2. Ambiguous Rings Based on a Polygon

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This Demonstration explores ambiguous rings.

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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.

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Contributed by: Erik Mahieu (May 2018)
Open content licensed under CC BY-NC-SA


Snapshots


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 the corresponding components equal. 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 θ 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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