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 - articulated motion, e.g. full-body animation 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 - articulated motion, e.g. full-body animation 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 - rigid body-simulation - character animation Physically correct - rigid body-simulation - natural phenomenon Natural - character animation Expressive - cartoon animation

Keyframe Animation

Keyframe Interpolation t=50ms t=0 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 in-between 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 t=50ms t=0

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.

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 constraints do we need to determine a cubic curve?

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

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

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

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

Natural Cubic Curves Q(t1) Q(t2) Q(t3) Q(t4)

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

Interpolation Find a polynomial that passes through specified values

2D Trajectory Interpolation Perform interpolation for each component separately Combine result to obtain parametric curve

2D Trajectory Interpolation Perform interpolation for each component separately Combine result to obtain parametric curve

2D Trajectory Interpolation Perform interpolation for each component separately Combine result to obtain parametric 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

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 G: C= MG

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 G: C= MG MG

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 G: C= MG M G

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 G: C= MG M? G?

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

Hermite Curve - endpoints P1 and P4 A Hermite curve is determined by - tangent vectors R1 and R4 at the endpoints R1 P4 R4 P1

Hermite Curve - endpoints P1 and P4 A Hermite curve is determined by - tangent vectors R1 and R4 at the endpoints Use these elements to control the curve, i.e. construct control vector R1 P4 R4 P1 Mh Gh

Hermite Basis Matrix - endpoints meet P1 and P4 Given desired constraints: - endpoints meet P1 and P4 Q(0) = [0 0 0 1 ] · Mh · Gh = P1 Q(1) = [1 1 1 1 ] · Mh · Gh = P4 - tangent vectors meet R1 and R4

Tangent Vectors

Tangent Vectors

Hermite Basis Matrix - endpoints meet P1 and P4 Given desired constraints: - endpoints meet P1 and P4 Q(0) = [0 0 0 1 ] · Mh · Gh = P1 Q(1) = [1 1 1 1 ] · Mh · Gh = P4 - tangent vectors meet R1 and R4 Q’(0) =[0 0 1 0] · Mh · Gh =R1 Q’(1) =[3 2 1 0] · Mh · Gh =R4

Hermite Basis Matrix - endpoints meet P1 and P4 Given desired constraints: - endpoints meet P1 and P4 Q(0) = [0 0 0 1 ] · Mh · Gh = P1 Q(1) = [1 1 1 1 ] · Mh · Gh = P4 - tangent vectors meet R1 and R4 Q’(0) =[0 0 1 0] · Mh · Gh =R1 Q’(1) =[3 2 1 0] · Mh · Gh =R4 So how to compute the basis matrix Mh?

Hermite Basis Matrix We can solve for basis matrix Mh Mh

Hermite Basis Matrix We can solve for basis matrix Mh Mh

Hermite Basis Matrix P1 R1 P4 R4

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

Bezier Curve Indirectly specify tangent vectors by specifying two intermediate points R1 P2 P3 P4 P1 R2

Bezier Curve Indirectly specify tangent vectors by specifying two intermediate points R1 P2 P3 P4 P1 R4

Bezier Curve Indirectly specify tangent vectors by specifying two intermediate points R1 P2 P3 P4 P1 R4 How to compute the basis matrix Mb?

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 Mhb Gb

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 P2 P1 P4 P3

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

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

Hermite Basis Function Let’s define B as a product of T and M Bh(t) indicates the weight of each element in Gh What’s function of this red curve? 2t3-3t2+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? -t3+3t2-3t+1

Bezier Basis Functions Bezier blending functions are also called Bernstein polynomials What’s function of this red curve? -t3+3t2-3t+1

How to interpolate a 3D curve y o x z

How to interpolate a 3D curve y o x z Bezier curve

Bezier java applet Try this online at - Move the interpolation point, see how the others (and the point on curve) move - Control points (can even make loops) http://www.cse.unsw.edu.au/~lambert/splines/

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

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