On the affine rectification of convex curves.
Leichtweiss, Kurt (1999)
Beiträge zur Algebra und Geometrie
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Displaying similar documents to “Remark on the equisingularity of families of affine plane curves”
On the affine rectification of convex curves.
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We study affine invariants of plane curves from the view point of the singularity theory of smooth functions. We describe how affine vertices and affine inflexions are created and destroyed.
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After having given the general variational formula for the functionals indicated in the title, the critical points of the integral of the equi-affine curvature under area constraint and the critical points of the full-affine arc-length are studied in greater detail. Notice. An extended version of this article is available on arXiv:0912.4075.
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We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups and , respectively. We define general-affine length parameter and curvatures and show how such invariants determine the curve up to general-affine motions. We then study the extremal problem of the general-affine length functional and derive a variational formula. We give several examples of curves and also discuss some relations with equiaffine treatment and...
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Maciej Borodzik, Henryk Żołądek (2008)
Annales Polonici Mathematici
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We consider the space Curv of complex affine lines t ↦ (x,y) = (ϕ(t),ψ(t)) with monic polynomials ϕ, ψ of fixed degrees and a map Expan from Curv to a complex affine space Puis with dim Curv = dim Puis, which is defined by initial Puiseux coefficients of the Puiseux expansion of the curve at infinity. We present some unexpected relations between geometrical properties of the curves (ϕ,ψ) and singularities of the map Expan. For example, the curve (ϕ,ψ) has a cuspidal singularity iff it...
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All Bézier curves are attractors of iterated function systems.
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The New York Journal of Mathematics [electronic only]
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An affine framework for analytical mechanics
Paweł Urbański (2003)
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An affine Cartan calculus is developed. The concepts of special affine bundles and special affine duality are introduced. The canonical isomorphisms, fundamental for Lagrangian and Hamiltonian formulations of the dynamics in the affine setting are proved.
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Annales Polonici Mathematici
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Self-affine fractals of finite type
Christoph Bandt, Mathias Mesing (2009)
Banach Center Publications
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In the class of self-affine sets on ℝⁿ we study a subclass for which the geometry is rather tractable. A type is a standardized position of two intersecting pieces. For a self-affine tiling, this can be identified with an edge or vertex type. We assume that the number of types is finite. We study the topology of such fractals and their boundary sets, and we show how new finite type fractals can be constructed. For finite type self-affine tiles in the plane we give an algorithm which...