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Bézier Curves: Integrating Math, Arts and Technology. Jomar F. Rabajante UPLB. Parametric Curves. Parametric Curves. Parametric Curves. Parametric Curves. Parametric Curves. Widely used in vector graphics and computer-aided designs Example of Parametric Curve: Bézier curve
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Bézier Curves: Integrating Math, Arts and Technology Jomar F. Rabajante UPLB
Parametric Curves • Widely used in vector graphics and computer-aided designs • Example of Parametric Curve: Béziercurve • Affine transformations on the curve can be done by just manipulating the “control points”
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:
Bézier Curves • We can generate a curve using the parametric form (Bernstein representation): Familiar?
Bézier Curves • For 3 points (Quadratic Bézier): Notice that if t=0 we get (x0,y0). If t=1 we get (x2,y2). As t takes on values between 0 & 1, a curve is traced but it may not pass through the central point.
Bézier Curves • For 4 points (Cubic Bézier):
TO DO: The following control points are used: .
Bézier Curves • 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 convexiff the line segment between any two points in the set lies entirely in the set. Examples of convex hull of four points:
Bézier Curves • 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.
de Casteljau’s Algorithm • Independently made by Paul de Faget de Casteljau to generate Bézier curves. • Uses barycenter coordinates. • Let’s use Geogebra
Bézier Curves: Integrating Math, Arts and Technology Jomar F. Rabajante UPLB
