Subdivision Curves Dr Scott Schaefer 1 What is
Subdivision Curves Dr. Scott Schaefer 1
What is subdivision? Set of rules S that take a curve as input and produce a more highly refined curve as output n Recursively applying S yields a sequence of curves which should converge to some limit shape n 2/96
Subdivision Rules Typically chosen to be linear combinations of neighboring vertices n Rules usually depend only on local topology of shape n Reposition Old Vertices Insert New Vertices 3/96
Applying Subdivision Rules 4/96
Applying Subdivision Rules 5/96
Applying Subdivision Rules 6/96
Applying Subdivision Rules 7/96
Applying Subdivision Rules 8/96
Applying Subdivision Rules 9/96
Applying Subdivision Rules 10/96
Applying Subdivision Rules 11/96
Applying Subdivision Rules 12/96
Applying Subdivision Rules 13/96
Applying Subdivision Rules 14/96
Applying Subdivision Rules 15/96
Subdivision Rules Via Blossoming Assume knot-spacing uniform 0, 2, 4, 6, 8, … n Find control points for refined knots 0, 1, 2, 3, … n 16/96
Subdivision Rules Via Blossoming Assume knot-spacing uniform 0, 2, 4, 6, 8, … n Find control points for refined knots 0, 1, 2, 3, … n Given Find 17/96
Subdivision Rules Via Blossoming Assume knot-spacing uniform 0, 2, 4, 6, 8, … n Find control points for refined knots 0, 1, 2, 3, … n Given Find 18/96
Subdivision Rules Via Blossoming Assume knot-spacing uniform 0, 2, 4, 6, 8, … n Find control points for refined knots 0, 1, 2, 3, … n Given Find 19/96
Subdivision Rules Via Blossoming Assume knot-spacing uniform 0, 2, 4, 6, 8, … n Find control points for refined knots 0, 1, 2, 3, … n Given Find Works for arbitrary degree B-splines!!! 20/96
Lane Reisenfeld Subdivision Linearly subdivide the curve by inserting the midpoint on each edge n Perform averaging by replacing each edge by its midpoint d times n 21/96
Lane Reisenfeld Subdivision 22/96
Lane Reisenfeld Subdivision 23/96
Lane Reisenfeld Subdivision 24/96
Lane Reisenfeld Subdivision 25/96
Lane Reisenfeld Subdivision 26/96
Lane Reisenfeld Subdivision 27/96
Lane Reisenfeld Subdivision 28/96
Lane Reisenfeld Subdivision 29/96
Lane Reisenfeld Subdivision 30/96
Interpolatory Subdivision Interpolating control vertices may be desirable n Catmull-Rom splines are not refinable!!! n Reposition Old Vertices Insert New Vertices 31/96
Four-Point Subdivision 32/96
Four-Point Subdivision 33/96
Four-Point Subdivision 34/96
Four-Point Subdivision 35/96
Four-Point Subdivision 36/96
Four-Point Subdivision 37/96
Four-Point Subdivision 38/96
Four-Point Subdivision 39/96
Four-Point Subdivision 40/96
Four-Point Subdivision 41/96
Four-Point Subdivision 42/96
Four-Point Subdivision 43/96
Four-Point Subdivision 44/96
Four-Point Subdivision 45/96
Four-Point Subdivision 46/96
Four-Point Subdivision 47/96
Four-Point Subdivision 48/96
Subdivision as Basis Function Refinement 49/96
Subdivision as Basis Function Refinement Cubic B-spline Basis Function 50/96
Subdivision as Basis Function Refinement Four-Point Basis Function 51/96
Subdivision as Basis Function Refinement 52/96
Subdivision as Basis Function Refinement 53/96
Subdivision as Basis Function Refinement 54/96
Subdivision as Basis Function Refinement 55/96
Subdivision as Basis Function Refinement 56/96
Subdivision as Basis Function Refinement 57/96
Subdivision as Basis Function Refinement 58/96
Subdivision as Basis Function Refinement 59/96
Limit Points of Curve Subdivision 60/96
Limit Points of Curve Subdivision 61/96
Limit Points of Curve Subdivision 62/96
Limit Points of Curve Subdivision 63/96
Limit Points of Curve Subdivision 64/96
Limit Points of Curve Subdivision 65/96
Limit Points of Curve Subdivision Symmetry!!! 66/96
Limit Points of Curve Subdivision 67/96
Limit Points of Curve Subdivision Zero!!! 68/96
Limit Points of Curve Subdivision 69/96
Limit Points of Curve Subdivision 70/96
Limit Points of Curve Subdivision 71/96
Limit Points of Curve Subdivision 72/96
Limit Points of Curve Subdivision 73/96
Limit Points of Curve Subdivision 74/96
Limit Points of Curve Subdivision 75/96
Limit Points of Curve Subdivision Evaluate scaling relationship at n Solve linear system of equations with constraint n 76/96
Limit Points of Curve Subdivision 77/96
Limit Points of Curve Subdivision Assume N(x) has finite support n Let y be the smallest parameter such that N(y)=0 and for all x>y N(x)=0 n 78/96
Limit Points of Curve Subdivision Assume N(x) has finite support n Let y be the smallest parameter such that N(y)=0 and for all x>y N(x)=0 n 79/96
Limit Points of Curve Subdivision Assume N(x) has finite support n Let y be the smallest parameter such that N(y)=0 and for all x>y N(x)=0 n 80/96
Limit Points of Curve Subdivision Assume N(x) has finite support n Let y be the smallest parameter such that N(y)=0 and for all x>y N(x)=0 n 81/96
Limit Points of Curve Subdivision 82/96
Limit Points of Curve Subdivision 83/96
Limit Points of Curve Subdivision 84/96
Limit Points of Curve Subdivision Limit mask is left-eigenvector corresponding to 1 85/96
Limit Points of Curve Subdivision 86/96
Limit Points of Curve Subdivision 87/96
Derivatives of Subdivision Curves 88/96
Derivatives of Subdivision Curves 89/96
Derivatives of Subdivision Curves 90/96
Derivatives of Subdivision Curves left-eigenvector corresponding to 1/2 91/96
Derivatives of Subdivision Curves 92/96
Four-Point Limits and Derivatives 93/96
Four-Point Limits and Derivatives 94/96
Four-Point Limits and Derivatives 95/96
Four-Point Limits and Derivatives 96/96
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