1. Ambiguous Rings

This Demonstration explores ambiguous rings.
An ambiguous ring is a three-dimensional composite space curve that can be viewed as a circle, a polygon, or 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 ring and a polygonal cylinder (a prism) with the same radius or circumradius and intersecting at a right angle.
This Demonstration considers intersections of a circular cylinder with triangular, square, pentagonal and hexagonal cylinders. For each case, closed curves are possible when there is an exact fit of the polygonal cross section inside the circular cylinder. These specific settings for the circumradius, axial rotation and axial offset of the polygonal cylinder can be derived by using the Wolfram Demonstration Intersection of Circular and Polygonal Cylinders.

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DETAILS

The parametric equation of a circular cylinder with radius inclined at an angle from the vertical is given by:
,
with parameters and .
Define the functions:
and
 .
The and functions define the composite curve of the -gonal base of the polygonal cylinder [1].
The parametric equation of a polygonal cylinder with sides and radius rotated by an angle around its axis is given by:
,
with parameters and .
To find the equation of the intersection curve, set . This gives the three equations:
,
,
,
with
.
These are equations with four variables, , , and . Eliminating , and by solving the equations gives the parametric equation of the intersection curve with as the only parameter:
.
Reference
[1] E. Chicurel-Uziel, "Single Equation without Inequalities to Represent a Composite Curve," Computer Aided Geometric Design, 21(1), 2004 pp. 23–42. doi:10.1016/j.cagd.2003.07.011.
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