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 R3 to be paths (i.e., simple open curves). We also show two contrasting results: The three shadows of a path in R3 can all be cycles (although not all convex) and, for every d ≥ 1, there exists a d-sphere embedded in Rd+2 whose d + 2 shadows have no holes (i.e., they deformation-retract onto a point).

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Conference 27th Canadian Conference on Computational Geometry, CCCG 2015
Citation
Bose, P, De Carufel, J.-L. (Jean-Lou), Dobbins, M.G. (Michael G.), Kim, H. (Heuna), & Viglietta, G. (Giovanni). (2015). The shadows of a cycle cannot all be paths. In Proceedings of the 27th Canadian Conference on Computational Geometry, CCCG 2015 (pp. 70–75).