Real algebraic curves of constant width
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Real algebraic curves of constant width Chatchawan Panraksa1 · Lawrence C. Washington2
Published online: 14 June 2016 © Akadémiai Kiadó, Budapest, Hungary 2016
Abstract Rabinowitz constructed a parametric curve of constant width and expressed it as a plane algebraic curve; however, the algebraic curve also contains isolated points separate from the original curve. We show how to modify his example in order to produce a curve with no isolated points. We then conjecture a method for producing a family of such curves and prove the conjecture in several cases. Keywords Constant width · Algebraic curves · Isolated points · Convex curves Mathematics Subject Classification 52A10 (primary) · 53A04 · 14H50
1 Introduction In [8], Rabinowitz constructed the plane algebraic curve defined by F(x, y) = (x 2 + y 2 )4 − 45(x 2 + y 2 )3 − 41283(x 2 + y 2 )2 + 7950960(x 2 + y 2 ) + 16(x 2 − 3y 2 )3 + 48(x 2 + y 2 )(x 2 − 3y 2 )2 + (x 2 − 3y 2 )(x) × 16(x 2 + y 2 )2 − 5544(x 2 + y 2 ) + 266382 − 7203 = 0.
(1.1)
The algebraic set {(x, y) ∈ R2 | F(x, y) = 0} includes a curve of constant width 18 (that is, the orthogonal projection onto each line in the plane is an interval of length 18). However, if we set y = 0, we obtain
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Lawrence C. Washington [email protected] Chatchawan Panraksa [email protected]
1
Applied Mathematics Program, Science Division, Mahidol University International College, 999 Phutthamonthon 4 Road, Salaya, Nakhonpathom 73170, Thailand
2
Department of Mathematics, University of Maryland, College Park, MD 20742, USA
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C. Panraksa, L. C. Washington
x 8 + 16x 7 + 22x 6 − 5544x 5 − 41283x 4 + 266382x 3 + 7950960x 2 − 7203 = 0. This polynomial factors as (x − 10)(x + 8)((x + 3)3 − 2187).
√ Therefore, the graph of (1.1) has an extra, isolated point at (x, y) = (−3 + 9 3 3, 0), so this point must be discarded from the algebraic set in order to obtain the desired curve. The question then arises, are there plane algebraic sets that are curves of constant width, or must they always have extra, isolated points that need to be discarded in order to obtain the desired curves? Of course, there is the circle x 2 + y 2 − 1 = 0, so the question really is whether there are curves of degree 3 or higher. In the following, we first show that the construction used by Rabinowitz usually produces algebraic sets with isolated points. But then we show how a special choice of parameters yields a real algebraic set of constant width with no isolated points. We then conjecture a method for producing a family of such curves and prove the conjecture in several cases. There is an extensive theory of sets of constant width, both in the plane and in higher dimensions. See, for example, the surveys [2,5].
2 Curves of constant width As in [8], curves of constant width can be obtained by choosing a support function p(θ ) satisfying p (θ + π) = − p (θ ) and setting x = p(θ ) cos θ − p (θ ) sin θ y = p(θ ) sin θ + p (θ ) cos θ.
(2.1)
Then p(θ + π) + p(θ ) gives the width of the curve. Rabinowitz used p(θ ) = 9
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