Animating Rotations and Using Quaternions

What We’ll Talk About Animating Translation Animating 2D Rotation Euler Angle representation 3D Angle problems Quaternions Animating with quaternions

Review: Translation P = X Y Z 1 T = 1 tx 1 ty 1tz 1 P’ = TP =x + tx y + ty z + tz 1 Animation: interpolate over tx,ty,tz in T Move from (10,20,30) to (10,50,40), in time = 0 to 10 ∆x = (50-20)/10 ∆y = (40-30)/ Time = 0 Time = 1Time = 2Time = 10 … Translation 2D Rotation 3D Rotation Euler Problems Quaternions Q. Animation

Review: 2D Rotation 2D Rotation 3D Rotation Euler Problems Quaternions Q. Animation R = cos Ө -sin Ө sin Ө cos Ө 1 P’ = RP Ө Animating 2D Rotations timeRotation Number of frames: 3 ∆R = (90-0)/3 = 30 cos 0 -sin 0 sin 0 cos 0 1 cos 30 -sin 30 sin 30 cos 30 1 cos 60 -sin 60 sin 60 cos 60 1 cos 90 -sin 90 sin 90 cos 90 1

Review: 3D Rotations 3D Rotation Euler Problems Quaternions Q. Animation Rx = 1 cos Ө -sin Ө sin Ө cos Ө 1 Ry = cos Ө sin Ө 1 - sin Ө cos Ө 1 Rz = cos Ө - sin Ө sin Ө cos Ө 1 1 Orientation specified by a combination of rotations in a predetermined order: RxRyRz

Euler Angles Euler’s Theorem: any orientation can be expressed as a single rotation about an axis Can lead to Gimbal lock Gives you the basic idea Euler representation is good basis for calculating quaternions, which work well Ideas: –Orientation represented by an angle and an (x,y,z) vector, e.g. (A 1, Ө 1 ) –Axes represent local coordinate system, not global Rotation order is reverse of global, i.e. RzRyRx Euler Quaternions Q. Animation

Determine axis of rotation from 1st line a to second line b: cross product a x b Determine angle between lines dot product = |a| |b| cos Ø Ø = acos ( a b/ (|a||b|) ) note: normalized angle Animation with k = 0..1 : axis k = rotate(k Ø ) a angle k = (1-k)Ө 1 +kӨ 2 Euler Quaternions Q. Animation Euler Angles Ø Ө1Ө1 Ө2Ө2

Can you do it? Line 1: P1 = (0,1,0) P2 = (1,0,1) Line 2: Q1 = (1,1,1) Q2 = (3,3,3) Cross product: (-4,0,-4) acos ab/|a||b| acos(2/(√3 * 2√3) = acos(1/3) = 70.5 degrees xyz

Motivation We would like to represent 3D rotations about an arbitrary axis We would like to be able to apply a series of arbitrary rotations and have it actually work.. –Direct interpolation of matrices leads to nonsense –Gimbal lock occurs when the axes of two of the three gimbals needed to compensate for rotations in 3D space are driven to the same direction, e.g. (0,90,0) Problems Euler Quaternions Q. Animation

What is a quaternion? Alternative to Euler angles for specifying orientation 4-tuple: use 3 numbers for axis of rotation + 1 for angle of rotation Let q be a quaternion: q = s,v = s, v x, v y, v z = s + v x i + v y j + V z k Quaternions Q. Animation

Quaternion algebra A.i 2 = j 2 = k 2 = -1 = i-j-k B.ij = k = -ji jk = i = -kj ki = j = -ik C.q 1 + q 2 = (s 1 + s 2, v 1 + v 2 ) D.q 1 *q 2 = q 3 if s 1 = s 2 = 0, then q 3 = (v 1 v 2,v 1 x v 2 ) general: q 3 = (s 1 s 2 - v 1 v 2, s 1 v 2 +s 2 v 1 + v 1 x v 2 ) E.q 1 q 2 = (s 1 s 2 + v 1 v 2 ) Quaternions are like complex numbers, with one normal component and 3 imaginary components, i,j,k. (a+bi)(c+di) = (ac-bd) +(cb-ad)i Quaternions Q. Animation

Not all Quaternions Represent Rotations Only unit-length quaternions are rotations || q || = sqrt( s 2 + v v) = 1 q = 1/||q|| [s,v] q -1 = 1/||q|| 2 [s,-v] The inverse rotation is rotation by the same amount by the negative axis Unit quaternion defined by q = (cos Ө/2, sin Ө/2[x,y,z]) Quaternions Q. Animation

Suppose I had an Euler Rotation… 1.q = (cos Ө/2, sin Ө/2[x,y,z]) where Ө and (x,y,z) are the Euler angle and axis respectively 2.Normalize using quaternion normalization rules: q

Note: q = -q = (-s,-v) = cos (-Ө/2), sin(-Ө/2)[-x,-y,-z] = cos (Ө/2), -sin(Ө/2)[-x,-y,-z] =( cos (Ө/2), sin(Ө/2)[x,y,z] ) Quaternions Q. Animation

Convert Q to matrix form M = 1-2y 2 -2z 2 2xy + 2sz 2xy -2sy 2xy + 2sz 1-2x 2 -2z 2 2yz – 2sx 2xz – 2sy 2yz – 2sx 1-2x 2 -2y 2 Quaternions Q. Animation

Rotating a point p by q Rq(p) = q p q where q = [s,-v] (conjugate of q) where p = [0,(p x, p y, p z )] Rq(p) = [s,v][0,p][s,-v] =[0,(s 2 -v v)p + 2v(v p) +2s(v x p)] Quaternions Q. Animation

Consecutive Rotations R 2 (R 1 (p)) q 2 (q 1 p q 1 )q 2 = (q 2 q 1 ) p (q 2 q 1 ) =R 21 (p) Very convenient…

Note about conjugates q = [s,-v] q -1 = 1/|q| 2 q (normalize!!) For unit quaternions, q -1 = q

Animating Rotations Using Quaternions How to find intermediate rotations? A.Linear interpolation: lerp( q 1, q 2, u) = q 1 (1-u) + uq 2 Advantage: easier to find q’ vs. animating rotation matrix Problem: will speed up in the midst of rotation Ө q1q1 q2q2 q: point on a 4D unit sphere

Animating Rotations Using Quaternions B.Spherical Interpolation s lerp( q 1, q 2, u) = [sin(1-u) Ө/sin Ө ] q 1 + [sin uӨ/sin Ө ] q 2 Where q 1 q 2 = cos Ө (note: use smaller Ө ) Why? Plane trigonometry a = b = c sin sin ß sin a b c ß Spherical Trigonometry Same formula, but a,b,c are arc lengths a b c ß