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Lie Group Approach to Envelope Surfaces

Michal Molnár, Zbyněk Šír, Jana Vráblíková

TL;DR

The paper develops a Lie-group–theoretic framework to compute envelope surfaces for one-parameter families of implicitly described surfaces by treating them as curves in homogeneous spaces. It leverages lifts to a Lie group and an implicit-representation space $Q$ to express characteristic curves via the tangent mapping $d\phi_{\mathbf1}$, yielding a main result that envelopes can be obtained from fixed elementary surfaces through low-dimensional derivative data. A key contribution is the explicit rational parameterization of envelopes for cones under rational motions, enabling efficient trimming and CNC-compatible representations (e.g., trimmed NURBS). The approach generalizes to diverse elementary surfaces and transformation groups, offering a systematic, symmetry-driven method for parameterizing envelopes with practical implications for geometric modeling and manufacturing.

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.

Lie Group Approach to Envelope Surfaces

TL;DR

The paper develops a Lie-group–theoretic framework to compute envelope surfaces for one-parameter families of implicitly described surfaces by treating them as curves in homogeneous spaces. It leverages lifts to a Lie group and an implicit-representation space to express characteristic curves via the tangent mapping , yielding a main result that envelopes can be obtained from fixed elementary surfaces through low-dimensional derivative data. A key contribution is the explicit rational parameterization of envelopes for cones under rational motions, enabling efficient trimming and CNC-compatible representations (e.g., trimmed NURBS). The approach generalizes to diverse elementary surfaces and transformation groups, offering a systematic, symmetry-driven method for parameterizing envelopes with practical implications for geometric modeling and manufacturing.

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.

Paper Structure

This paper contains 10 sections, 4 theorems, 50 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Envelope surfaces
  4. Lie groups and their homogeneous spaces
  5. Systems of surfaces as curves in homogeneous spaces
  6. Applications to envelope parameterization
  7. Rational parameterization of the envelope of moving cones
  8. The trimming problem
  9. Examples
  10. Conclusion

Key Result

Proposition 1

Let a one--parameter system of surfaces be given by the implicit equations where $f(x,y,z,t)$ is a smooth function. Then the characteristic curves are given as and the envelope surface as their union $\chi=\bigcup_{t\in I}\chi_t$.

Figures (9)

  • Figure 1: One parameter system of surfaces with their characteristics (black) and the envelope surface, which consists of two sheets (red and blue). The green cone shows the element of $F_t$ for $t = \frac{1}{2}$.
  • Figure 2: The surface $F_{t}$ for $t=\frac{1}{2}$ (green), the corresponding derivative surface (yellow), and the characteristic curve $\chi_t$ (black) as their intersection.
  • Figure 3: The mutual position of the one parameter system of cones and the elementary cone $\bar{F}$ (purple). The cone $F_t$ for $t=\frac{1}{2}$ is highlighted in green (left). Characteristic curves are transformations of curves on the elementary surface (right).
  • Figure 4: Truncated elementary cone (left) and the exact envelope of its rigid motion (right).
  • Figure 5: Surface patch corresponding to the square domain (left) and to the domain trimmed by a B-spline curve (center; right).
  • ...and 4 more figures

Theorems & Definitions (16)

  • Proposition 1
  • Example 2
  • Lemma 3
  • Definition 4
  • Example 5
  • Definition 6
  • Definition 7
  • Example 8
  • Definition 9
  • Definition 10
  • ...and 6 more