Implicitizing rational surfaces using moving quadrics constructed from moving planes

This paper presents a new algorithm for implicitizing tensor product surfaces of bi-degree (m,n)(m,n) with no base points, assuming that there are no moving planes of bi-degree (m−1,n−1)(m−1,n−1) following the surface. The algorithm is based on some structural results: (1) There are exactly 2n   linearly independent moving planes of bi-degree (m,n−1)(m,n−1) following the surface; (2) mn   linearly independent moving quadrics of bi-degree (m−1,n−1)(m−1,n−1) following the surface can be constructed from the 2n linearly independent moving planes; (3) The mn linearly independent moving quadrics form a compact determinant of order mn which exactly gives the implicit equation of the rational surface. Complexity analysis and experimental results show that the new algorithm is significantly more efficient than the previous methods.