CSCE 441 Keyframe Animation Smooth Curves Cont Jinxiang Chai Key frame Interpolation Given parameter values at key frames how to interpolate parameter values for inbetween frames t Key frame Interpolation Given parameter values at key frames how to interpolate parameter values for inbetween frames t Key frame Interpolation Given parameter values at key frames how to interpolate parameter values for inbetween frames t Nonlinear interpolation Review Natural cubic cruves a x t 3 bx t 2 cx t d x Qx t Review Natural cubic cruves a x t 3 bx t 2 cx t d x Qx t t 3 t2 ax bx t 1 Qx t c x d x Review Natural cubic cruves a x t 3 bx t 2 cx t d x Qx t t t t t t 3 3 1 3 2 3 3 3 4 ax bx t 1 Qx t c x d x t2 t1 2 t2 2 t3 2 t4 2 t1 1 a x Qx t1 t 2 1 bx Qx t 2 c Q t t 3 1 x x 3 t 4 1 d x Qx t 4 a x t13 t12 t1 3 2 bx t 2 t 2 t 2 c t 3 t 2 t x 33 3 2 3 d t x 4 t4 t4 1 1 1 1 1 Qx t1 Q x t 2 Q t x 3 Q t x 4 Review Natural cubic curves Q t t 3 t2 ax bx t 1 t3 c x d x t2 t13 t12 t1 3 t2 t2 2 t2 t 1 3 2 t t t3 3 3 t 3 t 2 t 4 4 4 1 1 1 1 1 Q t1 Q t 2 Q t 3 Q t 4 Review Hermite Curves P1 start position P4 end position R1 start derivative R4 end derivative R1 P1 P4 R4 Review Hermite Curves Q t t 3 t 2 1 p1 2 2 1 3 3 2 1 p4 t 1 0 0 1 0 R1 1 0 0 0 R4 R1 P1 P4 R4 Review Hermite Curves Q t t 3 t 2 1 p1 2 2 1 3 3 2 1 p4 t 1 0 0 1 0 R1 1 0 0 0 R4 Herminte basis matrix R1 P1 P4 R4 Review Hermite Curves Q t t 3 t 2 R1 1 p1 2 2 1 3 3 2 1 p4 t 1 0 0 1 0 R1 1 0 0 0 R4 P1 P 3 2 3 2 4 t 2t t t t R 1 R 4 Herminte basis matrix 2t 3 3t 2 1 2t 3 3t 2 P1 P4 R4 Review Hermite Curves Q t t 3 t 2 R1 1 p1 2 2 1 3 3 2 1 p4 t 1 0 0 1 0 R1 1 0 0 0 R4 2t 3 3t 2 1 2t 3 3t 2 P1 P1 P 3 2 3 2 4 t 2t t t t R 1 R 4 P4 R4 Review Hermite Curves Q t t 3 t 2 R1 1 p1 2 2 1 3 3 2 1 p4 t 1 0 0 1 0 R1 1 0 0 0 R4 2t 3 3t 2 1 2t 3 3t 2 P1 P1 P 3 2 3 2 4 t 2t t t t R 1 R 4 P4 R4 Review Hermite Curves Q t t 3 t 2 R1 1 p1 2 2 1 3 3 2 1 p4 t 1 0 0 1 0 R1 1 0 0 0 R4 2t 3 3t 2 1 2t 3 3t 2 P1 P1 P 3 2 3 2 4 t 2t t t t R 1 R 4 P4 R4 Review Hermite Curves Q t t 3 t 2 R1 1 p1 2 2 1 3 3 2 1 p4 t 1 0 0 1 0 R1 1 0 0 0 R4 2t 3 3t 2 1 2t 3 3t 2 P1 P1 P 3 2 3 2 4 t 2t t t t R 1 R 4 P4 R4 Review Hermite Curves Q t t 3 t 2 R1 1 p1 2 2 1 3 3 2 1 p4 t 1 0 0 1 0 R1 1 0 0 0 R4 2t 3 3t 2 1 2t 3 3t 2 P1 P1 P 3 2 3 2 4 t 2t t t t R 1 R 4 Hermite basis functions P4 R4 Review Hermite Curves Q t t 3 t 2 basis function 1 1 1 0 9 0 8 0 8 0 7 0 7 0 6 0 6 0 5 0 5 0 4 0 4 0 3 0 3 0 2 0 2 0 1 0 1 0 0 20 40 60 80 100 120 0 20 40 R4 P1 basis function 1 0 9 P4 P1 P 3 2 3 2 4 t 2t t t t R 1 R 4 2t 3 3t 2 1 2t 3 3t 2 0 R1 1 p1 2 2 1 3 3 2 1 p4 t 1 0 0 1 0 R1 1 0 0 0 R4 60 80 100 basis function 1 120 basis function 1 0 16 0 0 14 0 02 0 12 0 04 0 1 0 06 0 08 0 08 0 06 0 1 0 04 0 12 0 02 0 14 0 0 20 40 60 80 100 120 0 16 0 20 40 60 80 100 120 Review Hermite Curves Q t t 3 t 2 R1 1 p1 2 2 1 3 3 2 1 p4 t 1 0 0 1 0 R1 1 0 0 0 R4 P4 R4 P1 P1 P 3 2 3 2 4 t 2t t t t R 1 R 4 2t 3 3t 2 1 2t 3 3t 2 P1 2t 3 3t 2 1 P4 2t 3 3t 2 R1 t 3 2t 2 t R4 t 3 t 2 1 1 P1 0 9 0 8 0 7 0 7 0 6 0 5 0 5 0 4 0 4 0 3 0 3 0 2 0 2 0 1 0 1 0 0 20 40 60 80 P4 0 8 0 6 0 0 16 0 9 100 120 0 20 40 60 80 100 120 0 R1 0 14 0 12 0 04 0 1 0 06 0 08 0 08 0 06 0 1 0 04 0 12 0 02 0 14 0 0 20 40 60 80 100 120 R4 0 02 0 16 0 20 40 60 80 100 120 Review Bezier Curves R1 3 v1 v0 R2 3 v3 v2 P1 v0 P4 v3 Review Bezier Curves Q t t 3 t2 1 3 3 3 6 3 t 1 3 3 0 1 0 0 1 v0 0 v1 0 v2 0 v3 Review Bezier Curves Q t t 3 t2 1 3 3 3 6 3 t 1 3 3 0 1 0 0 t 3 3t 2 3t 1 3t 3 6t 2 3t …