CSCE 441 Computer Graphics Keyframe Animation Smooth Curves Jinxiang Chai Outline Keyframe interpolation Curve representation and interpolation natural cubic curves Hermite curves Bezier curves Required readings HB 8 8 8 9 8 10 Computer Animation Animation making objects moving Compute animation the production of consecutive images which when displayed convey a feeling of motion Animation Topics Rigid body simulation bouncing ball millions of chairs falling down Animation Topics Rigid body simulation bouncing ball millions of chairs falling down Natural phenomenon water fire smoke mud etc Animation Topics Rigid body simulation bouncing ball millions of chairs falling down Natural phenomenon water fire smoke mud etc Character animation articulated motion e g full body animation deformation e g face Animation Topics Rigid body simulation bouncing ball millions of chairs falling down Natural phenomenon water fire smoke mud etc Character animation articulated motion e g full body animation deformation e g face Cartoon animation Animation Criterion Physically correct rigid body simulation natural phenomenon Natural character animation Expressive cartoon animation Keyframe Animation Keyframe Interpolation 3 c3 2 A c2 c1 t 0 1 t 50ms What s the inbetween motion Outline Process of keyframing Key frame interpolation Hermite and bezier curve Splines Speed control 2D Animation Highly skilled animators draw the key frames Less skilled lower paid animators draw the inbetween frames Time consuming process Difficult to create physically realistic animation 3D Animation Animators specify important key frames in 3D Computers generates the in between frames Some dynamic motion can be done by computers hair clothes etc Still time consuming Pixar spent four years to produce Toy Story The Process of Keyframing Specify the keyframes Specify the type of interpolation linear cubic parametric curves Specify the speed profile of the interpolation constant velocity ease in ease out etc Computer generates the in between frames A Keyframe In 2D animation a keyframe is usually a single image In 3D animation each keyframe is defined by a set of parameters Keyframe Parameters What are the parameters position and orientation body deformation facial features hair and clothing lights and cameras Outline Process of keyframing Key frame interpolation Hermite and bezier curve Splines Speed control Inbetween Frames Linear interpolation Cubic curve interpolation Keyframe Interpolation 3 c3 2 A c2 c1 t 0 1 t 50ms Linear Interpolation Linearly interpolate the parameters between keyframes x1 x x0 t0 t t1 Linear Interpolation Limitations Requires a large number of key frames when the motion is highly nonlinear y t Cubic Curve Interpolation We can use a cubic function to represent a 1D curve Qx t a x t 3 bx t 2 c x t d x Smooth Curves Controlling the shape of the curve 2 Qx t 1 t t t 3 Smooth Curves Controlling the shape of the curve 2 Qx t 3 t t t 3 Smooth Curves Controlling the shape of the curve 2 Q x t 1 t t t 3 Smooth Curves Controlling the shape of the curve 2 Qx t 1 t t t 3 Smooth Curves Controlling the shape of the curve 2 Qx t 1 t t t 3 Smooth Curves Controlling the shape of the curve 2 Qx t 1 t 3t t 3 Constraints on the cubics How many constraints do we need to determine a cubic curve a x t 3 bx t 2 c x t d x Qx t Constraints on the Cubic Functions How many constraints do we need to determine a cubic curve a x t 3 bx t 2 c x t d x Qx t t 3 t2 ax bx t 1 Qx t c x d x Constraints on the Cubic Functions How many constraints do we need to determine a cubic curve 4 a x t 3 bx t 2 c x t d x Qx t t t13 3 t2 3 t3 t 3 4 t1 2 t2 2 t3 2 t4 2 3 t2 ax bx t 1 Qx t c x d x 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 Constraints on the Cubic Functions How many constraints do we need to determine a cubic curve 4 a x t 3 bx t 2 c x t d x Qx t t t13 3 t2 3 t3 t 3 4 t1 2 t2 2 t3 2 t4 2 3 t2 ax bx t 1 Qx t c x d x 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 Constraints on the Cubic Functions How many constraints do we need to determine a cubic curve 4 Natural Cubic Curves Q t t 3 t2 ax bx t 1 t3 c x d x t2 Q t1 Q 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 Q t3 Q t4 1 1 1 1 1 Q t1 Q t 2 Q t 3 Q t 4 Interpolation Find a polynomial that passes through specified values 2 y t a bt ct dt 3 y t Interpolation Find a polynomial that passes through specified values 2 3 y t a bt ct dt y y 0 a 3 y 1 a b c d 1 y 2 a 2b 4c 8d 3 y 3 a 3b 9c 27 d 1 t Interpolation Find a polynomial that passes through specified values 2 y t a bt ct dt 1 1 1 1 3 0 a 3 1 b 1 2 4 8 c 3 3 9 27 d 1 y 0 0 1 1 t Interpolation Find a polynomial that passes through specified values 2 y t a bt ct dt a 3 b 20 3 c 6 d 1 3 3 y t Interpolation Find a polynomial that passes through specified values y t 2D Trajectory Interpolation Perform interpolation for each component separately Combine result to obtain parametric curve y p t x t y t x 2D Trajectory Interpolation Perform interpolation for each component separately Combine result to obtain parametric curve y p t x t y t x 2D Trajectory Interpolation Perform interpolation for each component separately Combine result to obtain parametric curve y p t x t y t x Constraints on the Cubic Curves How many constraints do we need to determine a cubic curve 4 does not provide local control of the curve Constraints on the Cubic Curves How many constraints do we need to determine a cubic curve 4 does not provide local control of the curve Redefine C as a product of the basis matrix M and the control vector …