BEZIER CURVES AND SURFACES

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1 Department of Applied Mathematics and Computational Sciences University of Cantabria UC-CAGD Group COMPUTER-AIDED GEOMETRIC DESIGN AND COMPUTER GRAPHICS: BEZIER CURVES AND SURFACES Andrés Iglesias Web pages: Andrés Iglesias. See:

2 Andrés Iglesias. See: BEZIER CURVES n i= P i B n i (t) Bézier curves Let P={P,P,...,P n } be a set of points P i IR d, d=,. The Bézier curve associated with the set P is defined by: B n i (t) where represent the Bernstein polynomials, which are given by: ( ) Bi n (t) = n ( t) n i t i i i =,..., n n being the polynomial degree. Bézier curve with n=5 (six control or Bézier points) Bernstein polynomials B i (t)

3 Bézier curves Control polygon 6 Control points Bézier curve 7 control points n= n i= P i B n i (t) Andrés Iglesias. See:

4 Given by: Properties: Extreme values: Normalizing property: B n i (t) = Bernstein polynomials ( ) n i B in ()= B in ()= i=,...,n- B n ()= B nn ()= B in ()= B in ()= n i= B n i (t)= ( t) n i t i i =,..., n Maxima: n the degree i the index t the variable Positivity: B in (t) in [,] Simmetry: Andrés Iglesias. See: B in (t) = B n-in (-t) B in (t) attains exactly one maximum on the interval [,], at t=i /n.

5 Properties of the Bézier curves The Bézier curve generally follows the shape of the control polygon, which consists of the segments joining the control points. D curve Control polygon Bézier scheme is useful for design. D curve Andrés Iglesias. See:

6 Properties of the Bézier curves Bézier curves exhibit global control: moving a control point alters the shape of the whole curve. Control point traslation. LOCAL vs. GLOBAL CONTROL B-splines allow local control: only a part of the curve is modified when changing a control point. Control point traslation. (,) (,) (,) (,) The curve changes here The curve does not change here Andrés Iglesias. See:

7 Properties of the Bézier curves Interpolation. A Bézier curve always interpolates the end control points. Tangency. The endpoint tangent vectors are parallel to P - P and P n - P n- Convex hull property. The curve is contained in the convex hull of its defining control points. Variation disminishing property. No straight line intersects a Bézier curve more times than it intersects its control polygon. Intersections Curve: Polygon: Curve: Polygon: Curve: Polygon: For a three-dimensional Bézier curve, replaces the words straight line with the word plane. Andrés Iglesias. See:

8 Properties of the Bézier curves A given Bézier curve can be subdivided at a point t=t into two Bézier segments which join together at the point corresponding to the parameter value t=t. Left-hand side: control points Andrés Iglesias. See: Original curve: control points 5 6 t=. t=. t=. Subdivided curve: 7 control points Right-hand side: control points

9 Properties of the Bézier curves Degree raising: any Bézier curve of degree n (with control points P i ) can be expressed in terms of a new basis of degree n+. The new control points Q i are given by: i Q i = P i- + (- ) P n+ n+ i i=,...,n+ P - = P n+ = Original cubic curve: control points Final quartic curve: 5 control points 5 6 Andrés Iglesias. See: 5 6

10 Bézier curves Degree raising of the Bézier curve of degree n= to degree n= Andrés Iglesias. See:

11 Rational Bézier curves There are a number of important curves and surfaces which cannot be represented faithfully using polynomials, namely, circles, ellipses, hyperbolas, cylinders, cones, etc. All the conics can be well represented using rational functions, which are the ratio of two polynomials. Rational Bézier curve R(t) = n i= P i w i B n i (t) n i= w i B n i (t) w i weights If all w i =, we recover the Bézier curve. Andrés Iglesias. See: Farin, G.: Curves and Surfaces for CAGD, Academic Press, rd. Edition, 99 (Chapters and 5). Hoschek, J. and Lasser, D.: Fundamentals of CAGD, A.K. Peters, 99 (Chapter ). Anand, V.: Computer Graphics and Geometric Modeling for Engineers, John Wiley & Sons, 99 (Chapter ).

12 Andrés Iglesias. See: Changing the weights: Rational Bézier curves w i > -> the curve approximates to P i w i < -> the curve moves away from P i

13 Rational Bézier curves Influence of the weights: The effect of changing a weight is different from that of moving a control point. Moving a control point: a nonrational Bézier curve with a change in one control point. Original curve Final curve Changing a weight: a rational Bézier curve with one weight changed. Andrés Iglesias. See:

14 Andrés Iglesias. See: Rational Bézier curves Rational Bézier curves are useful to represent conics, which become an important tool in the aircraft industry. Let c(t) be a point on a conic. Then, there exist numbers w, w and w and two-dimensional points P, P and P such that: c(t) = s = s < s > w P B (t) +w P B (t) +w P B (t) w B (t) +w B (t) +w B (t) If we take w = w = and we define : gives a parabolic arc gives an elliptic arc gives a hyperbolic arc w = s = w + w hyperbola w = parabola elipse w =

15 Example: the circle.75 (, ) Rational Bézier curves /.5.5 (/, /).75 (/, /).5 /.5 (,).5.5 (,) w = w = / w = (,) Andrés Iglesias. See: /

16 BEZIER SURFACES Let P={{P,P,...,P n }, {P,P,...,P n },..., {P m,p m,...,p mn }} be a set of points P ij IR (i=,,...,m ; j=,,...n) Bézier surfaces The Bézier surface associated with the set P is defined by: m n S(u, v) = P ij B m i (u)b n j (v) i= B m i (u) j= where and represent the Bernstein polynomials of degrees m and n and in the variables u and v, respectively. Bn j (v) x y z x y z x y z x y z x y z x y z Andrés Iglesias. See:

17 Bézier surfaces Note that along the isoparametric lines u=u and v=v, the surface reduces to Bézier curves: n m S(u,v) = with control points: j= b j = b j B n j (v) S(u, v ) = m i= P ij B m i (u ) i= c i = Andrés Iglesias. See: c i B m i (u) n j= P ij B n j (v ) Isoparametric lines u=u } v DOMAIN Z D space Y S(u,v) } u Isoparametric lines v=v X

18 From the equation: S(u, v) = Bézier surfaces is obtained from a control point and the product of two univariate Bernstein polynomials. Each product makes up a basis function of the surface. For instance: Function B (u). B (v): m i= n j= P ij B m i (u)b n j (v) it is clear that each term.8.6 B (u). B (v) B (v) B (u) Andrés Iglesias. See:

19 If a single surface does not approximate enough a given set of points, we may use several patches joined together. Bézier surfaces BLENDING SURFACES Two Bézier patches F and F are connected with C -continuity by using a Bézier F patch. F F F BLENDING Andrés Iglesias. See:

20 Rational Bézier surfaces If we introduce weights w ij to a nonrational Bézier surface, we obtain a rational Bézier surface: m n S(u, v) = i= j= m i= j= P ij w ij B m i (u)bn j (v) n w ij B m i (u)b n j (v).5.5 w ij = Example: Andrés Iglesias. See: ( ) ( ) 7.

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