Polygons in Three Dimensions

Abstract This paper studies polygons and polygonal arcs in three dimensions, as well as their orthographic projections. We define a class of "nondegenerate" orthographic projections which we call Wirtinger projections, and show that most of the orthographic projections of a polygon(al arc) are Wirtinger. We also show that a polygon(al arc) can be reconstructed from three Wirtinger projections in noncoplanar directions. Finally, we consider polygon(al arcs) that are isothetic (e.g., all their sides are parallel to the coordinate axes); we show how to represent such a polygon(al arc), up to translation and rotation, by a sequence of side lengths and two-bit numbers representing 90° rotations, and how to derive its properties from this representation. Wirtinger-like projections are used in knot theory to obtain group presentations of knots; for completeness, we review this topic in the Appendix.