Shifting Planes to Follow a Surface of Revolution

摘要

A degree n rational plane curve rotating about an axis in the plane creates a degree 2n rational surface. Two formulas are given to generate 2n moving planes that follow the surface. These 2n moving planes lead to a 2n×2n implicitization determinant that manifests the geometric revolution algebraically in two 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 almost an n-row rotation of the left half. As an aside, it is observed that rational parametrizations of a surface of revolution due to a symmetric rational generatrix must be improper.