Mitering a Closed 3D Path
 When constructing a path in space by mitering together beams of the same cross section, it is not always the case that the edges nicely match up all the way around. Snapshot 1 shows a situation where the edges do match up. In Snapshot 2, the path is slightly altered by reducing its height and depth, resulting in a mismatch. Rotating the cross section is to no avail, as shown in Snapshot 3. The twist is more visible when highlighting a seam of the beam. It is interesting to explore the difference between using miter joints and fold joints, in particular for an odd number of vertices. Use of a 2-gon as cross section (i.e., a strip) avoids self-intersection for fold joints. In the case of miter joints, all beams rotate in the same direction, but with fold joints adjacent, beams rotate in the opposite direction. This is illustrated in Snapshots 4 to 8. Also see T. Verhoeff and K. Verhoeff, "The Mathematics of Mitering and Its Artful Application," in Bridges Leeuwarden: Mathematical Connections in Art, Music, and Science, Bridges 2008 Conference Proceedings (R. Sarhangi and C. Séquin, eds.), Hertfordshire, UK: Tarquin Publications, 2008. This paper presents the Miter Joint Rotation Invariance theorem, Even Fold Joint Rotation Invariance theorem, and the Odd Fold Joint Matching theorem for closed 3D paths.  "Mitering a Closed 3D Path" from The Wolfram Demonstrations Project http://demonstrations.wolfram.com/MiteringAClosed3DPath/ Contributed by: Tom Verhoeff (Eindhoven University of Technology) Inspired by the artwork of: Koos Verhoeff | ||||||||||||||
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