1. Bézier Curves: Integrating Math, Arts and Technology Jomar F. Rabajante UPLB

2. Parametric Curves x 10 21 2 13 t 0 1 2 3

3. Parametric Curves y 1 3 8 2 t 0 1 2 3

4. xy 101 213 28 132 t 0 1 2 3

5. Parametric Curves

7. Widely used in vector graphics and computer- aided designs Example of Parametric Curve: Bézier curve Affine transformations on the curve can be done by just manipulating the control points Parametric Curves

8. Bézier Curves Named after the French engineer Pierre Bézier of the Renault Automobile Company. Free form curves Suppose we are given a set of control/Bézier points:

9. We can generate a curve using the parametric form (Bernstein representation): Familiar? Bézier Curves

10. For 3 points (Quadratic Bézier): Notice that if t=0 we get (x 0,y 0 ). If t=1 we get (x 2,y 2 ). As t takes on values between 0 & 1, a curve is traced but it may not pass through the central point. Bézier Curves

11. Source: Wikipedia

12. For 4 points (Cubic Bézier): Bézier Curves

13. You can use MS Excel, GraphCalc or any graphing software…

19. TO DO:

20. The Bézier curve lies entirely inside the convex hull containing all the control points. Convex hull of a set of points is the smallest convex set that contains the points. A set is convex iff the line segment between any two points in the set lies entirely in the set. Examples of convex hull of four points: Bézier Curves

21. Some curves that seem simple, such as the circle, cannot be described exactly by a Bézier or piecewise Bézier curve; RATIONAL BEZIER curves can do this. Bézier Curves

22. de Casteljaus Algorithm Independently made by Paul de Faget de Casteljau to generate Bézier curves. Uses barycenter coordinates. Lets use Geogebra

23. Bézier Curves: Integrating Math, Arts and Technology Jomar F. Rabajante UPLB