Subdivision Subdivision Bezier Curves (Recall) Subdivision using - PowerPoint PPT Presentation

Subdivision

Subdivision Bezier Curves (Recall) Subdivision using control points yields the curve

Curves Bezier Curves Subdivision b 2 b 1 Curve P(t) 0 t 1 is ≤ ≤ subdivided into two curves Q(u) 0 u 1 • ≤ ≤ R(v) 0 v 1 • ≤ ≤ R(v) Q(u) b 3 b 0

Curves Bezier Curves Subdivision Using de Casteljau Algorithm b 2 c 0 =b 0 b 1 c 1 =(b 0 +b 1 )/2 d 1 c 3 =d 0 c 2 =(b 0 +2b 1 +b 2 )/4 c 2 d 2 c 3 =(b 0 +3b 1 +3b 2 +b 3 )/8 c 1 d 0 =c 3 R(v) Q(u) d 1 =(b 1 +2b 2 +b 3 )/4 d 2 =(b 2 +b 3 )/2 b 0 =c 0 d 3 =b 3 b 3 =d 3

Curves Bezier Curves Subdivision d 1 c 3 =d 0 c 2 d 2 c 1 R(v) Q(u) c 0 d 3

Subdivision (Curves) Idea: [Zorin and Schröder,2000] �

Subdivision (Surfaces) Idea: [Zorin and Schröder,2000] �

Subdivision Limit Surface Input Mesh - Output refined (continuous) Mesh M i = f(M i ) S = lim(M i ) [DeRose et al.,1998]

Subdivision • Do away with the explicit parametric representation • Base a curve or surface solely on its control points and their connectivity. • Provide a simple mechanism which produces a larger, more refined set of control points from the current set. • Iterate refinement until the appropriate level of detail is achieved.

Subdivision Curves • Set of rules S that take a curve as input and produce a more highly refined curve as output • Recursively applying S yields a sequence of curves which should converge to some limit shape

Subdivision Rules • Typically chosen to be linear combinations of neighboring vertices • Rules usually depend only on local topology of shape 3 1 1 4 2 2 1 1 8 8

Subdivision Curves For Example S S S

Example

Example 1 1 2 2

Example 3 1 4 8 1 8

Example 1 2 1 2

Example 1 8 3 1 4 8

Example 1 1 2 2

Example 1 8 3 1 4 8

Example 1 2 1 2

Example 3 1 4 8 1 8

Example

Example

Example

Subdivision Chaiken ’ s Algorithm [Real Time Rendering]

Subdivision Chaiken ’ s Algorithm Approximation using initial control points Basically follow ¾ ¼ rule C1 continuous Limiting curve is quadratic B-spline Extension for surfaces: Doo-Sabin Subdivision [Real Time Rendering]

Subdivision Interpolating Curve: Dyn et. al 1987 [Real Time Rendering]

Subdivision Interpolating Curve: Dyn et. al 1987 P 2i k+1 = P i k P 2i+1 k+1 = (1/2 + w) (P i k + P i+1 k ) – w (P i-1 k + P i+2 k ) w: tension parameter [Real Time Rendering]

Subdivision: Surfaces Loop Subdivision [Zorin and Schröder,2000] �

Subdivision: Surfaces Loop Subdivision Rules [Zorin and Schröder,2000] �

Subdivision: Surfaces Loop Subdivision Rules New Vertex Updated Vertex Vertex Mask Edge Mask [Zorin and Schröder,2000] �

Subdivision: Surfaces Catmull Clark Subdivision • Introduce new points - At face centre - At mid edges • Adjust position of original points • Repeat until sufficient details [Zorin and Schröder,2000] �

Subdivision: Surfaces Catmull Clark Subdivision Face � Edge � Update �

Subdivision: Surfaces Some Results Loop Catmull Clark [Zorin and Schröder,2000] �

Subdivision: Surfaces Some Results Loop Catmull Clark [Zorin and Schröder,2000] �

Subdivision: Surfaces Manipulation/Deformation [Zorin and Schröder,2000] �

Subdivision: Surfaces Very popular in Computer Graphics: Pixar ’ s: Geri ’ s Game 1998 Monster Inc 2000