Corpus ID: 2552029

The Shadows of a Cycle Cannot All Be Paths

@article{Viglietta2015TheSO,
  title={The Shadows of a Cycle Cannot All Be Paths},
  author={Giovanni Viglietta and Prosenjit Bose and Jean-Lou De Carufel and Michael Gene Dobbins and Heuna Kim},
  journal={ArXiv},
  year={2015},
  volume={abs/1507.02355}
}
  • Giovanni Viglietta, Prosenjit Bose, +2 authors Heuna Kim
  • Published in CCCG 2015
  • Computer Science, Mathematics
  • ArXiv
  • 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… CONTINUE READING

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