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 : S 1 → 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 ) = e i ϑ . The plane curve determined by the smooth function h is γ ( ϑ ) = ( h ( ϑ ) + ih ′ ( ϑ )) e i ϑ . Example. The curve whose support fonction is h ( ϑ ) = 2 cos 2 ϑ is the standard astroid γ ( ϑ ) = 3 e − i ϑ − e 3 i ϑ . 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 , γ c ( 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 C 2 = S 1 × R ⊂ R 2 × R It is parametrised by Γ h ( 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 h (of the graph Γ ) on any horizontal hyperplane R 2 × { s } is front Γ ∨ 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 of singularities of type X of the evolute of a smooth manifold γ ⊂ R n is isomorphic to the set of singularities of type X in the front formed by the hyperplanes of R n +1 which are tangent to the image of γ (in the unit cylinder C n ) by the support map and do not contain the vertical direction . Ricardo Uribe-Vargas Lagrange and Legendre singularities - support function