Algebraic surfaces invariant under scissor shears

Alcázar, Juan Gerardo; Goldman, Ron; Hermoso, Carlos

doi:10.1016/j.gmod.2016.09.001

Cited by 5 publications

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“…stands for lower order terms. Since the form of highest order of an affine surface of rotation has a very specific structure (see Theorem 6 in [8], Theorem 6 in [10], Theorem 6 in [9]), we deduce that S 1 is not an affine surface of rotation. In order to compute cubic equations defining the variety V, we consider Equation (15) for the points corresponding to (t i , s i ) with t i , s i ranging from −3 to 3.…”

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confidence: 89%

“…These surfaces can be of three different subtypes, elliptic, hyperbolic and parabolic, the first of them being the classical surfaces of revolution. Theoretical properties of algebraic affine surfaces of rotation are treated in some recent papers: the elliptic case is studied in [8], the hyperbolic case is addressed in [9], and the parabolic in [10]. Furthermore, an algorithm for recognizing algebraic affine surfaces of revolution is provided in [11].…”

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confidence: 99%

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On the Affine Image of a Rational Surface of Revolution

Alcázar

2020

Mathematics

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We study the properties of the image of a rational surface of revolution under a nonsingular affine mapping. We prove that this image has a notable property, namely that all the affine normal lines, a concept that appears in the context of affine differential geometry, created by Blaschke in the first decades of the 20th century, intersect a fixed line. Given a rational surface with this property, which can be algorithmically checked, we provide an algorithmic method to find a surface of revolution, if it exists, whose image under an affine mapping is the given surface; the algorithm also finds the affine transformation mapping one surface onto the other. Finally, we also prove that the only rational affine surfaces of rotation, a generalization of surfaces of revolution that arises in the context of affine differential geometry, and which includes surfaces of revolution as a subtype, affinely transforming into a surface of revolution are the surfaces of revolution, and that in that case the affine mapping must be a similarity.

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confidence: 89%

Section: Introductionmentioning

confidence: 99%

On the Affine Image of a Rational Surface of Revolution

Alcázar

2020

Mathematics

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“…There are three types of affine rotation surfaces corresponding to the three types of affine rotation groups: elliptic rotation surfaces correspond to surfaces invariant under classical rotations; hyperbolic rotation surfaces correspond to surfaces invariant under hyperbolic rotations, also called scissor shears [4]; and parabolic rotation surfaces correspond to surfaces invariant under transformations which are composites of certain classical shears. The elliptic affine rotation surfaces are the classical surfaces of revolution, which have been studied extensively in the algebraic case; see for example [2,3].…”

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“…The elliptic affine rotation surfaces are the classical surfaces of revolution, which have been studied extensively in the algebraic case; see for example [2,3]. The hyperbolic affine rotation surfaces are what the authors have previously called scissor shear invariant surfaces [4], because these surfaces are invariant under certain scissor shears that leave an axis line unchanged. Algebraic surfaces of this type have been studied recently by the authors in [4].…”

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Affine differential geometry and affine rotation surfaces: algebraic surfaces invariant under non-Euclidean affine rotations

Alcázar,

Goldman

2018

Preprint

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Affine rotation surfaces are a generalization of the well-known surfaces of revolution. Affine rotation surfaces arise naturally within the framework of affine differential geometry, a field started by Blaschke in the first decades of the past century. Affine rotations are the affine equivalents of Euclidean rotations, and include certain shears as well as Euclidean rotations. Affine rotation surfaces are surfaces invariant under affine rotations. In this paper, we analyze several properties of algebraic affine rotation surfaces and, by using some notions and results from affine differential geometry, we develop an algorithm for determining whether or not an algebraic surface given in implicit form, or in some cases in rational parametric form, is an affine rotation surface. We also show how to find the axis of an affine rotation surface. Additionally, we discuss several properties of affine spheres, analogues of Euclidean spheres in the context of affine differential geometry. Introduction.Classical differential geometry, initiated by Gauss in the first decades of the 19 th century, studies Euclidean invariants -normal vectors and normal lines, mean curvature and Gaussian curvature -under rigid motions. In contrast, affine differential geometry, initiated by Blaschke in the first decades of the 20 th century, studies the corresponding affine invariants -affine normals and affine curvatures -which are invariant under the unimodular affine group generated by the special linear group SL 3 (R), i.e. the group of matrices with determinant equal to 1.Classical surfaces of revolution are surfaces invariant under Euclidean rotations about a fixed axis. Affine rotation surfaces are surfaces invariant under affine rotations, groups of linear transformations that leave a line in 3-space

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Algebraic affine rotation surfaces of parabolic type

Alcázar

Goldman

2019

J. Geom.

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Algebraic surfaces invariant under scissor shears

Cited by 5 publications

References 18 publications

On the Affine Image of a Rational Surface of Revolution

On the Affine Image of a Rational Surface of Revolution

Affine differential geometry and affine rotation surfaces: algebraic surfaces invariant under non-Euclidean affine rotations

Algebraic affine rotation surfaces of parabolic type

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