Caustics of convex curves

It is well-known that the focal set (i.e. the image of the caustic) of a given convex closed curve γ admits singular points. In this paper, we classify the diffeomorphic type of focal sets of convex curves which admit at most four cusps.

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Axes for symmetric convex curves

The Mathematical Gazette ◽

10.1017/mag.2018.4 ◽

2018 ◽

Vol 102 (553) ◽

pp. 23-30

Author(s):

David L. Farnsworth

Keyword(s):

Line Segment ◽

Closed Curve ◽

Polar Coordinates ◽

Symmetric Convex ◽

Minkowski Geometry ◽

Convex Curves ◽

Rectangular Coordinates

Curves are given in polar coordinates (r, θ)by equations of the form r = f (θ), where for f (θ) > 0 all θ. Consider curves which are symmetric about the origin O, so that, f(θ + π) = f (θ) for all θ. For such a curve, its interior is the set {(r, θ) : 0 ≤ r ≤ f (θ)}. Further, assume that the curve is convex. Recall that a closed curve is convex if a line segment between any two of its points has no points exterior to the curve [1], [2, pp. 198-203]. We call these curves M-curves, because the curves are fundamental objects in Minkowski geometry, where they are called Minkowski circles or simply circles [3, 4]. That application is briefly discussed in the Section 4 but is not required for our purposes.Examples of M-curves are displayed in Figures 1 to 6. In order to express these curves as functions in rectangular coordinates, we need axes.

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On Differentiable Arcs and Curves, VI: Singular Osculating Spaces of Curves of Order n + 1 in Projective n-Space

Canadian Journal of Mathematics ◽

10.4153/cjm-1967-095-7 ◽

1967 ◽

Vol 19 ◽

pp. 1042-1061

Author(s):

Peter Scherk

Keyword(s):

Algebraic Geometry ◽

Closed Curve ◽

Singular Points ◽

Osculating Spaces

A closed curve Kn+1 of order n + 1 in real projective n-space Rn has a maximum number of n + 1 points in common with any (n — 1)-space. These curves are subjected to certain differentiability assumptions which make it possible to describe their singular points and to provide them with multiplicities in analogy with algebraic geometry.

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On the Projective Centres of Convex Curves

Canadian Journal of Mathematics ◽

10.4153/cjm-1960-050-7 ◽

1960 ◽

Vol 12 ◽

pp. 568-581 ◽

Cited By ~ 3

Author(s):

Paul Kelly ◽

E. G. Straus

Keyword(s):

Projective Plane ◽

Closed Curve ◽

Full Line ◽

Convex Curves ◽

The Relationship

We consider a closed curve C in the projective plane and the projective involutions which map C into itself. Any such mapping γ, other than the identity, is a harmonic homology whose axis η we call a projective axis of C and whose centre p we call an interior or exterior projective centre according as it is inside or outside C. The involutions are the generators of a group Γ, and the set of centres and the set of axes are invariant under Γ. The present paper is concerned with the type of centre sets which can exist and with the relationship between the nature of C and its centre set.If C is a conic, then every point which is not on C is a projective centre. Conversely, it was shown by Kojima (4) that if C has a chord of interior centres, or a full line of exterior centres, then C is a conic.

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