Ordinary Differential Equations by Vladimir I. Arnol'd

By Vladimir I. Arnol'd

Good differential equaitons are all approximately switch, and this booklet replaced my existence. I learn this greater than 30 years in the past, and all of the arithmetic i do know, I suggest quite recognize, I realized from this booklet. in addition to Aristotle's ethics, it truly is the most vital booklet in my existence.

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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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