Algebraic affine rotation surfaces of parabolic type
TL;DR: In this paper, the algebraic parabolic affine rotation surfaces of elliptic, hyperbolic and parabolic surfaces of revolution are characterized in terms of the structure of their implicit equation.
Abstract: Affine rotation surfaces, which appear in the context of affine differential geometry, are generalizations of surfaces of revolution. These affine rotation surfaces can be classified into three different families: elliptic, hyperbolic and parabolic. In this paper we investigate some properties of algebraic parabolic affine rotation surfaces, i.e. parabolic affine rotation surfaces that are algebraic, generalizing some previous results on algebraic affine rotation surfaces of elliptic type (classical surfaces of revolution) and hyperbolic type (hyperbolic surfaces of revolution). In particular, we characterize these surfaces in terms of the structure of their implicit equation, we describe the structure of the form of highest degree defining an algebraic parabolic affine rotation surface, and we prove that these surfaces can have either one, or two, or infinitely many axes of affine rotation. Additionally, we characterize the surfaces with more than one parabolic axis.
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TL;DR: The method is efficient, and works generally better than known algorithms for implicitizing the whole surface, in the absence of base points blowing up to a curve at infinity.
1 citations
19 Nov 2020
TL;DR: In this paper, the authors studied the properties of the image of a rational surface of revolution under a nonsingular affine mapping and showed that the affine normal lines of the rational surface intersect a fixed line.
Abstract: 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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TL;DR: The problem of rationality and unirationality of surfaces of revolution is investigated and it is shown that this question can be efficiently answered discussing the rationality of a certain associated planar curve.
22 citations
TL;DR: An algorithm for extracting the axis of revolution from the implicit equation of an algebraic surface of revolution based on three distinct computational methods: factoring the highest order form into quadrics, contracting the tensor of thehighest order form, and using univariate resultants and gcds is presented.
Abstract: We present an algorithm for extracting the axis of revolution from the implicit equation of an algebraic surface of revolution based on three distinct computational methods: factoring the highest order form into quadrics, contracting the tensor of the highest order form, and using univariate resultants and gcds. We compare and contrast the advantages and disadvantages of each of these three techniques and we derive conditions under which each technique is most appropriate. In addition, we provide several necessary conditions for an implicit algebraic equation to represent a surface of revolution.
15 citations
TL;DR: Given an implicit polynomial equation or a rational parametrization, algorithms are developed to determine whether the set of real and complex points defined by the equation is a cylindrical surface, a conical surface, or a surface of revolution.
Abstract: Given an implicit polynomial equation or a rational parametrization, we develop algorithms to determine whether the set of real and complex points defined by the equation, i.e., the surface defined by the equation, in the sense of Algebraic Geometry, is a cylindrical surface, a conical surface, or a surface of revolution. The algorithms are directly applicable to, and formulated in terms of, the implicit equation or the rational parametrization. When the surface is cylindrical, we show how to compute the direction of its rulings; when the surface is conical, we show how to compute its vertex; and when the surface is a surface of revolution, we show how to compute its axis of rotation directly from the defining equations.
12 citations
TL;DR: In this paper, the authors discuss affine rotation surfaces in R3 and projective rotation surface in RP3 from a group-theoretical point of view, which pave a way toward generalized affine rotations.
Abstract: From group-theoretical point of view, we discuss affine rotation surfaces in R3 and projective rotation surfaces in RP3. These pave a way toward generalized affine rotation surfaces in R3. We will follow closely the modern approach introduced by Nomizu in the study affine differential geometry [N].
11 citations