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An Introduction to NURBS: With Historical Perspective (The by David F. Rogers

By David F. Rogers

For one week after receiving this publication I agreed with an past very severe evaluation. I replaced my brain. the topic isn't effortless yet written by means of somebody who understands his company. Having acquired used to his notation i locate this e-book an increasing number of worthwhile and refer again to it every time an issue arises and typically find the solution 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)

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Is defined by vertices Bi and an by vertices Ci, then first-derivative the curves join is given by P'(1)-gQ'(O) where g is a scalar. Using Eqs. , Bn need only lie somewhere on the line between B n - i and C1. If both direction and magnitude of the tangent vectors at the join are to be equal, then g - 1. In this case, parametric continuity is also achieved. e. , for two cubic B~zier curves. 7 First-derivative continuity between cubic B~zier curves. Second-derivative parametric continuity between adjacent B~zier curves is given by m ( m - 1)(C0 - 2C1 + C2) - n ( n - 1)(Bn-2 - 2 B n - 1 + Bn) Using the conditions for C o and C 1 continuity at the join given on page 31 yields C2- n ( n - 1) Bn-2 m(m 1) -2 n ( n - 1) 1) ~)Bn-l+ (1+2n m + n(n-1))Bn m ( m - 1) Thus, the position of the third defining vertex in the second control polygon is given in terms of the last three vertices in the first control polygon.

3 B~zier C u r v e D e r i v a t i v e s Although it is not necessary to numerically specify the tangent vectors at the ends of an individual B~zier curve, maintaining slope and curvature continuity when joining B~zier curves, determining surface normals for lighting or numerical control tool path calculation, or local curvature for smoothness or fairness calculations requires a knowledge of both the first and second derivatives of a B~zier curve. Recalling Eq. 12) i=0 The derivatives of the basis function are obtained by formally differentiating Eq.

Robin Forrest was responsible for the mechanical design and construction of the Cambridge CAD Group's Mark II foam-cutting machine, with Charles Lang and David Wheeler responsible for the electrical and electronic design. The machine was controlled along each of the three z y z - a x e s , with two additional rotational axes (one about the center of the cutter). The cutting volume was approximately 20 x 14 x 10 inches. 6 in/sec, with a maximum speed of 16 in/sec on all three axes. The machine was used to cut rigid foam, wood and rigid, high-density polyethylene models, two examples of which, produced on the earlier Mark I machine, are held by Robin Forrest in the picture on page 13.

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