Shifting planes always implicitize a surface of revolution Original Research Article

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 image 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 image determinant. Secondly the right half of the image 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.