Smooth Nonprojective Equivariant Completions     of Affine Space

AbstractThe present essay investigates the potential of generative representation applied to the study of relief perspective architectures realized in Italy between the sixteenth and seventeenth centuries. In arts, and architecture in particular, relief perspective is a three-dimensional structure able to create the illusion of great depths in small spaces. A method of investigation applied to the case study of the Avila Chapel in Santa Maria in Trastevere in Rome (Antonio Gherardi 1678) is proposed. The research methodology can be extended to other cases and is based on the use of a Relief Perspective Camera, which can create both a linear perspective and a relief perspective. Experimenting mechanically and automatically the perspective transformations from the affine space to the illusory space and vice versa has allowed us to see the case study in a different light.

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Almost vanishing polynomials and an application to the Hough transform

Journal of Algebra and Its Applications ◽

10.1142/s0219498814500571 ◽

2014 ◽

Vol 13 (08) ◽

pp. 1450057 ◽

Cited By ~ 7

Author(s):

Maria-Laura Torrente ◽

Mauro C. Beltrametti

Keyword(s):

Hough Transform ◽

Affine Space ◽

Bounded Region ◽

Pattern Recognition Technique ◽

Local Study ◽

Affine Hypersurface ◽

Automated Recognition ◽

Real Affine ◽

Necessary And Sufficient ◽

Recognition Technique

We consider the problem of deciding whether or not an affine hypersurface of equation f = 0, where f = f(x1, …, xn) is a polynomial in ℝ[x1, …, xn], crosses a bounded region 𝒯 of the real affine space 𝔸n. We perform a local study of the problem, and provide both necessary and sufficient numerical conditions to answer the question. Our conditions are based on the evaluation of f at a point p ∈ 𝒯, and derive from the analysis of the differential geometric properties of the hypersurface z = f(x1, …, xn) at p. We discuss an application of our results in the context of the Hough transform, a pattern recognition technique for the automated recognition of curves in images.

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Random nested tetrahedra

Advances in Applied Probability ◽

10.1017/s0001867800008508 ◽

1998 ◽

Vol 30 (03) ◽

pp. 619-627 ◽

Cited By ~ 1

Author(s):

Gérard Letac ◽

Marco Scarsini

Keyword(s):

Random Walk ◽

Convex Hull ◽

Affine Space ◽

Barycentric Coordinates ◽

Geometrical Application

In a real n-1 dimensional affine space E, consider a tetrahedron T 0, i.e. the convex hull of n points α1, α2, …, α n of E. Choose n independent points β1, β2, …, β n randomly and uniformly in T 0, thus obtaining a new tetrahedron T 1 contained in T 0. Repeat the operation with T 1 instead of T 0, obtaining T 2, and so on. The sequence of the T k shrinks to a point Y of T 0 and this note computes the distribution of the barycentric coordinates of Y with respect to (α1, α2, …, α n ) (Corollary 2.3). We also obtain the explicit distribution of Y in more general cases. The technique used is to reduce the problem to the study of a random walk on the semigroup of stochastic (n,n) matrices, and this note is a geometrical application of a former result of Chamayou and Letac (1994).

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