Lecture_Chapter3_b

# Ecs 175 chapter 3 object representation 51 bzier

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Unformatted text preview: ints i=0 cubic Bernstein basis polynomials ￿￿ ni bin (u) = u (1 − u)n−i i ECS 175 Chapter 3: Object Representation 50 Bézier Curves •  Convex hull property ￿￿ ni bin (u) = u (1 − u)n−i i n ￿ bin (u) = 1 partition of unity i=0 0 ≤ bin (u) ≤ 1 Bézier polynomial is convex combination of control points; stays Within convex hull of control polygon. ECS 175 Chapter 3: Object Representation 51 Bézier Curves •  Subdivision – example: cubic Bézier curve −1 pk (u) = (1 − u) · pk−1 + u · pk−1 i i i f ( u ) = p3 ( u ) 3 cubic Bézier curve (p0 , p1 , p2 , p3 ) 0123 (p3 , p3 , p3 , p3 ) 3210 ECS 175 Chapter 3: Object Representation Cubic curve 1 Cubic curve 2 52 Bézier Curves •  Derivatives f ( u) = n ￿ i=0 bin (u) · pi Derivatives are tangents. ECS 175 Chapter 3: Object Representation Lighting computations require derivatives. Example: Utah Teapot consists of bicubic Bézier patches 53 Bézier Curves •  Derivatives f ( u) = n ￿ i=0 bin (u) · pi n n ￿ dbin (u) df (u) d￿ = bin (u) · pi = · pi du du i=0 du i=0 dbin (u) = n(bi−1,n−1 (u) − bi,n−1 (u)) du n df (u) ￿ = n(bi−1,n−1 (u) − bi,n−1 (u)) · pi du i=0 ECS 175 Chapter 3: Object Representation 54 Bézier Curves •  Derivatives n df (u) ￿ = n(bi−1,n−1 (u) − bi,n−1 (u)) · pi du i=0 n− 1 df (u) ￿ = bi,n−1 (u) · n(pi+1 − pi ) du i=0 Derivative is a Bézier curve of order (n-1) with “combined” control points ECS 175 Chapter 3: Object Representation 55 Bézier Curves •  Joining Bézier curves Discontinuous C0 f continuous f’ discontinuous C1 f continuous f’ continuous ECS 175 Chapter 3: Object Representation 56...
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