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Algebraic surfaces of revolution and algebraic surfaces invariant under scissor shears: similarities and differences Goldman, Ron


<i>Scissor shears</i> are space transformations sharing certain properties with rotations in 3-space. In fact, the formulas for scissor shears are, up to sign, the same as the formulas for rotations with sines and cosines replaced by hyperbolic sines and hyperbolic cosines. Thus one might consider scissor shears as <i>hyperbolic</i> versions of 3D rotations. While algebraic surfaces of revolution, which are well-known in Computer Aided Geometric Design, are algebraic surfaces invariant under all the rotations about a fixed axis (the axis of revolution of the surface), algebraic <i>scissor shear invariant surfaces</i> (or SSI for short) are invariant under all the scissor shear transformations about a fixed axis. Hence, both types of surfaces can be constructed from an axis, and an algebraic space curve.
Interestingly, there are a number of analogies, but also differences, between these two types of surfaces. In both cases, the intersections of the surface with a plane normal to the axis are curves of the same nature, circles in the case of surfaces of revolution, and hyperbolas in the case of ssi surfaces. While surfaces of revolution can have either one axis or infinitely many axes (when the surface is a union of spheres), ssi surfaces can have one, three, or infinitely many axes (when the surface is the union of hyperboloids of one sheet and cones with the same axis, or the union of hyperboloids of two sheets and cones with the same axis). Furthermore, in both cases the form of highest degree of the implicit equation of the surface has a special structure, where again circles are replaced by hyperbolas in the case of ssi surfaces. Finally, the axis (or axes) can be detected by similar methods in both cases: factoring the form of highest degree, and ontracting the tensor corresponding to the highest degree form.
[1] Alcazar J.G., Goldman R., (2016), Finding the axis of revolution of a surface of revolution, to appear in IEEE Transactions in Visualization and Computer Graphics.
[2] Alcazar J.G., Goldman R., (2016), Algebraic surfaces invariant under scissor shears, submitted.

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