CS 475 / CS 675 Computer Graphics. Lecture 16 : Interpolation for Animation

Transcription

1 CS 475 / CS 675 Computer Graphics Lecture 16 : Interpolation for

2 Keyframing Selected (key) frames are specified. Interpolation of intermediate frames. Simple and popular approach. May give incorrect (inconsistent) results. In between Frames KF1 KF KF3 KF4 KF5 Time

3 Keyframing

4 Keyframing Interpolate Position

5 Keyframing Interpolate Orientation

6 Interpolating orientation Fixed Angle Representation - Ordered triple of rotations about global axes. Any triple is valid that doesn't immediately repeat an axis, e.g., x-y-z or z-x-y. But not x-x-y. Let us assume a z-y-x order for now.,, P ' =R z. R y. R x. P

7 Interpolating orientation Fixed Angle Representation Make a rotation matrix from the angles and interpolate [ 1 1 R z 9 = 1 1 [ ] 1 R z 9 = ] [ ] R z? = 1 1

8 Interpolating orientation Fixed Angle Representation Interpolate the angles and then form the matrix. Suffers from the Gimbal lock! ±, 9,,,, 9,, 9,±

9 Interpolating orientation Fixed Angle Representation When to form and apply the matrix if rotation has to be incremented by in each frame? Form a rotation matrix for and apply repeatedly to rotated object in each frame. Update the rotation matrix R axis by multiplying with R axis in each frame. Apply updated matrix to the object. Update the rotation angle, by the increment and form the new matrix R axis in each frame. Apply this matrix to the object.

10 Interpolating orientation Euler Angle Representation - Ordered triple of rotations about local axes. Any triple is valid that doesn't immediately repeat an axis, e.g., x-y-z or z-x-y. But not x-x-y. Let us assume a x-y-z order for now.,, P ' =R x. R y. R z. P Rotation given by a triad of Euler angles is the same as given by a triad of Fixed angles considered in opposite order. So it has the same Gimbal Lock problem.

11 Interpolating orientation Axis Angle Representation - Specified as an axis of rotation A x, y, z and an angle of rotation, around it. Euler's Theorem Any orientation can be derived from another by a single rotation about and axis. A x, y, z

12 Interpolating orientation - Axis Angle Representation To interpolate between two orientations A 1, 1 and A, B= A1 A 1 =cos A1 A A1 A 1 A k =R B k. A1 k = 1 k 1 k with k 1 A1 B A

13 Interpolating orientation Unit Quaternions Have the same information as the axis-angle representation but in a more convenient form. A q=[ s, x, y, z ] =[ s, v ] =[ cos /, sin / a ], where a= A A x, y, z

14 Interpolating orientation Quaternions A non commutative number system that extends complex numbers Defined as: q=s x i y j z k =[ s, v ] where 1, i, j, k are called the Hamilton basis The product of the basis elements is defined as: i = j = k = i j k = 1 i j= k, j k = i, k i = j j i = k, k j = i, i k = j Quaternions form a four dimensional normed division algebra, H over the real numbers.

15 Interpolating orientation Quaternions A non commutative number system that extends complex numbers Defined as: q=s x i y j z k =[ s, v ] where 1, i, j, k are called the Hamilton basis The product of the basis elements is defined as: i = j = k = i j k = 1 i j= k, j k = i, k i = j j i = k, k j = i, i k = j Note that the multiplication being defined here is a quaternion multiplication and not the inner or out product of vectors. It is not commutative.

16 Interpolating orientation Quaternions Quaternion Arithmetic q1 q =[ s1 s, v 1 v ] Addition: Scalar Multiplication: k q=[ k s, k v ]=ks kx i ky j kz k Quaternion Multiplication: q1 q = a1 b1 i c 1 j d 1 k a b i c j d k =a 1 a b i c j d k b 1 i a b i c j d k c 1 j a b i c j d k d 1 k a b i c j d k = a 1 a b 1 b c 1 c d 1 d a1 b b 1 a c 1 d d 1 c i a 1 c b 1 d c 1 a d 1 b j a 1 d b1 c c1 b d 1 a k

17 Interpolating orientation Quaternions * q Conjugate Quaternion: =s x i y j z k Quaternion Norm: q = q q = q q= s x y z If is real then, q = q The norm is multiplicative: p q = p q Unit Quaternion: q = q 1 Quaternion Inverse: q = q * * q q *

18 Keyframing Interpolate Orientation Interpolate Position Interpolate Shape Interpolate Colour Light Intensity Camera Zoom Any other parameter Real Time Shape Editing using Radial Basis Functions. Mario Botsch, Leif Kobbelt. Computer Graphics Forum 4(3), Proc. Eurographics 5

19 Keyframing Moving on curves. Specify spatial position to fix the curve In addition, we specify the speed at which we travel along the curve D, t=6 B, t=1 A, t= C, t=35

20 Controlling speed on curves Typically parametrization is not arc length. Arc length is the distance along the curve. Arc length parametrization can be computed using Analytical Computation Table-based Summed linear distances (forward differencing) Gaussian quadrature (numerical integration)

21 Controlling speed on curves Given a parametric curve, P(u) = (x(u), y(u), z(u)) We may have to solve two versions of the problem: Given parameters u1 and u, find arc length, LENGTH(u1, u) Given an arc length s and parameter u1, find u so that LENGTH(u1, u) =s

22 Controlling speed on curves Given parameters u1 and u, find arc length, LENGTH(u1, u) Can we compute s = G(u) = distance from start of curve to point at u? With G, arc length parameterization can be obtained by inversion as P(G-1(s)), where G-1(s) gives the parameter u up to which distance travelled on the curve is s. Parameter values obtained from G-1(s) Equal arc lengths s over the curve. Re-parameterize to have equal spacing in the parameteric interval.

23 Controlling speed on curves Given an arc length s and parameter u1, find u so that LENGTH(u1, u) =s Generally, neither of the two forms of the problem admit analytical solutions. Arc Length u LENGTH u 1, u =s= u1 u dp u du= du u 1 dp u du du dp u = du dx u dy u dz u du du du

24 Controlling speed on curves The arc length integral can be approximated using a forward differencing method. Create a piece wise linear approximation of the curve from many parameter evaluations and sum these to form the arc length. Store these values into a table. s s Actual Curve Linear Approximation

25 Controlling speed on curves The inversion can then be calculated using bisection s=g u is a monotonically increasing function. i.e., if u1 < u then s1 < s s So we can do a bisection or a binary search for u, given a value of s. u

26 Controlling speed on curves Space curves we have seen till now give the path. What if we want to control the speeds Accelerates from stop position Reaches maximum speed Decelerates to a stop. Given the speed as a plot of s vs. t. v = ds/dt

27 Controlling speed on curves A slow in slow out curve may look like:

28 Controlling speed on curves Given next time instant t Distance-time curve gives total distance s travelled up to time t. P(G-1(s)) gives the position on the path space curve. Or solve a space-time optimization for the whole path.

29 Interpolating orientation Quaternions Quaternion and the Geometry of R3 i, j, k denote both the basis vector of H and a basis for R 3 Vectors in R 3 can be written as pure imaginary quaternions v= x i y j z k =[, u ] Inner product of vectors in R 3 1 * 1 * * * v 1 v = x 1 x y 1 y z 1 z = v 1 v v v 1 = v 1 v v v 1 Cross product of vectors in 1 * * v 1 v = v 1 v v v 1 Quaternion multiplication can be written as: q1 q =[ s1 s v 1 v, s v 1 s1 v v 1 v ]

30 Interpolating orientation Unit Quaternions A unit quaternion denotes a rotation by an angle θ about an axis A A q=[ s, x, y, z ]=[ s, v ]=[ cos /, sin / a ], where a = A Multiplication with a unit quaternion q can be used to rotate a vector v 1 1 R q v =q v q =q [, v ] q Composition of rotations is equivalent to quaternion 1 1 multiplication R q R q v =R q q v q 1 =q q [, v ] q 1 q = q 1 q [, v ] q 1 q =R q q v Rotating by a scalar multiple of a unit quaternion is the same as rotating by the unit quaternion R q v =R k q v Only Unit Quaternions represent rotations! 1

31 Interpolating orientation Unit Quaternions Antipodal Unit Quaternions q=[ cos /, a sin / ] If we rotate by θ-π instead of θ [ cos /, a sin / ] =[ cos /, a sin / ]=[ cos /, a sin / ] = q So both q and q represent the same rotation and are called antipodal points. R v =R v q q If < θ < π then the positive rotation is the shorter one else the negative rotation is the shorter one, i.e., the quaternion with the positive value of the s coordinate will give the shorter path I am abusing notation here for convenience. Please remember we are talking about unit quaternions.

32 Interpolating orientation Unit Quaternions Linear Interpolation q= 1 k q 1 k q How to take equi-distant steps along orientation path? How to pass through orientations smoothly? With dual unit quaternion representations, which one to use? Dual representation: For Interpolation between q1 and q, compute cosine between q1 and q and between q1 and q; choose smallest angle. Computer : Algorithms and Techniques, Rick Parent, Morgan Kaufmann, 1

33 Interpolating orientation Unit Quaternions Linear Interpolation q= 1 k q 1 k q This is not equally spaced. q q q1 Computer : Algorithms and Techniques, Rick Parent, Morgan Kaufmann, 1

34 Interpolating orientation Unit Quaternions Spherical Linear Interpolation or SLERP We write, q =[cos /, a sin / ] We want to interpolate between two rotations q1 and q Rotation that takes us from 1 to is given by q q 1 Now we start at 1, and go to in α steps as q q 1 q Slerp q 1, q, = q q 1 q1 sin 1 sin q1 q sin sin cos =q 1 q =s1 s x 1 x y 1 y z 1 z Slerp q 1, q, =

35 Interpolating orientation Unit Quaternions Interpolating between multiple quaternions Let us define two operators Double p, q = p q q p p q Bisect p, q = p q Double p, q q q Bisect p, q p p

36 Interpolating orientation Unit Quaternions Interpolating between multiple quaternions Given three successive quaternions qn-1, qn, qn+1, we define a n =Bisect Double q n 1, q n, q n 1 b n =Double a n, q n q n 1 bn an qn q n 1 q n 1

37 Interpolating orientation Unit Quaternions Interpolating between multiple quaternions Now we can join successive Bèzier spline segments. And find points on the Bèzier curves we use de Casteljau's algorithm to find in-between rotations. bn qn an b n 1 a n 1 q n 1 q n 1