2. Ambiguous Rings Based on a Polygon
2. Ambiguous Rings Based on a Polygon
This Demonstration explores ambiguous rings.
An ambiguous ring is a three-dimensional space curve or set of space curves that can be viewed as either a circle, a polygon, a shape like a lemniscate or the letter S, depending on the viewpoint.
Such a ring or ring-set can be defined as the intersection curve of a circular cylinder and a generalized cylinder over a regular polygon that cross at a right angle.
This Demonstration considers the intersections of a circular cylinder with a cylinder with triangular, square, pentagonal or hexagonal cylinders. You can vary the specific settings for the radius and axial offset of the circular cylinder and the number of vertices and axial rotation of the polygonal cylinder.
For each case, closed curves are possible when the polygonal cylinder's cross-section fits exactly inside the circular cylinder; click "A" or "B" for the two solutions.
A single ring that looks like a circle or polygon can be generated using the "single ring cutoff angle" slider to control the range of the angular parameter in the parametric equation of "ring 1" or "ring 2". This is shown in the last snapshot.