By David F. Rogers
For one week after receiving this booklet I agreed with an previous very severe evaluate. I replaced my brain. the topic isn't effortless yet written by means of an individual who understands his company. Having received used to his notation i locate this booklet an increasing number of invaluable and refer again to it every time an issue arises and typically uncover the answer or a few pointer to the reply.
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Extra info for An Introduction to NURBS: With Historical Perspective (The Morgan Kaufmann Series in Computer Graphics)
Given a parametric vector-vMued function, P(t), over some parameter range describing a curve, then any given value of t represents a specific point on the curve. The derivative, P~(t), represents the velocity of a point as it moves along the curve; and P"(t) represents the acceleration. If the curve is C 1 continuous at a join, then both the direction and magnitude of the tangent vector are equal across the join; and the point smoothly transitions from one curve segment to the next. If, however, the curve is only G 1 continuous at the join, then as the point transitions across the join the direction of motion does not change; but there is an instantaneous change in speed (magnitude of the velocity) t h a t represents an instantaneous acceleration as the point transitions across the join.
So it had the effect of promulgating the uniform floating case and leaving largely unexploited the much more powerful and general nonuniform case. It was not until several years later that the Oslo Algorithm and subdivision interacted to focus attention on nonuniform B-splines and their importance. On a visit to Ivan Sutherland at the University of Utah, Steve presented the B-spline method for curves and surfaces that I was developing. ), implementation of an interactive design system using the B-spline method for uniform floating surfaces.
3) i=1 Each basis function is positive or zero for all p a r a m e t e r values, t h a t is, N~,k _> O. Except for first-order basis functions, k = 1, each basis function has precisely one m a x i m u m value. The m a x i m u m order of the curve equals the n u m b e r of control polygon vertices. The m a x i m u m degree is one less. The curve exhibits the variation-diminishing property. Thus, the curve does not oscillate about any straight line more often t h a n its control polygon oscillates about the line.