Generating Solids by Sweeping Polyhedra

A. Elsonbaty ; H. Stachel

Journal for geometry and graphics, Tome 1 (1997) no. 1, pp. 013-022.

Résumé

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.