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 deformation-retract onto a point).
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2015
- DOI:
- 10.48550/arXiv.1507.02355
- arXiv:
- arXiv:1507.02355
- Bibcode:
- 2015arXiv150702355B
- Keywords:
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- Computer Science - Computational Geometry;
- Computer Science - Computer Vision and Pattern Recognition;
- Mathematics - Metric Geometry
- E-Print:
- 6 pages, 10 figures