Discrete gliding along principal curves. (English) Zbl 1222.53012

Schröcker, Hans-Peter

J. Geom. Graph. 14, No. 2, 171-180 (2010).

The author considers \(n\)-dimensional discrete motions such that any two neighbouring positions correspond in a pure rotation (“rotating motions”). In the Study quadric model of Euclidean displacements, these motions correspond to quadrilateral nets with edges contained in the Study quadric (“rotation nets”). The basic direction of this research is towards a connection between rotation nets and discrete principal contact element nets. It is shown that every principal contact element net occurs in infinitely many ways as a trajectory of a discrete rotating motion (a discrete gliding motion on the underlying surface). Moreover, discrete rotating motions with two non-parallel principal contact element net trajectories are constructed. Rotation nets with this property can be consistently extended to higher dimensions.
This work consists of the following basic parts: curvature line discretizations; kinematics and dual quaternions; rotation quadrilaterals; discrete rotating motions; principal contact element nets and rotating motions; pairs of principal contact element nets.

Reviewer: Vladislav Nikolaevich Dumachev (Voronezh)

MSC:

53A17	Differential geometric aspects in kinematics
53A05	Surfaces in Euclidean and related spaces

Keywords:

discrete differential geometry; kinematics; rotating motion; curvature line discretization; principal contact element net; gliding motion; Study quadric