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Properties of Bezier Curves

Properties of Bezier Curves. Invariance under affine parameter transformation P i B i,n (u) = P i B i,n ((u –a)/(b-a)) Invariance under barycentric combinations (weighted average): ( Q i + R i )B i,n (u) = Q i B i,n (u) + R i B i,n (u), + = 1

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Properties of Bezier Curves

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  1. Properties of Bezier Curves • Invariance under affine parameter transformation PiBi,n (u) = PiBi,n ((u –a)/(b-a)) • Invariance under barycentric combinations (weighted average): ( Qi + Ri)Bi,n (u) = Qi Bi,n (u) + RiBi,n(u), + = 1 • Pseudo-local control: Bi,n (u)has a max at u = i/n. If we move the control point Pi , then the curve is most affected in the region around the parameter value i/n. , Dinesh Manocha, COMP258

  2. Derivatives of Bezier Curve • Derivative of a Bezier curve: P(u) = n PiBi,n-1 (u) = P’(u), where Pi = Pi+1 - Pi. P’(u) is alsocalled the hodograph curve • Higher order derivatives can also be defined in terms of lower order Bezier curves • Based on the derivatives, we can place constraints on the control points for C1 or G1 continuity. Dinesh Manocha, COMP258

  3. Degree Elevation • Geometric representation of a degree n curve in terms of n+1 degree curve • Compute the control points (Pi) of the elevated curve PiBi,n+1 (u) =PiBi,n (u) where Pi = (i/(n+1))Pi-1 + (1 -i/(n+1))Pi, where i=0,….,n+1 , • What happens if degree elevation is applied repeatedly? Dinesh Manocha, COMP258

  4. Truncating a Bezier Curve • Truncation and subsequent reparametrization: Given a Bezier curve, find the new set of control points of a Bezier curve that define a segment of this curve in the parametric interval: u [ui, uj] • Subdivision: Given a Bezier curve, P(u), subdivide at a parameter value ui. Compute the control points of two Bezier curves: P1(s) and P2(t), so that P1(s), s [0,1] corresponds to P(u), u [0,ui], and P2(t), t [0,1] corresponds to P(u), u [ui,1]. • Subdivision can be used to truncate a curve. The control points of the subdivided curve are computed using de Casteljau’s algorithm. Dinesh Manocha, COMP258

  5. Subdividing a Bezier Curve • Subdivision doesn’t change the shape of a Bezier curve • It can be used for local refinement: subdivide a curve and change the control point(s) of one of the subdivided curve • The union of convex hulls of the subdivided curve is a subset of the convex hull of the original curve (i.e. the convex hulls are a better approximation of the Bezier curve). • Asymptotically the control polygons of the subdivided curve converge to the actual curve (at a quadratic rate) • Subdivision and convex hulls are frequently used for intersection computations Dinesh Manocha, COMP258

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