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BezierCurves

BezierCurves - Bzier Curves e James Keesling 1 Denition of...

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ezier Curves James Keesling 1 Definition of a B´ ezier Curve A B´ ezier curve is a parametric curve that is used widely in computer graphics. They were originally used as an aid in design in the automobile industry. The B´ ezier curves are based on the Bernstein polynomials. The Bernstein polynomial of degree n is given by the following formula. B n ( t ) = n X i =0 n i t i · (1 - t ) n - i In themselves the Bernstein polynomials are not very interesting. They are just the binomial expansion of ( t + (1 - t )) n 1. However, for B´ ezier curves we will restrict t to the interval [0 , 1]. Then the individual terms in the expansion will be non-negative and will sum to one. We will use them to produce a convex combination of points in R k . Let { P 0 , P 1 , . . . , P n } be a set of n + 1 points in R k . The B´ ezier curve with control points { P 0 , P 1 , . . . , P n } is given by the following formula. B ( t ) = n X i =0 n i t i · (1 - t ) n - i · P i The curve begins at the point P 0 and ends at point P n , but will not generally go through the other control points.

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BezierCurves - Bzier Curves e James Keesling 1 Denition of...

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