Lagrange and Legendre singularities related to support function - - PowerPoint PPT Presentation

Lagrange and Legendre singularities related to support function

Ricardo Uribe-Vargas Goryunov 60- 2016 at Liverpool

Ricardo Uribe-Vargas Lagrange and Legendre singularities - support function

• I. Support Function of a Cooriented Curve

Definition The supoort function of a cooriented smooth curve γ at γ(ϑ) is the algebraic distance from its tangent line at γ(ϑ) to the “origin” O.

• Example. The support fonction of the circle of radius R and center

(a, b), is h(ϑ) = R + a cos ϑ + b sin ϑ.

Ricardo Uribe-Vargas Lagrange and Legendre singularities - support function

Construction of the curve from its support function A smooth function on the circle h : S1 → R is the support function a plane curve. To make it easy, we will use the complex notation writing the coorienting unit normal vector as n(ϑ) = eiϑ. The plane curve determined by the smooth function h is γ(ϑ) = (h(ϑ) + ih′(ϑ))eiϑ .

• Example. The curve whose support fonction is

h(ϑ) = 2 cos 2ϑ is the standard astroid γ(ϑ) = 3e−iϑ − e3iϑ.

Ricardo Uribe-Vargas Lagrange and Legendre singularities - support function

Support function and properties of the curve * h + h′′ = R is the radius of curvature * h′ + h′′′ = 0 vertices of the curve (cusps ot its evolute). * If h is the support function of γ, then for each c ∈ R h + c is the support function of the equidistant curve at distance c, γc(ϑ) = γ(ϑ) + cn(ϑ) .

Ricardo Uribe-Vargas Lagrange and Legendre singularities - support function

We have the same construction in the higherdimensional case

Ricardo Uribe-Vargas Lagrange and Legendre singularities - support function

• II. The Graph of the Support Function

The Graph of the Support Function is a space curve on the unit cylinder C2 = S1 × R ⊂ R2 × R It is parametrised by Γh(ϑ) = (cos ϑ, sin ϑ, h(ϑ)) .

Ricardo Uribe-Vargas Lagrange and Legendre singularities - support function

Properties of the Curve γ and of the Graph Γ. * det(Γ, Γ′, Γ′′) = h + h′′ = R the radius of curvature of γ * det(Γ′, Γ′′, Γ′′′) = h′ + h′′′ = 0 cusps of the evolute of γ. * The osculating plane of Γ at Γ(ϑ) intsersects the z-axis at the point (0, 0, h + h′′). * If γ has inflections, then Γ has cusps and is the “graph” of a multivalued function.

Ricardo Uribe-Vargas Lagrange and Legendre singularities - support function

• III. Polar Duality

Let E, E two planes with coordinates (x, y; ( x, y).

Ricardo Uribe-Vargas Lagrange and Legendre singularities - support function

Ricardo Uribe-Vargas Lagrange and Legendre singularities - support function

Ricardo Uribe-Vargas Lagrange and Legendre singularities - support function

• Theorem. The projection of the cuspidal edge of the polar dual

front Γ∨

h (of the graph Γ) on any horizontal hyperplane R2 × {s} is

the evolute (caustic) of γ.

Ricardo Uribe-Vargas Lagrange and Legendre singularities - support function

Ricardo Uribe-Vargas Lagrange and Legendre singularities - support function

Ricardo Uribe-Vargas Lagrange and Legendre singularities - support function

Our constructions and results hold also in higher dimensions :

• Theorem. For a class of simple singularities X ∈ {A, D, E} the set
• f singularities of type X of the evolute of a smooth manifold

γ ⊂ Rn is isomorphic to the set of singularities of type X in the front formed by the hyperplanes of Rn+1 which are tangent to the image of γ (in the unit cylinder Cn) by the support map and do not contain the vertical direction.

Ricardo Uribe-Vargas Lagrange and Legendre singularities - support function