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Axes for symmetric convex curves

Published online by Cambridge University Press:  08 February 2018

David L. Farnsworth
David L. Farnsworth*
Affiliation:
School of Mathematical Sciences, 84 Lomb Memorial Drive, Rochester Institute of Technology, Rochester, NY 14623USA e-mail: DLFSMA@rit.edu
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Curves are given in polar coordinates (r, θ)by equations of the form r = f (θ), where for f (θ) > 0 all θ. Consider curves which are symmetric about the origin O, so that, f(θ + π) = f (θ) for all θ. For such a curve, its interior is the set {(r, θ) : 0 ≤ r ≤ f (θ)}. Further, assume that the curve is convex. Recall that a closed curve is convex if a line segment between any two of its points has no points exterior to the curve [1], [2, pp. 198-203]. We call these curves M-curves, because the curves are fundamental objects in Minkowski geometry, where they are called Minkowski circles or simply circles [3, 4]. That application is briefly discussed in the Section 4 but is not required for our purposes.

Examples of M-curves are displayed in Figures 1 to 6. In order to express these curves as functions in rectangular coordinates, we need axes.

Type
Articles
Information
The Mathematical Gazette , Volume 102 , Issue 553 , March 2018 , pp. 23 - 30
Copyright
Copyright © Mathematical Association 2018 

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References

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