this paper, we develop a new and efficient approach to the computation of envelope surfaces. We interpret one-parameter systems of surfaces as curves in the homogeneous spaces of suitable Lie groups. Using the formalism of Lie groups and Lie algebras, we rigorously capture the inherent symmetry and linearity in the computation of envelopes. In particular, the possible set of characteristic curves (which constitute the envelope surface) can be precomputed as the intersection of a fixed canonical surface and a low-dimensional set of its possible "derivatives." To demonstrate the effectiveness of our approach, we present several examples of surfaces undergoing transformations from various Lie groups. As a remarkable side result, we show that the characteristic curves and the envelopes of cones undergoing rational motions are themselves rational. Furthermore, we provide an explicit rational parameterization of these envelopes and use it to solve the trimming problem.