The Shadows of a Cycle Cannot All Be Paths

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).Comment: 6 pages, 10 figure

Application of LANDSAT data to delimitation of avalanche hazards in Montane, Colorado

The author has identified the following significant results. Photointerpretation of individual avalanche paths on single band black and white LANDSAT images is greatly hindered by terrain shadows and the low spatial resolution of the LANDSAT system. Maps produced in this way are biased towards the larger avalanche paths that are under the most favorable illumination conditions during imaging; other large avalanche paths, under less favorable illumination, are often not detectable and the smaller paths, even those defined by sharp trimlines, are only rarely identifiable