Shifting planes always implicitize a surface of revolution

摘要

A degree n rational plane curve revolving in space around an axis in its plane yields a degree 2n rational surface. Two formulas are presented to generate 2n moving planes that follow the surface. These 2n moving planes give a 2n× 2n implicitization determinant that manifests conspicuously the geometric action of revolution in two algebraic aspects. Firstly the moving planes are constructed by successively shifting terms of polynomials from one column to another of a spawning 3×3 determinant. Secondly the right half of the 2n× 2n implicitization determinant is an n-row rotation of the left half with some sign flipping. Additionally, it is observed that rational parametrizations for a surface obtained as a surface of revolution with a symmetric generatrix must be improper.