In this article, we study discrete curvature and torsion for spatial polygonal lines of unit sides. We express geometric conditions on polygons by using inner products of oriented sides. As a application, we prove a generalization of van der Waerden’s theorem. The theorem given in this paper and its proof clarify how the conditions on the sides affect the polygon being planar from the discrete torsion point of view.
"Discrete torsion and its application for a generalized van der Waerden’s theorem." Proc. Japan Acad. Ser. A Math. Sci. 87 (10) 209 - 214, December 2011. https://doi.org/10.3792/pjaa.87.209