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Inversion of Degree n + 2

Vladimir Benic
Sonja Gorjanc
Journal / Anthology

Acta Mathematica Hungarica
Year: 2009
Volume: 122
Issue: 3
Page range: 237-253

By the method of synthetic geometry, we define a seemingly new transformation of a three-dimensional projective space where the corresponding points lie on the rays of the first order, nth class congruence C1 n and are conjugate with respect to a proper quadric . We prove that this transformation maps a straight line onto an n + 2 order space curve and a plane onto an n + 2 order surface which contains an n-ple (i.e. n-multiple) straight line. It is shown that in Euclidean space the pedal surfaces of the congruences C1 n can be obtained by this transformation. The analytical approach enables new visualizations of the resulting curves and surfaces with the program Mathematica. They are shown in four examples.