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

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

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

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xy 101 213 28 132 t 0 1 2 3

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Parametric Curves

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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

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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:

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

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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

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Source: Wikipedia

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For 4 points (Cubic Bézier): Bézier Curves

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

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TO DO:

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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

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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

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

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

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