Bezier Curves and Surfaces (Redbook ch12)

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Parametric representation Recall a 1D curve in 3D can be represented as:[ xup u= yuzu]xp u=[ ' 'y ' z u] ' p'(u) is the tangent (velocity) vector● Usually limit u to interval [0,1] for simplicity For surfaces we have two parameters (u, v):pu ,v=[xu , v]yu , vzu , v∂ p[ ∂ x/∂u]∂ u u , v= ∂ y/∂ u∂ z/∂ u∂ p[ ∂ x/∂v]∂ v u , v= ∂ y/∂ v∂ z/∂ vCMPT370: <strong>Bezier</strong> curves 27 Mar 20074
Polynomial curves Restrict the functions x(u), y(u), z(u) to bepolynomial (of degree n) in u:● Each coefficient c kis a 3-vector● u k are the n+1 basis functions● Often choose n=3: cubic polynomial k=0..3, (x,y,z): need 12 numbers Similarly for surfaces:np u ,v=∑j=0np u=∑k =0n∑k =0c ku kc jku j v kCMPT370: <strong>Bezier</strong> curves 27 Mar 20075
Page 1 and 2: Bezier Curves and Surfaces(Redbook
Page 3: What's on for today Polynomial curv
Page 7 and 8: Geometry matrix Invert this matrix
Page 9 and 10: Hermite polynomial curves Another w
Page 11 and 12: Bezier curves Widely used, provided
Page 13 and 14: PR EFACIO.cuentran continuamente en

Bezier Curves and Surfaces (Redbook ch12)

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