Journal for Geometry and Graphics, Vol. 1, No. 1, pp. 13 - 21 (1997)

Generating Solids by Sweeping Polyhedra

Ahmed Elsonbaty, Hellmuth Stachel

Institute of Geometry, Vienna University of Technology,
Wiedner Hauptstr. 8-10/113, A-1040 Wien, Austria

Abstract: Let a one-parametric motion $\beta$ and the boundary representation of a polyhedron $P$ be given. Our goal is to determine the solid $S$ swept by $P$ under $\beta$: The complete boundary $\partial S$ of $S$ contains a subset of the enveloping surface $\Phi$ of the moving polyhedron's boundary $\partial P$ together with portions of the boundaries of the initial and the final positions of $P$. For each intermediate position of $P$ the curve of contact $c_{\partial P}$ between $\partial P$ and $\Phi$ is called the characteristic curve $c_{\partial P}$ of the surface $\partial P$. However, in general only a subset of $c_{\partial P}$ gives the characteristic curve $c_P$ of the solid $P$ which is defined as the curve of contact between $\partial P$ and $\partial S$.
After a short introduction into instantaneous spatial kinematics, these two characteristic curves $c_{\partial P}$ and $c_P$ are characterized locally. Then some global problems are discussed that arise when the boundary representation of a polyhedral approximation of $S$ is derived automatically. The crucial point here is the determination of self-intersections at the envelope $\Phi$. For the global point of view the motion $\beta$ is restricted to the case of a helical motion with fixed axis and parameter.

Keywords: Kinematics, enveloping surfaces, constructive solid geometry

Classification (MSC2000): 53A17

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