3. Ambiguous Rings Based on a Rose Curve

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This Demonstration further 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 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 rose curve that cross at a right angle.

In this Demonstration, we consider rose curves with 2, 3, 4, 5 or 6 petals.

You can vary the specific settings for the radius and axial offset of and the number of petals and axial rotation of .

For each case, closed curves are possible when the cross section of fits exactly inside ; click "A" or "B" for the two solutions.

A single ring with the same view properties (circle or rose) can be generated using the "single ring cutoff angles" sliders and to control the range of the angular parameter in the parametric equation of the full ring set.

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


Snapshots


Details

The parametric equation for with radius inclined at an angle from the vertical is given by:

.

The parametric equation of in the - plane with petals and an angular offset of from the axis is given by:

.

To find the intersection, set the corresponding components equal. This gives three equations in four unknowns: .

Eliminating , and by solving the equations gives the parametric curve of the intersection with as the only parameter:

,

with

.

This composite curve (ring set) can be split into two rings. Therefore, the parameter range for , from to , is divided into sections using the cutoff angles and .

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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