A Geometric Approach to Bézier Curves
Using techniques from classical geometry we present a purely geometric approach to Bézier curves and B-splines. The approach is based on the intersection of osculating flats: The osculating 1-flat is simply the tangent line, the osculating 2-flat is the osculating plane, etc. The intersection of osculating flats leads to the so-called polar form. We discuss the main properties of the polar form and show how polar forms lead to a simple new labeling scheme for Bézier curves and B-splines.
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- W. Dahmen, C.A. Micchelli and H.-P. Seidel. Blossoming Begets B-Splines Built Better by B-Patches. Research Report RC 16261(#72182), IBM Research Division, Yorktown Heights, 1990. To appear in Math. Comp..Google Scholar
- C. de Boor. B-form basics. In G. Farin, editor, Geometric Modeling, Algorithms and New Trends, 131–148, SIAM, 1987.Google Scholar
- P. de Casteljau. Courbes et surfaces à pôles. Technical Report, André Citroen, Paris, 1963.Google Scholar
- P. de Casteljau. Formes it Pôles. Hermes, Paris, 1985.Google Scholar
- P. de Casteljau. Outillages méthodes calcul. Technical Report, André Citroen, Paris, 1959.Google Scholar
- L. Ramshaw. Béziers and B-splines as multiaffine maps. In Theoretical Foundations of Computer Graphics and CAD, 757–776, Springer, 1988.Google Scholar
- L. Ramshaw. Blossoming: A connect-the-dots approach to splines. Technical Report, Digital Systems Research Center, Palo Alto, 1987.Google Scholar
- L. Ramshaw. The polar forms of polynomial curves. In Blossoming: The New Polar-Form Approach to Spline Curves and Surfaces, SIGGRAPH ‘81 Course Notes #26, 1.1–1.29, ACM SIGGRAPH, 1991.Google Scholar
- H.-P. Seidel. Geometric Constructions and Knot Insertion for Geometrically Continuous Spline Curves of Arbitrary Degree. Technical Report CS-90–24, Dept. of Computer Science, University of Waterloo, 1990.Google Scholar
- H.-P. Seidel. Polar forms and triangular B-Spline surfaces. In Blossoming: The New Polar-Form Approach to Spline Curves and Surfaces, SIGGRAPH ‘81 Course Notes #26, 8.1–8.52, ACM SIGGRAPH, 1991.Google Scholar