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: 12-6 & , & 14-7

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 looking - character animation Expressive - cartoon animation

Keyframe Animation

Keyframe Interpolation Key frame interpolation in after effects (Click here)here

Spatial Key Framing Demo video (click here)here

Keyframe Interpolation IK can be used to create Key poses t=0 t=50ms

Keyframe Interpolation What’s the inbetween motion? t=0 t=50ms

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 in- between frames Time consuming process Difficult to create physically realistic animation

Animating a Bouncing Ball Key frames

Animating Three Walking Steps Key frames

Playing Basketball

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 producing Toy Story

3D Bouncing Ball

3D Jumping

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 t=0 t=50ms

Linear Interpolation Linearly interpolate the parameters between keyframes t x t0t0 t1t1 x0x0 x1x1

Linear Interpolation: Limitations Requires a large number of key frames when the motion is highly nonlinear.

Cubic Curve Interpolation We can use a cubic function to represent a 1D curve

Smooth Curves Controlling the shape of the curve

Smooth Curves Controlling the shape of the curve

Smooth Curves Controlling the shape of the curve

Smooth Curves Controlling the shape of the curve

Smooth Curves Controlling the shape of the curve

Smooth Curves Controlling the shape of the curve

Constraints on the cubics How many points do we need to determine a cubic curve?

Constraints on the Cubic Functions How many points do we need to determine a cubic curve?

Constraints on the Cubic Functions How many points do we need to determine a cubic curve? 4

Constraints on the Cubic Functions How many points do we need to determine a cubic curve? 4

Natural Cubic Curves Q(t 1 )Q(t 2 )Q(t 3 ) Q(t 4 )

Interpolation Find a polynomial that passes through specified values

Interpolation Find a polynomial that passes through specified values

Interpolation Find a polynomial that passes through specified values

Interpolation Find a polynomial that passes through specified values

2D Trajectory Interpolation Each point on the trajectory is associated with a time stamp t. Perform interpolation for each component separately Combine result to obtain parametric curve

2D Trajectory Interpolation Each point on the trajectory is associated with a time stamp t. Perform interpolation for each component separately Combine result to obtain parametric curve

2D Trajectory Interpolation Each point on the trajectory is associated with a time stamp t. Perform interpolation for each component separately Combine result to obtain parametric curve

Limitation? What’s the main limitation of interpolation using natural cubic curves?

Limitation? What’s the main limitation of interpolation using natural cubic curves? - does not provide local control of the curve

Constraints on the Cubic Curves What’s the main limitation of interpolation using natural cubic curves? - does not provide local control of the curve

Constraints on the Cubic Curves What’s the main limitation of interpolation using natural cubic curves? - does not provide local control of the curve Changes of any control point will change the shape of the whole curve!

Constraints on the Cubic Curves What’s the main limitation of interpolation using natural cubic curves? - does not provide local control of the curve Redefine C as a product of the basis matrix M and the control vector G: C= MG

Constraints on the cubic curves What’s the main limitation of interpolation using natural cubic curves? - does not provide local control of the curve Redefine C as a product of the basis matrix M and the control vector G: C= MG MG

Constraints on the Cubic Curves What’s the main limitation of interpolation using natural cubic curves? - does not provide local control of the curve Redefine C as a product of the basis matrix M and the control vector G: C= MG M G

Constraints on the Cubic Curves What’s the main limitation of interpolation using natural cubic curves? - does not provide local control of the curve Redefine C as a product of the basis matrix M and the control vector G: C= MG M? G?

Two Issues How to select a new set of control points for local control of the curve? How to determine M? M?

Outline Process of keyframing Key frame interpolation Hermite and bezier curve Splines Speed control

Hermite Curve A Hermite curve is determined by - endpoints P 1 and P 4 - tangent vectors R 1 and R 4 at the endpoints P1P1 R1R1 P4P4 R4R4

Hermite Curve A Hermite curve is determined by - endpoints P 1 and P 4 - tangent vectors R 1 and R 4 at the endpoints Use these elements to control the curve, i.e. construct control vector P1P1 R1R1 P4P4 R4R4 MhMh GhGh 0<=t<=1

Hermite Basis Matrix Given desired constraints: - endpoints meet P 1 and P 4 Q(0) = [ ] · M h · G h = P 1 Q(1) = [ ] · M h · G h = P 4 - tangent vectors meet R 1 and R 4

Tangent Vectors

Hermite Basis Matrix Given desired constraints: - endpoints meet P 1 and P 4 Q(0) = [ ] · M h · G h = P 1 Q(1) = [ ] · M h · G h = P 4 - tangent vectors meet R 1 and R 4 Q’(0) =[ ] · M h · G h =R 1 Q’(1) =[ ] · M h · G h =R 4

Hermite Basis Matrix Given desired constraints: - endpoints meet P 1 and P 4 Q(0) = [ ] · M h · G h = P 1 Q(1) = [ ] · M h · G h = P 4 - tangent vectors meet R 1 and R 4 Q’(0) =[ ] · M h · G h =R 1 Q’(1) =[ ] · M h · G h =R 4 So how to compute the basis matrix M h ?

Hermite Basis Matrix Taking them together Q(0) = [ ] · Mh · Gh = P1 Q(1) = [ ] · Mh · Gh = P4 Q’(0) = [ ] · Mh · Gh = R1 Q’(1) = [ ] · Mh · Gh = R4

Hermite Basis Matrix We can solve for basis matrix M h MhMh

Hermite Basis Matrix We can solve for basis matrix M h MhMh

Hermite Basis Matrix P1P1 R1R1 P4P4 R4R4

Bezier Curve Indirectly specify tangent vectors by specifying two intermediate points P1P1 P3P3 P2P2 P4P4

Bezier Curve Indirectly specify tangent vectors by specifying two intermediate points -The curve starts at P1 going towards P2 and Arrives at P4 coming from the direction P3 - usually does not pass through P2 or P3; these points are only there to provide directional information. P1P1 P3P3 P2P2 P4P4

Bezier Curve Indirectly specify tangent vectors by specifying two intermediate points P1P1 P3P3 P2P2 P4P4

Bezier Curve Indirectly specify tangent vectors by specifying two intermediate points How to compute the basis matrix M b ? P1P1 P3P3 P2P2 P4P4

Bezier Basis Matrix Establish the relation between Hermite and Bezier control vectors

Bezier Basis Matrix Establish the relation between Hermite and Bezier control vectors

Bezier Basis Matrix Establish the relation between Hermite and Bezier control vectors M hb GbGb

Bezier Basis Matrix For Hermite curves, we have For Bezier curves, we have

Bezier Basis Matrix For Hermite curves, we have For Bezier curves, we have

Bezier Basis Matrix P1P1 P3P3 P2P2 P4P4

Hermite Basis Function Let’s define B as a product of T and M B h (t) indicates the weight of each element in G h

Hermite Basis Function Let’s define B as a product of T and M B h (t) indicates the weight of each element in G h What’s function of this red curve?

Hermite Basis Function Let’s define B as a product of T and M B h (t) indicates the weight of each element in G h What’s function of this red curve? 2t 3 -3t 2 +1

Bezier Basis Functions Bezier blending functions are also called Bernstein polynomials

Bezier Basis Functions Bezier blending functions are also called Bernstein polynomials What’s function of this red curve?

Bezier Basis Functions Bezier blending functions are also called Bernstein polynomials What’s function of this red curve? B 1 = -t 3 +3t 2 -3t+1

Bezier Basis Functions Bezier blending functions are also called Bernstein polynomials What’s function of this red curve? B 2 = -t 3 +3t 2 -3t+1

Bezier Basis Functions Bezier blending functions are also called Bernstein polynomials What’s function of the Bezier curve? B 1 P 1 +B 2 P 2 +B 3 P 3 +B 4 P 4

How to interpolate a 3D curve x y z o

x y z o Bezier curve

Try this online at - Move the interpolation point, see how the others (and the point on curve) move - Control points (can even make loops) Bezier java applet

Cubic curves: Hermite curves: Bezier curves: Different basis functions

Complex curves Suppose we want to draw a more complex curve

Complex curves Suppose we want to draw a more complex curve How can we represent this curve?

Complex curves Suppose we want to draw a more complex curve Why not use a high-order Bézier? - Wiggly curves - No local control

Complex curves Suppose we want to draw a more complex curve Why not use a high-order Bézier? - Wiggly curves - No local control Instead, we’ll splice together a curve from individual segments that are cubic Béziers

Complex curves Suppose we want to draw a more complex curve Why not use a high-order Bézier? - Wiggly curves - No local control Instead, we’ll splice together a curve from individual segments that are cubic Béziers

Complex curves Suppose we want to draw a more complex curve Why not use a high-order Bézier? - Wiggly curves - No local control Instead, we’ll splice together a curve from individual segments that are cubic Béziers

Complex curves Suppose we want to draw a more complex curve Why not use a high-order Bézier? - Wiggly curves - No local control Instead, we’ll splice together a curve from individual segments that are cubic Béziers Why cubic? - Lowest dimension with control for the second derivative - Lowest dimension for non-planar polynomial curves

Next lecture Spline curve and more key frame interpolation