Algebraic surfaces invariant under scissor shears

• Published in: Graphical models Vol. 87; pp. 23 - 34
• Main Authors: Alcázar, Juan G., Goldman, Ron, Hermoso, Carlos
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.09.2016
• ISSN: 1524-0703; 1524-0711
• DOI: 10.1016/j.gmod.2016.09.001

[Display omitted] Scissor shears are affine transformations in 3-space that, in analogy with the usual rotations, can be understood as hyperbolic rotations about a fixed line, in a fixed coordinate frame. We study algebraic surfaces invariant under scissor shears, and investigate their similarities and differences with the algebraic surfaces invariant under the usual rotations, namely the algebraic surfaces of revolution. In particular, we provide a necessary condition for an algebraic surface to be invariant under scissor shears, and we prove that such shear invariant surfaces can have either one, three, or infinitely many scissor axes. Furthermore, we characterize the surfaces with either three or infinitely many scissor axes. Additionally, in each case we show how to calculate the location of these scissor axes as well as the rest of the coordinate frame.