Flat Curves

Langer, Joel; Singer, David A.

doi:10.1007/s00009-017-1037-0

Cited by 3 publications

(2 citation statements)

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“…We note that arc length parameterization is almost never defined by elliptic (or elementary) functions-precisely because the analytic continuation of the arc length parameterization of k R develops branch points at singularities of Q k [13,14]. A rare exception is provided by the circle (where C happens to coincide with A in the case of sin t).…”

Section: Introductionmentioning

confidence: 95%

“…The latter results involve ruler and compass constructibility and Fermat primes; some modern analogues involve origami constructibility and Pierpont primes [6][7][8][9][10]. In particular, the Kiepert trefoil admits such a result [11], and the related fact that this exceptional sextic curve has unit speed parameterization by Dixon elliptic functions [12] is explained in [13] using the clinant quadratic differential Q k .…”

Section: Introductionmentioning

confidence: 99%

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Orthogonal Families of Bicircular Quartics, Quadratic Differentials, and Edwards Normal Form

Langer,

Singer

2023

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Orthogonal families of bicircular quartics are naturally viewed as pairs of singular foliations of C^ by vertical and horizontal trajectories of a non-vanishing quadratic differential. Yet the identification of these trajectories with real quartics in CP2 is subtle. Here, we give an efficient, geometric argument in the course of updating the classical theory of confocal families in the modern language of quadratic differentials and the Edwards normal form for elliptic curves. In particular, we define a parameterized Edwards transformation, providing explicit birational equivalence between each curve in a confocal family and a fixed curve in normal form.

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Section: Introductionmentioning

confidence: 95%