A global theory of flexes of periodic functions

Thorbergsson, Gudlaugur; Umehara, Masaaki

doi:10.1017/s0027763000008734

Cited by 9 publications

(13 citation statements)

References 10 publications

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“…A similar result is proved in [17] and [18] for clean sextactic points on a strictly convex curve in the affine plane. It says that such a curve has three inscribed osculating conics and three circumscribed osculating conics.…”

Section: Figure 4 a Simple Closed Curve With No Clean Inflection Pointssupporting

confidence: 77%

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Inflection points and double tangents on anti-convex curves in the real projective plane

Thorbergsson

Umehara²

2008

Tohoku Math. J. (2)

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A simple closed curve γ in the real projective plane P 2 is called anti-convex if for each point p on the curve, there exists a line which is transversal to the curve and meets the curve only at p. We shall prove the relation i(γ) − 2δ(γ) = 3 for anti-convex curves, where i(γ) is the number of independent (true) inflection points and δ(γ) the number of independent double tangents. This formula is a refinement of the classical Möbius theorem. We shall also show that there are three inflection points on a given anticonvex curve such that the tangent lines at these three inflection points cross the curve only once. Our approach is axiomatic and can be applied in other situations. For example, we prove similar results for curves of constant width as a corollary.

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Section: Figure 4 a Simple Closed Curve With No Clean Inflection Pointssupporting

confidence: 77%

“…Then A m is the kernel of the operator L m . The following proposition is proved in the appendix of [18], p. 135. A function f : R → R is called π-antiperiodic if it satisfies f (t + π) = −f (t).…”

Section: Anti-periodic Functions and Curves Of Constant Widthmentioning

confidence: 92%

Inflection points and double tangents on anti-convex curves in the real projective plane

Thorbergsson

Umehara²

2008

Tohoku Math. J. (2)

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“…See Example (i) that we give after the axioms and the next section for full details of this application. Generalizations to higher order intrinsic systems and applications to Fourier series of periodic functions will be given in [TU2]. We explain a special case of the construction in [TU2] in Example (ii) after the axioms.…”

Section: B Intrinsic Conic Systemsmentioning

confidence: 99%

“…(ii) For a real valued C 4 -function u on S 1 , one defines the osculating polynomial ϕ s (of order five) at a point s ∈ S 1 to be the unique trigonometric polynomial of degree two, ϕ s (t) = a 0 + a 1 cos t + b 1 sin t + a 2 cos 2t + b 2 sin 2t, whose value and first four derivatives at s coincide with those of u at s. If u is C 5 and the value and the first five derivatives of u and ϕ s coincide in s, i.e., if ϕ s hyperosculates u in s, then we call s a flex of u (of order five). The existence of six flexes of order five can easily be proved as a consequence of the well-known fact that a function has at least six zeros if its Fourier coefficients a i and b i vanish for i ≤ 2, see [TU2]. One can use intrinsic conic systems to prove the much stronger result that there are six such flexes satisfying the global property that the osculating polynomials ϕ s in the flexes support u, i.e., either ϕ s ≤ u or u ≤ ϕ s .…”

Section: B Intrinsic Conic Systemsmentioning

confidence: 99%

Sextactic points on a simple closed curve

Thorbergsson¹,

Umehara²

2002

Nagoya Mathematical Journal

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Abstract. We give optimal lower bounds for the number of sextactic points on a simple closed curve in the real projective plane. Sextactic points are after inflection points the simplest projectively invariant singularities on such curves. Our method is axiomatic and can be applied in other situations. §1. IntroductionIn analogy with tangent lines and inflection points of regular curves in the real or complex projective plane, one can consider their osculating conics and sextactic points. Choose five points on a curve γ in a neighborhood of a point p on γ that is not an inflection point. There is a unique regular conic passing through the five points. Letting the five points all converge to p, the conics converge to a uniquely defined regular conic that is called the osculating conic of γ in p. The osculating conic meets γ with multiplicity at least five in p. If it meets with multiplicity at least six in p, then p is called a sextactic point.Inflection and sextactic points on curves in the complex projective plane were well understood already in the nineteenth century. We will make some historic remarks on this towards the end of the introduction. It is the case of curves in the real projective plane that still poses problems.In the present paper we will be dealing with closed C ∞ -parameterized curves γ : S 1 → P 2 that are simple (free of self-intersections) and regular (nowhere vanishing tangent vector). Here and elsewhere in the paper we let P 2 denote the real projective plane. The existence of inflection and sextactic points on such curves has of course been much studied. Of importance for us is the result of Möbius [Mö] that a simple regular curve in P 2 that is not null-homotopic has at least three inflection points. As far as we know,

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“…Of recent results, we would like to mention [60] in which the Sturm theorem is extended from Fourier series to Fourier integrals and [218] where analogs of extactic points for trigonometric polynomials are studied.…”

Section: Commentmentioning

confidence: 99%

Projective Differential Geometry Old and New

Ovsienko

Tabachnikov

2004

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Metrical geometry is a part of descriptive geometry 1 , and descriptive geometry is all geometry. Arthur Cayley On October 5-th 2001, the authors of this book typed in the word "Schwarzian" in the MathSciNet database and the system returned 666 hits. Every working mathematician has encountered the Schwarzian derivative at some point of his education and, most likely, tried to forget this rather scary expression right away. One of the goals of this book is to convince the reader that the Schwarzian derivative is neither complicated nor exotic, in fact, this is a beautiful and natural geometrical object. The Schwarzian derivative was discovered by Lagrange: "According to a communication for which I am indebted to Herr Schwarz, this expression occurs in Lagrange's researches on conformable representation 'Sur la construction des cartes géographiques' " [117]; the Schwarzian also appeared in a paper by Kummer in 1836, and it was named after Schwarz by Cayley. The main two sources of current publications involving this notion are classical complex analysis and one-dimensional dynamics. In modern mathematical physics, the Schwarzian derivative is mostly associated with conformal field theory. It also remains a source of inspiration for geometers. The Schwarzian derivative is the simplest projective differential invariant, namely, an invariant of a real projective line diffeomorphism under the natural SL(2, R)-action on RP 1. The unavoidable complexity of the formula for the Schwarzian is due to the fact that SL(2, R) is so large a group (three-dimensional symmetry group of a one-dimensional space). Projective geometry is simpler than affine or Euclidean ones: in projective geometry, there are no parallel lines or right angles, and all nondegenerate conics are equivalent. This shortage of projective invariants is This curve is non-degenerate at point γ(0). The plane, dual to point γ(t), is given, in an appropriate affine coordinate system (a 1 , a 2 , a 3) in RP 3 * , by 1.2. DISCRETE INVARIANTS AND CONFIGURATIONS 1.5. SCHWARZIAN DERIVATIVE AS A COCYCLE OF DIFF(RP 1) 19 counterpart of the Huygens construction of the involute of a plane curve using a non-stretchable string: the role of the tangent line is played by the osculating conic and the role of the Euclidean length by the projective one. Affine differential geometry and the corresponding differential invariants appeared later than the projective ones. A systematic theory was developed between 1910 and 1930, mostly by Blaschke's school. 1.5 Schwarzian derivative as a cocycle of Diff(RP 1) The oldest differential invariant of projective geometry, the Schwarzian derivative, remains the most interesting one. In this section we switch gears and discuss the relation of the Schwarzian derivative with cohomology of the group Diff(RP 1). This contemporary viewpoint leads to promising applications that will be discussed later in the book. To better understand the material of this and the next section, the reader is recommended to consult Section 8.4. Invariant and relative ...

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A global theory of flexes of periodic functions

Cited by 9 publications

References 10 publications

Inflection points and double tangents on anti-convex curves in the real projective plane

Inflection points and double tangents on anti-convex curves in the real projective plane

Sextactic points on a simple closed curve

Projective Differential Geometry Old and New

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