By Gerald Farin, Dianne Hansford
Placing the G into CAGD, the authors supply a much-needed useful and uncomplicated creation to computer-aided geometric layout. This booklet can assist readers comprehend and use the weather of computer-aided geometric layout, curves and surfaces, with no the mathematical luggage that's valuable just for extra complex paintings. even though purely minimum heritage in arithmetic is required to appreciate the bookís innovations, the booklet covers an awesome array of issues equivalent to Bezier and B-spline curves and their corresponding surfaces, subdivision surfaces, and NURBS (Non-Uniform Rational B-Splines). additionally integrated are innovations akin to interpolation and least squares equipment.
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Additional resources for The Essentials of CAGD
3 We define a parametric curve by setting x(t) = [ 2 ~ ] -(1- t)3 + t3 ] [ 3(1- t) 2 t - 3(1 - t)t 2 Piecewise schemes are introduced in Chapter 9. · 3. Cubic Bezier Curves From this definition, it is not at all obvious what the curve might look like. Let's rewrite it: This way of rewriting the polynomial expresses the polynomial in terms of a combination of points. The four points form a polygon, and it roughly resembles the curve segment from x(O) to x(l). See Sketch 25. 5) = [ ~ ] . /In general, we define a cubic Bezier curve by r curve.
Construct the Bezier control points for the four parametric curves defined over [0, 1], and sketch each polygon and curve. 6. 3? 7. What is the Bezier form of the 3D curve segment corresponding totE [0, 1], when the curve is given in monomial form by x(t) = [ :: ] ? 3. Cubic Bezier Curves 8. Let a Bezier curve be given by bo = [ ~ ], b1 =[ ~ ], Sketch the curve fortE [-1, 2]. b2 = [ ~ ] , b3 = [ ~ ]. Bezier Curves: Cubic and Beyond Remargue 1 - Le poin~ 4 L peut ttnv-«rt6rleur k I. le e La oonstl elleest l Remargua P:.
31:!!.. 2ho = )(h2- 2hl + ho). Thus, ative vector. x(o) = ~( [ -~ J -2 [ ~ J + [ -~ J) = [ -~~ J. Sketch 29 illustrates. \_ _ _ _ _ _. . . 9 for more on Pascal's triangle. 3 The de Casteljau Algorithm Revisited 47 One nice feature of Bezier curves is the geometric interpretation of the first and second derivatives at the endpoints. The first derivative has the direction of the control polygon leg. The second derivative is in the direction of the diagonal of the parallelogram formed by the three control points at the endpoint.