# Generalized Solids of Revolution

Peter Čech

peter@cech.cjb.net

Faculty of Mathematics, Physics and Informatics

Comenius University

Bratislava / Slovakia

## Abstract

Modeling technique based on sweeping is introduced. Generalizations of solids of revolution by axis bending, usage of multiple axes and usage of various metrics are presented. Generalized solids of revolution are 2.5D objects and depending on input parameters, they may contain gaps. Constrained variants offer additional modeling possibilities and easier input. Finally some examples are given.

**Keywords:** geometric modeling, sweeping, solids of revolution

## Table of Contents

1. Introduction

2. Generalization

2.1 Axis Bending

2.2 Multiple Axes

2.3 Various Metrics

2.4 Segments

3. Generalized Conics and Thier Properties

3.1 Reducing Number of Foci

3.2 Degenrated Generalized Conic

4. Properties of Generalized Solids of Revolution
4.1 Generalized Solids of Revolution as 2.5D Object

4.2 Gaps
4.2.1 Discontinuity of leading curve

4.2.2 Tangent of leading curve does not exist

4.2.3 No foci in slice

4.2.4 Infinite number of intersection points in slice

4.2.5 No intersection of rotated curve and slice

4.2.6 Value of distance function is too small

4.2.7 Change in number of foci

4.2.8 Discontinuous change in weights of foci

5. Special Variants of Generalized Solids of Revolution
5.1 Simplified Variant of Generalized Solids of Revolution

5.2 Lofted Variant of Generalized Solids of Revolution

5.3 Slice choosing Curve

6. Modeling Examples
6.1 Modeling a Shell

6.2 Square to Circle Transition Using Two Metrics

7. Conclusion

8. References