Lecture_Chapter3_c

33 b13 067 b13 1 b13 0 b13 033 b13 067 b13 1 b23

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Unformatted text preview: 3 (1) p0 b03 (0) p1 b03 (0.33) = p2 b03 (0.67) p3 b03 (1) ECS 175 f (0.33) = f1 Chapter 3: Object Representation b13 (0) b13 (0.33) b13 (0.67) b13 (1) b13 (0) b13 (0.33) b13 (0.67) b13 (1) b23 (0) b23 (0.33) b23 (0.67) b23 (1) b23 (0) b23 (0.33) b23 (0.67) b23 (1) b33 (0) p0 b33 (0.33) p1 b33 (0.67) p2 b33 (1) p3 − 1 b33 (0) f0 b33 (0.33) f1 b33 (0.67) f2 b33 (1) f3 64 Bézier Curves •  Summary: •  Control point approximation (Endpoint interpolation) •  Convex hull property/Bounding-box property •  Compact representation •  Subdivision for rendering •  Easy manipulation •  Affine invariance (affine transformation of control points results in affine transformation of Bézier curve) ECS 175 Chapter 3: Object Representation 65 B-Splines •  Joining Bézier curves is cumbersome •  Number of control points directly influences degree •  High degree: Unstable curves •  Solution: B-Splines are defined on local sets of points ECS 175 Chapter 3: Object Representation 66 B-Splines •  Solution: B-Splines are defined on local sets of points Given m + 1 control points Construct B-Spline with degree pi ∈ {p0 , . . . , pm } n Knot vector ui ∈ {u0 , . . . , um+n+1 } f ( u) = m ￿ i=0 ECS 175 Bin (u) · pi defined on Chapter 3: Object Representation defines parameter values [un , . . . , um+1 ] 67 B-Splines •  B-Spline base functions f ( u) = m ￿ i=0 Bin (u) · pi Bk 0 ( u) = ￿ Bkn (u) = u − uk uk+n+1 − u Bk,n−1 (u) + Bk+1,n−1 (u) uk+n − uk uk+n+1 − uk+1 ECS 175 1, 0, uk ≤ u ≤ uk+1 otherwise Chapter 3: Object Representation 68 B-Splines •  B-Spline base functions have local influence/support ￿ 1, uk ≤ u ≤ uk+1 Bk 0 ( u) = 0, otherwise m=0 Bkn (u) = u − uk uk+n+1 − u Bk,n−1 (u) + Bk+1,n−1 (u) uk+n − uk uk+n+1 − uk+1 n=1 ECS 175 Chapter 3: Object Representation 69 B-Splines •  Knot vector examples (for m = 3, n = 3) uniform knot vector [0, 1, 2, 3, 4, 5, 6, 7] B-Spline defined on u ∈ [3, 4] end knot multiplicity of n+1 leads to interpolation Non-uniform knot vector [0, 0, 0, 0, 1, 1, 1, 1] u ∈ [0, 1] B-Spline defined on cubic B-Spline identical to cubic Bézier curve ECS 175 Chapter 3: Object Representation 70 B-Spline Properties •  B-Spline has (local) convex hull property m ￿ Bin (u) = 1 i=0 0 ≤ Bin (u) ≤ 1 •  B-Spline is piecewise curve with local control •  Cn-k continuous at knot with multiplicity k •  Bézier curv...
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This document was uploaded on 03/12/2014 for the course ECS 175 at UC Davis.

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