Combining complementary methods for implicitizing rational tensor product surfaces. (November 2018)

• Record Type: Journal Article
• Authors: Shen, Li-Yong; Goldman, Ron
• Is Part Of: Computer aided design. Volume 104(2018)
• Page Start: 100
• Page End: 112
• Publication Date: 2018-11
• Subjects: Base point -- Implicitization -- A-resultant -- Moving plane -- Resultant -- Extraneous factor; Computer-aided design -- Periodicals; Engineering design -- Data processing -- Periodicals; Computer graphics -- Periodicals; Conception technique -- Informatique -- Périodiques; Infographie -- Périodiques; Computer graphics; Engineering design -- Data processing; Periodicals; Electronic journals; 620.00420285
• DOI: 10.1016/j.cad.2018.07.002
• Languages: English
• ISSNs: 0010-4485

Abstract

Abstract: We present an algorithm to implicitize rational tensor product surfaces that works correctly and efficiently even in the presence of base points by combining three complementary approaches, some classical and some novel, to implicitization. One straightforward method is to implicitize these surfaces using the Dixon A -resultant of three obvious syzygies. For surfaces that cannot be implicitized using this resultant due, for example, to the presence of base points, we try the method of moving planes and moving quadrics. We show that this method works very well on surfaces with simple base points. For general rational surfaces that cannot be implicitized by these two methods, we refine the method that employs the resultant matrix of three low bidegree moving planes by providing a simple technique to construct this resultant matrix. To avoid computing the base points explicitly, we compute the extraneous factors associated to the base points by solving a system of polynomial equations with a finite number of solutions. Highlights: Recall some classical methods for implicitizing rational tensor product surfaces. Introduce the latest method for implicitizing rational surfaces by moving planes. Combine the methods into a general approach for implicitizing rational surfaces.