The Shadows of a Cycle Cannot All Be Paths
Abstract
A "shadow" of a subset $S$ of Euclidean space is an orthogonal projection of $S$ into one of the coordinate hyperplanes. In this paper we show that it is not possible for all three shadows of a cycle (i.e., a simple closed curve) in $\mathbb R^3$ to be paths (i.e., simple open curves). We also show two contrasting results: the three shadows of a path in $\mathbb R^3$ can all be cycles (although not all convex) and, for every $d\geq 1$, there exists a $d$sphere embedded in $\mathbb R^{d+2}$ whose $d+2$ shadows have no holes (i.e., they deformationretract onto a point).
 Publication:

arXiv eprints
 Pub Date:
 July 2015
 arXiv:
 arXiv:1507.02355
 Bibcode:
 2015arXiv150702355B
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Computer Vision and Pattern Recognition;
 Mathematics  Metric Geometry
 EPrint:
 6 pages, 10 figures