Fundamentals of Computer Animation

Transcription

1 Fundamentals of Computer Animation Quaternions as Orientations () page 1

2 Multiplying Quaternions q1 = (w1, x1, y1, z1); q = (w, x, y, z); q1 * q = ( w1.w - v1.v, w1.v + w.v1 + v1 X v) where v1 = (x1, y1, z1) v = (x, y, z) Order of multiplication is important! Quaternion multiplication is not commutative: q1 * q does not equal q * q1 Unit quarternian rotated around axis (x,y,z) by amount θ q = Rot θ,(x,y,z) = [cos(θ/), sin(θ/).(x,y,z)] page

3 page 3 Quaternions as Rotations Concatenating rotations rotate using q 1 and then using q is like rotation using q *q ) * *( )* * ( ) * *( )* * ( )* * * *( = = q q P q q q q P q q q q P q q

4 Quaternion Cheat Sheet [s1,v1] + [s,v] = [s1+s,v1+v] [s1,v1] * [s,v] = [s1*s-v1.v,s1*v+s*v1+v1xv] q = sqrt(s*s + x*x + y*y + z*z) q * [1,0,0,0] = q q -1 = [-s,v]/ q q * q -1 = [1,0,0,0]

5 Quaternions q =[s,v]=[s,x,y,z] Α θ (cos(θ/),sin(θ/)*a) page 5

6 Rotations in Reality It s easiest to express rotations in Euler angles or Axis/angle We can convert to/from any of these representations Choose the best representation for the task input:euler angles interpolation: quaternions composing rotations: quaternions, orientation matrix page 6

7 Quaternions to rotate a point v = (x,y,z) => [0,v] Rot q (v) = v = q * [0,v] * q -1 page 7

8 Conversion From Quaternions Quaternion to Matrix ( w, x, y z) q =, To form a matrix equivalent to a unit quarternian: (w +x +y +z )=1 useful as orientation is independent of magnitude The columns are rotations of the principal axes by the unit quarternian, to rotate an axis v by the quarternian q: v rot = qvq -1 thus it is relatively simple to calculate the conversion matrix below. Matrix = 1 y xy xz + y zw z yw xy 1 x yz + zw z xw xz yz 1 x + yw xw y page 8

9 Converting quarternian matrix M 0,0 M 0,1 M 0, M 1,0 M 1,1 M 1, Matrix M,0 M,1 M, = 1 y y xy + zw xz yw z xy zw 1 x z yz + xw xz + yw yz xw 1 x y Sum of diagonals= 1-y -z +1-x -z + 1-x -y = 3-4x 4y -4z But (w +x +y +z )=1 so sum = 3 4(1-w ) = 4w 1 w = sqrt(1+m 0,0 +M 1,1 + M, )/ M 0,0 =1-y -z =1 - (y +z ) = 1 (1 - x +w ) = x +w 1 Converting from Matrix Quarternion similarly for y and z x = sqrt(1/(m 0,0 +1-w )) y = sqrt(1/(m 1,1 +1-w )) Z = sqrt(1/(m, +1-w )) page 9

10 Conversion From Quaternions Quaternion to Axis Angle To change a quaternion to a rotation around an arbitrary axis in 3D space: Axis of rotation (ax, ay, az) angle theta (radians) then angle= * acos(w) ax= x / scale ay= y / scale az= z / scale where scale = x + y + z if the scale is 0 no rotation and the axis will be infinite. In this case: Set the rotation axis to any unit vector with a rotation angle of 0. page 10

11 Example Camera Orientation Euler glrotatef( anglex, 1, 0, 0) glrotatef( angley, 0, 1, 0) glrotatef( anglez, 0, 0, 1) // translate Quaternion // convert Euler to quaternion // convert quaternion to axis angle glrotate(theta, ax, ay, az) // translate Gimbal Lock? YES YES page 11

12 Conversion To Quaternians Axis Angle to Quaternian Axis of rotation (ax, ay, az) /* must be unit vectors */ angle theta (radians) then w = cos (theta/) x = ax * sin (theta/) y = ay * sin (theta/) z = az * sin (theta/) If the axis is a zero vector no rotation In this case: Set the quaternion to the rotation identity quaternion q= [1,(0, 0, 0)] page 1

13 Conversion To Quaternions Euler to Quaternion if you have three Euler angles (a, b, c), then you can form three independent quaternions: qx = [ cos(a/), (sin(a/), 0, 0) ] qy = [ cos(b/), (0, sin(b/), 0) ] qz = [ cos(c/), (0, 0, sin(c/)) ] And the final quaternion is obtained by qx * qy * qz. page 13

14 How can quaternions avoid Gimbal Lock? Basic Idea: 1. Use a quaternion to represent the rotation.. Generate a temporary quaternion for the change from the current orientation to the new orientation. 3. PostMultiply the temp quaternion with the original quaternion. This results in a new orientation that combines both rotations. 4. Convert the quaternion to a matrix and use matrix multiplication as normal. page 14

15 CameraEuler.exe In the while loop: There are 3 angles for rotation in the X, Y, and Z axis. Euler With every key press, the corresponding rotation variable is adjusted. Translation is performed and then the 3 Euler angles are adjusted to rotation matrices and multiplied into the final transformation matrix. This program suffers from gimbal lock. If you want to see it in action, rotate the camera so that the yaw is 90 deg. Then try rotating in the X and Z direction. See what happens. page 15

16 CameraQuat.exe In the while loop: When a key is pressed, a temporary quaternion is generated corresponding to the key for a small rotation in that particular axis. The temporary quaternion is then multiplied into the camera quaternion. This concatenation of rotations in 4D space will avoid gimbal lock. Try it! Quarternion The orientation of the camera is a quaternion. There are 3 angles corresponding to the keypress. Note the angles are meant to be an on/off switch (not accumulative). The 3 angles converted to a temporary quaternion. Temporary quaternion is multiplied to the camera quaternion to obtain the combined orientation. Note the order of multiplication. The camera rotation is then converted to the Axis Angle representation for transforming the final matrix. page 16

17 Linearly interpolating fixed angles from (0,90,0) to (90,0,90) Interpolating quaternions from (0.5,0.0,1.0,0.0) to (0.5,0.5,0.5,0.5) using spherical linear interpolation q1 = [ cos(φ/), (sin(φ/), 0, 0) ] q 1 = [0.5, (0, 1, 0)] =[cos(π/4), sin(π/4)(0, 1, 0)] q = [0.5, (0.5, 0.5, 0.5)] =[cos(p/4), sin(p/4)(0.5, 0.5, 0.5)] page 17

18 Quaternion Interpolation Linearly interpolating fixed angles from (0,90,0) to (90,0,90) Fixed angles (0,90,0) (90,0,90) quaternions [0.7,0.0,0.7,0.0] [0.5,0.5,0.5,0.5] Interpolating quaternions from (0.5,0.0,1.0,0.0) to (0.5,0.5,0.5,0.5) using spherical linear interpolation page 18

19 Interpolation Scalar: K = (1-α) A + α B (linear interpolation: lerp) Euler Angles: [x,y,z] = (1- α)[x a,y a,z a ] + α[x b,y b,z b ] Quaternions?: Linear interpolation of quaternion values would give unequal rotation increments - need to slerp (spherical linear interpolation) page 19

20 Quaternion Interpolation If the components of q 1 and q are Linearly interpolated: Can be viewed as 4D points on a straight line between q 1 and q not constant speed rotation. B Equally spaced linear interpolation of straight-line path between two points on a circle generate unequal spacing of points after projecting onto a circle. A Need to interpolate on the surface Of the sphere. page 0

21 Quaternions as points on a 4D unit sphere Unit quaternion: q=(s,x,y,z), q = 1 Want equal increment along arc connecting two quaternions on surface of sphere spherical linear interpolation

22 Interpolation of Rotations How can we interpolate between two quaternion rotations along the shortest arc? Spherical Linear interpolation (SLERP) Find angle between q 1 and q and interpolate SLERP ( q q, t) cosθ = q q = q sin (( 1 t) θ ) + q sin( tθ ) sin( θ ) 1 1, 1 s 1,s + v 1 v If u=1/ can compute q 1 +q and normalize Nick Bobick Rotating Objects Using Quaternions GameDeveloper, Feb 1999 (available at course website) page

23 Orientation two representations Interpolate q1 to q q1 is closer to q So interpolate 180-θ The closer of the two representations of orientation is the better choice to use in interpolation. page 3

24 SLERP q form a sphere of unit length in the 4D space SLERP ( q q, t) q sin (( 1 t) θ ) + q sin( tθ ) sin( θ ) 1 1, = page 4

25 SLERP Given several orientations q 1,q,q 3 q n We get 1st derivative discontinuities use cubic Bezier Interpolation (see Parent appendix and Shoemake Siggraph 1985 p ) A Catmull-Rom spline will interpolate points and tangents but may not give youexactly what you want. This is discussed in Mortensen, Rogers, and Killer B s books. See Parent for references. (or borrow from my library books to be returned or suffer horrible and unusual punishment.) page 5

26 Bezier construction using quaternion operators Need quaternion-friendly operations to interpolate cubic Bezier curve using quaternion points de Casteljau geometric construction algorithm page 6

27 Convex Hull What is the Convex Hull? Given a set of points (in ) The smallest convex polyhedral that includes all the points given page 7

28 de Casteljau Algorithm Avoid direct evaluating of polynomials Geometric interpretation Consider a planer cubic at u 1-u column by column updating rule page 8

29 Pseudo Code Input P[j], d, u //P[j] : control point, d : degree, u : parameter //output will be Q(u) for i = 1 to d for j = 0 to d-i P[j] =(1-u)*P[j] + u*p[j+1] end end Output??? page 9

30 Some Observations on Cubic Case After one step of decasteljau algorithm for Obtaining 4 green points 4 blue points Where Small left Bezier curve: Small right Bezier curve: Original Bezier curve Divided and conquer method page 30

31 Matrix Relation Unit summation of any row. Non negative elements. Banded structure. page 31

32 Bezier construction using quaternion operators t 1 =slerp(q n, a n,1/3) t =slerp(a n, b n+1,1/3) t 3 =slerp(b n+1, q n+1,1/3) t 1 =slerp(t 1, t,1/3) t 3 =slerp(t 1, t 3,1/3) q=slerp(t 1, t 3,1/3) page 3

33 Practical use for quaternions (1) Camera rotations in third-person-perspective games

34 Practical use for quaternions (1) Camera rotations in third-person-perspective games page 34

35 Practical use for quaternions () Prerecorded (but not prerendered) animations Instead of recording camera movements by playing the game (as many games do today), you could prerecord camera movements and rotations using a commercial package such as Softimage 3D or 3D Studio MAX. Then, using an SDK, export all of the keyframed camera/object quaternion rotations. This would save both space and rendering time. Then you could just play the keyframed camera motions whenever the script calls for cinematic scenes. page 35