Animation Dr. Amy Zhang Lecture 8

Reading 2  Hill, Chapters 5 / 7 / 10  Red Book, Chapter 3, “Viewing”  Red Book, Chapter 12, “Evaluators and NURBS”

Outline 3  Computer-aided animation  Keyframing and interpolation of position  Hierarchical modeling.  Interpolation of rotation

Conventional Animation 4  Draw each frame of the animation  great control  tedious  Reduce burden with cel animation  Layer, keyframe, inbetween  cel panoramas (Disney’s Pinocchio) ... ACM © 1997 “Multiperspective panoramas for cel animation”

Computer-Assisted Animation 5  Keyframing  automate the inbetweening  good control  less tedious  creating a good animation still requires considerable skill and talent

Computer-Assisted Animation 6  Procedural animation  describes the motion algorithmically  express animation as a function of small number of parameters  Example: a clock with second, minute and hour hands  hands should rotate together  express the clock motions in terms of a “seconds” variable  the clock is animated by varying the seconds parameter  Example 2: A bouncing ball  Abs(sin( ω t+ Θ ))*e -kt

Computer-Assisted Animation 7  Physically Based Animation  Assign physical properties to objects (masses, forces, inertial properties)  Simulate physics by solving equations  Realistic but difficult to control

Computer-Assisted Animation 8  Motion Capture  Captures style, subtle nuances and realism  You must observe someone

Motion Capture 9

Motion Replay 10

Outline 11  Computer-aided animation  Keyframing and interpolation of position  Hierarchical modeling.  Interpolation of rotation

Keyframing 12  Describe motion of objects as a function of time from a set of key object positions. In short, compute the inbetween frames.

Interpolating Positions 13  Given positions:  find curve such that

Linear Interpolation 14  Simple problem: linear interpolation between first two points assuming :  The x-coordinate for the complete curve in the figure:

Polynomial Interpolation 15  A polynomial is a function of the form:  n is the degree of the polynomial  The order of the polynomial is the number of coefficients it has  Always: order = degree + 1

Polynomial Interpolation 16  What degree polynomial can interpolate n+1 points?  An n-degree polynomial can interpolate any n+1 points  wikipedia: Lagrange interpolation  The resulting interpolating polynomials are called Lagrange polynomials  We already saw the Lagrange formula for n = 1

Spline Interpolation 17  Lagrange polynomials of small degree are fine but high degree polynomials are too wiggly  Technical term: overfitting

Spline Interpolation 18  Spline (piecewise cubic polynomial) interpolation produces nicer interpolation t tt x(t)

Spline Curves 19  The word spline comes from ship building with wood  A wooden plank is forced between fixed posts, called “ducks”  Real-world splines are still being used for designing ship hulls, automobiles, and aircraft fuselage and wings

Splines 20

Spline Curves 21  Spline curve - any composite curve formed with piecewise parametric polynomials subject to certain continuity conditions at the boundary of the pieces.  Huh?

Parametric Polynomial Curves 22  A parametric polynomial curve is a parametric curve where each function x(t), y(t) is described by a polynomial:  Polynomial curves have certain advantages:  Easy to compute  Indefinitely differentiable

Why Parametric Curves? 23  Parametric curves are very flexible  They are not required to be functions  Curves can be multi-valued with respect to any coordinate system

Piecewise Curves 24  To avoid overfitting we will want to represent a curve as a series of curves pieced together  A piecewise parametric polynomial curve uses different polynomial functions for different parts of the curve  Advantages: Provides flexibility  Problem: How do we fit the pieces together?

Parametric Continuity 25  C0: Curves are joined “watertight” curve / mesh  C1: First derivative is continuous, d/dt Q(t) = velocity is the same. “looks smooth, no facets”  C2: Second derivative is continuous, d2/dt2 Q(t) = acceleration is the same (important for animation and shading)

Specifying Splines 26  Control Points - a set of points that influence the curve's shape  Hull - the lines that connect the control points  Interpolating – curve passes through the control points  Approximating – control points merely influence shape

Parametric Cubic Curves 27  In order to assure C2 continuity our functions must be of at least degree three.  Here's what a 2D parametric cubic function looks like: In matrix form:

Parametric Cubic Curves 28  This is a cubic function in 3D:  To avoid the dependency on the dimension we will use the following notation:

Parametric Cubic Curves 29  What does the derivative of a cubic curve look like?

Hermite Splines 30  We want to support general constraints: not just smooth velocity and acceleration. For example, a bouncing ball does not always have continuous velocity:  Solution: specify position AND velocity at each point

Hermite Splines 31  Given:  Two control points (P 1, P 2 ).  Tangents (derivatives) at the knot points (P’ 1, P’ 2 ):  “Control knob” vector: [P 1, P 2, P’ 1, P’ 2 ]

Hermite Spline 32  4 segments, (P 1, P 2 ) and (P’ 1, P’ 2 ) for each segment

Building it up… 33  Cubic curve equations:  Boundary constraints: or:

Solve for the c’s 34

Cubic Hermite Curves 35  Curve equation in matrix form:  T = Power basis  B = Spline basis, characteristic matrix  G = Geometry (control points)

Bézier Curves 36  Developed simultaneously by Bézier (at Renault) and de Casteljau (at Citroen), circa  Without specify consecutive tangents  A cubic Bézier curve has four control points, two of which are knots (P 1, P 4 ).

Bézier Specification 37  Four control points (P 1, P 2, P 3, P 4 ).  Gradients (tangents) at the knot points (P’ 1, P’ 4 ) are the line segments between adjacent control points:  Tangents: P’ 1 = 3 ( P 2 - P 1 ); P’ 2 = 3 ( P 4 - P 3 )  Scale factor (3) is chosen to make “velocity” approximately constant  Endpoint interpolation: P(0) = P 1 and P(1) = P 4  P 2 & P 3 control the magnitude and orientation of the tangent vectors at the end points  Geometry vector G: [P 1, P 2, P 3, P 4 ]

Cubic Bézier Curves 38  Repeating the same derivation gives:

Bézier Curves 39  Consider the cubic curve:  The basis functions  Bi,3(u)are given by:

Bézier Curves 40  De Casteljau representation: where r= 1,…,k; i= 0,…,k–r; p i 0 (u)=p i  A point on the curve with parameter value u is given by,i.e.,

Bézier Curves 41  Using the de Casteljau algorithm, a Bézier curve can be subdivided anywhere along its length into 2 smaller curves  Note that the convex hull of the smaller section lies closer to the curve than the original convex hull, and therefore repeated subdivision will produce a closer approximation to the curve

Bézier Curves 42  Subdivision formulae about midpoint (i.e.,u=0.5):  Let original control points be pi  Let the control points of the small sections be q i & r i  The 2 small curve segments join at

Properties of Bézier Curves 43  Convex hull: Curve is contained in the hull [P 1, P 2, P 3, P 4 ]  Affine Invariance:  Transforming the control points is computationally more efficient than transforming each point on the curve  Useful in ensuring smoothness while scaling a curve segment  Symmetry: P(t); [P 1, P 2, P 3, P 4 ] ≡ P(1-t); [P 4, P 3, P 2, P 1 ]

Catmull-Rom Splines 44  With built-in C 1 continuity.  Compared to Hermite/Bezier: fewer control points required, but less freedom.  Given n control points in 3-D: p 1, p 2, …, p n,  Tangent at p i given by s(p i+1 –p i-1 ) for i=2..n-1, for some s  Curve between p i and p i+1 is determined by p i-1, p i, p i+1, p i+2

Hungry for more? 45  You got your Hermite and Bézier splines, Catmull- Rom splines  Once you know it's a piecewise cubic, you know what game you're playing — putting conditions on the segments and how they fit together  You should know the basic properties of Hermite and Bézier splines

Interpolating Key Frames 46  Interpolation is not fool proof. The splines may undershoot and cause interpenetration. The animator must also keep an eye out for these types of side- effects.

Outline 47  Computer-aided animation  Keyframing and interpolation of position  Hierarchical modeling  Interpolation of rotation

Articulated Models 48  Articulated models:  rigid parts  connected by joints  They can be animated by specifying the joint angles as functions of time.

Forward Kinematics 49  Describes the positions of the body parts as a function of the joint angles 1 DOF: knee 2 DOF: wrist 3 DOF: arm

Skeleton Hierarchy 50  Each bone transformation described relative to the parent in the hierarchy:  Tree traversal

Forward Kinematics 51  Transformation matrix for an effecter v s is a matrix composition of all joint transformation between the effecter and the root of the hierarchy

Inverse Kinematics 52  Forward Kinematics  Given the skeleton parameters p and the position of the effecter in local coordinates v s, what is the position of the sensor in the world coordinates v w ?  Inverse Kinematics  Given the position of the effecter in local coordinates v s and the desired position v w in world coordinates, what are the skeleton parameters p?  Much harder; requires solving the inverse  Underdetermined problem with many solutions

Outline 53  Computer-aided animation  Keyframing and interpolation of position  Hierarchical modeling  Interpolation of rotation

3D rotations 54  How many degrees of freedom for 3D orientations?  3 degrees of freedom:  e.g. direction of rotation and angle

3D Rotation Representations 55  Rotation Matrix  orthornormal columns/rows  bad for interpolation  Fixed Angle  rotate about global axes  bad for interpolation, gimbal lock  Euler Angle  rotate about local axes  bad for interpolation, gimbal lock

Fixed Angle 56  (0,90,0) in x-y-z order  (90,45,90) in x-y-z order (0,90,0) (90,45,90)

Interpolation Problem 57  The rotation from (0,90,0) to (90,45,90) is a 45-degree x- axis rotation  Directly interpolating between (0,90,0) and (90,45,90) produces a halfway orientation (45, 67.5, 45)  Desired halfway orientation is (90, 22.5, 90)

Euler Angles 58  An Euler angle is a rotation about a single axis. Any orientation can be described composing three rotation around each coordinate axis.  Roll, pitch and yaw

Euler Angles 59  Gimbal lock: two or more axis align resulting in a loss of rotation dof.

Demo 60  See res/qpowers/page7.asp res/qpowers/page7.asp  See also mbal.jpg.html mbal.jpg.html

3D Rotation Representations 61  Axis angle  rotate about A by , (Ax,Ay,Az,  )  good interpolation, no gimbal lock  cannot compose rotations efficiently

Solution: Quaternion Interpolation 62  4-tuple of real numbers  q=(s,x,y,z) or [s,v]  s is a scalar; v is a vector  Same information as axis/angle but in a different form  Right-hand rotation of Θ radians about v is: q = {cos( Θ /2); v sin( Θ /2)}  Often noted (s, v) or {a; b, c, d}

Quaternions 63  q = {cos( /2); v sin( /2)}  What if we use -v?  Rotation of – around –v  The quaternion of the identity rotation: q i = {1; 0; 0; 0}  q{1; 0; 0; 0}= q  Is there exactly one quaternion per rotation?  No.  q={cos( /2); v sin( /2)} is the same rotation as -q={cos(( +2 )/2); v sin(( +2 )/2)}  The inverse q -1 of quaternion q {a, b, c, d}: {a, -b, -c, -d}  qq -1 = {1; 0; 0; 0}

Quaternion Algebra 64  Multiplication:  UsingToRepresentRotation.htm UsingToRepresentRotation.htm  Quaternions are associative:  But not commutative:

Rotation Example 65  To rotate 3D point/vector p by a rotation q, compute:

Compose Rotations 66

Quaternion Interpolation 67  A quaternion is a point on a 4D unit sphere  Unit quaternion: q=(s,x,y,z), ||q|| = 1  Interpolating rotations means moving on 4D sphere

Quaternion Interpolation 68  Linear interpolation generates unequal spacing of points after projecting to circle

Quaternion Interpolation 69  Spherical linear interpolation (slerp)  interpolats along the arc lines instead of the secant lines.  produce equal increment along arc connecting two quaternions on the spherical surface

Useful Analogies 70  Euclidean Space  Position  Linear interpolation 4D Spherical Space Orientation Spherical linear interpolation (slerp)