Computer Graphics and Mathematics by Maharaj Mukherjee, George Nagy, Shashank Mehta (auth.),

By Maharaj Mukherjee, George Nagy, Shashank Mehta (auth.), Bianca Falcidieno, Ivan Herman, Caterina Pienovi (eds.)

Since its very life as a separate box inside desktop technology, special effects needed to make large use of non-trivial arithmetic, for instance, projective geometry, reliable modelling, and approximation conception. This interaction of arithmetic and machine technology is interesting, but in addition makes it tricky for college students and researchers to assimilate or keep a view of the mandatory arithmetic. the probabilities provided via an interdisciplinary process are nonetheless now not absolutely applied. This ebook offers a variety of contributions to a workshop held close to Genoa, Italy, in October 1991, the place a gaggle of mathematicians and desktop scientists accumulated to discover methods of extending the cooperation among arithmetic and laptop graphics.

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4 B-splines We have seen above that a quadratie polynomial F and its palar form f is eompletely defined by its Bezier points b o = f(r,r), b l = f(r,s), and b 2 = f(s,s). Let be a non-deereasing sequenee of real numberso We wish to show that F ean equally well be defined by its de Boor points do=f(q,r), dl=f(r,s), andd 2 =f(s,t). t [r, sl, [q, sl, and [r, tl. a similar argument as above shows F(u) f(u, u) U) s-r ( S - (~) f(q,r) s-q +{(s - U) (u - q) s-r s-q + (U - r) (tt-r - U)}f(r,s) s-r U- rr) (Ut _- rr) f(s, t), + (s - and F is in fact eompletely determined by the points f(q,r), f(r,s), and f(s,t).

An analysis of the finite element method. Prentice-Hall, Englewood Clifrs, 1973. 3 A Geometric Approach to Bezier eUryeS Hans-Peter Seidel ABSTRACT Using techniques from elassical geometry we present a purely geometric approaeh to Bezier curves and B-splines. The approaeh 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 palar form. We discuss the main properties of the polar form and show how polar forms lead to a simple new labeling scheme for Bezier curves and B-splines.

J. Approx. , 6:50-62, 1972. [5] C. de Boor. A Praetieal Guide to Splines. Springer, New York, 1978. [6] P. de CasteIjau. Courbes et surfaees 1963. [7] P. de CasteIjau. Formes aPoles. apoles. Technical Report, Andre Citroen, Paris, Hermes, Paris, 1985. [8] P. de CasteIjau. Outillages methodes ealeul. Technical Report, Andre Citroen, Paris, 1959. N. Goldman. Blossoming and knot insertion algorithms for B-spline curves. Computer-Aided Geom. Design, 7:69-81, 1990. [10] L. Ramshaw. Beziers and B-splines as multiaffine maps.

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