CS-184: Computer Graphics Lecture #5: 3D Transformations and - PowerPoint PPT Presentation

CS-184: Computer Graphics Lecture #5: 3D Transformations and Rotations Prof. James O’Brien University of California, Berkeley V2013-F-05-1.0 1 Today • Transformations in 3D • Rotations • Matrices • Euler angles • Exponential maps • Quaternions 2 2 Tuesday, September 17, 13

3D Transformations • Generally, the extension from 2D to 3D is straightforward • Vectors get longer by one • Matrices get extra column and row • SVD still works the same way • Scale, Translation, and Shear all basically the same • Rotations get interesting 3 3 Translations   1 0 t x ˜ A = 0 1 t y For 2D   0 0 1   1 0 0 t x 0 1 0 t y ˜   A = For 3D   0 0 1 t z   0 0 0 1 4 4 Tuesday, September 17, 13

Scales   s x 0 0 ˜ A = 0 s y 0 For 2D   0 0 1   s x 0 0 0 0 s y 0 0 ˜   A = For 3D   0 0 s z 0   0 0 0 1 (Axis-aligned!) 5 5 Shears   1 h xy 0 ˜ A = h yx 1 0 For 2D   0 0 1   1 h xy h xz 0 h yx 1 h yz 0 ˜   A = For 3D   h zx h zy 1 0   0 0 0 1 (Axis-aligned!) 6 6 Tuesday, September 17, 13

Shears   1 h xy h xz 0 h yx 1 h yz 0 ˜   A =   h zx h zy 1 0   0 0 0 1 Shears y into x 7 7 Rotations • 3D Rotations fundamentally more complex than in 2D • 2D: amount of rotation • 3D: amount and axis of rotation -vs- 2D 3D 8 8 Tuesday, September 17, 13

Rotations • Rotations still orthonormal • Det ( R ) = 1 6 = � 1 • Preserve lengths and distance to origin • 3D rotations DO NOT COMMUTE! • Right-hand rule DO NOT COMMUTE! • Unique matrices 9 9 Axis-aligned 3D Rotations • 2D rotations implicitly rotate about a third out of plane axis 10 10 Tuesday, September 17, 13

Axis-aligned 3D Rotations • 2D rotations implicitly rotate about a third out of plane axis   cos ( θ ) − sin ( θ ) 0  � cos ( θ ) − sin ( θ ) R = sin ( θ ) cos ( θ ) 0 R =   sin ( θ ) cos ( θ ) 0 0 1 Note: looks same as ˜ R 11 11 Axis-aligned 3D Rotations   1 0 0 R = 0 cos ( θ ) − sin ( θ )   x ˆ 0 sin ( θ ) cos ( θ ) “Z is in your face”   cos ( θ ) 0 sin ( θ ) y ˆ R = 0 1 0   y ˆ − sin ( θ ) 0 cos ( θ )   cos ( θ ) − sin ( θ ) 0 x ˆ R = sin ( θ ) cos ( θ ) 0   z ˆ 0 0 1 z ˆ 12 12 Tuesday, September 17, 13

Axis-aligned 3D Rotations   1 0 0 R = 0 cos ( θ ) − sin ( θ )   x ˆ 0 sin ( θ ) cos ( θ ) Also right handed “Zup”   cos ( θ ) 0 sin ( θ ) z ˆ R = 0 1 0   y ˆ − sin ( θ ) 0 cos ( θ ) y ˆ   cos ( θ ) − sin ( θ ) 0 x ˆ R = sin ( θ ) cos ( θ ) 0   z ˆ 0 0 1 13 13 Axis-aligned 3D Rotations • Also known as “direction-cosine” matrices     1 0 0 cos ( θ ) 0 sin ( θ ) R = 0 cos ( θ ) − sin ( θ ) R = 0 1 0     y ˆ x ˆ 0 sin ( θ ) cos ( θ ) − sin ( θ ) 0 cos ( θ )   cos ( θ ) − sin ( θ ) 0 R = sin ( θ ) cos ( θ ) 0   z ˆ 0 0 1 14 14 Tuesday, September 17, 13

Arbitrary Rotations • Can be built from axis-aligned matrices: R = R ˆ z · R ˆ y · R ˆ x • Result due to Euler... hence called Euler Angles • Easy to store in vector • But NOT a vector. R = rot ( x , y , z ) 15 15 Arbitrary Rotations R = R ˆ z · R ˆ y · R ˆ x R ˆ R ˆ R ˆ x y z R 16 16 Tuesday, September 17, 13

Arbitrary Rotations • Allows tumbling • Euler angles are non-unique • Gimbal-lock • Moving -vs- fixed axes • Reverse of each other 17 17 Exponential Maps • Direct representation of arbitrary rotation • AKA: axis-angle, angular displacement vector • Rotate degrees about some axis θ • Encode by length of vector θ r ˆ θ = | r | θ 18 18 Tuesday, September 17, 13

Exponential Maps • Given vector , how to get matrix r R • Method from text: 1. rotate about x axis to put r into the x-y plane 2. rotate about z axis align r with the x axis 3. rotate degrees about x axis θ 4. undo #2 and then #1 5. composite together 19 19 Exponential Maps r x x r x ⊥ ⊥ x • Vector expressing a point has two parts x • does not change x • rotates like a 2D point ⊥ 20 20 Tuesday, September 17, 13

Exponential Maps r x x r x ⊥ ⊥ x x ` = ˆ r ⇥ x x 0 θ − x ⊥ = ˆ r × ( ˆ r × x ) x ⊥ x x 0 = x || + x ` sin ( θ )+ x ? cos ( θ ) x ` sin ( θ ) − x ⊥ cos ( θ ) 21 21 Exponential Maps • Rodriguez Formula x 0 = ˆ r ( ˆ r · x ) + sin ( θ )( ˆ r ⇥ x ) � cos ( θ )( ˆ r ⇥ ( ˆ r ⇥ x )) x Linear in x x r ! ! x Actually a minor variation ... 22 22 Tuesday, September 17, 13

Exponential Maps • Building the matrix x 0 = (( ˆ r t )+ sin ( θ )( ˆ r ˆ r ⇥ ) � cos ( θ )( ˆ r ⇥ )( ˆ r ⇥ )) x   0 − ˆ r z r y ˆ ( ˆ r × ) = r z ˆ 0 − ˆ r x   − ˆ r y r x 0 ˆ Antisymmetric matrix ( a × ) b = a × b Easy to verify by expansion 23 23 Exponential Maps • Allows tumbling • No gimbal-lock! • Orientations are space within π -radius ball • Nearly unique representation • Singularities on shells at 2 π • Nice for interpolation 24 24 Tuesday, September 17, 13

Exponential Maps • Why exponential? • Instead of rotating once by θ , let’s do n small rotations of θ /n r • Now the angle is small, so the rotated x is approximately x 0 x + ( θ /n )ˆ r × x ✓ I + (ˆ r × ) θ ◆ = x x n • Do it n times and you get ( θ /n )ˆ ✓ ◆ n I + (ˆ r × ) θ r × x x 0 = x n 25 25 Exponential Maps ✓ ◆ n I + (ˆ r × ) θ x 0 = lim x n !1 n • Remind you of anything? ⇣ ⌘ n 1 + a is a definition of e a lim n →∞ n • So the rotation we want is the exponential of ! (ˆ r × ) θ • In fact you can just plug it into the infinite series... 26 26 Tuesday, September 17, 13

Exponential Maps • Why exponential? • Recall series expansion of e x 1! + x 2 2! + x 3 e x = 1 + x 3! + ··· 27 27 Exponential Maps • Why exponential? e x • Recall series expansion of • Euler: what happens if you put in for x i θ 1! + − θ 2 2! + − i θ 3 3! + θ 4 e i θ = 1 + i θ 4! + ··· ✓ θ 1 + − θ 2 2! + θ 4 1! + − θ 3 ✓ ◆ ◆ + i = 4! + ··· 3! + ··· = cos ( θ )+ i sin ( θ ) 28 28 Tuesday, September 17, 13

Exponential Maps • Why exponential? r × ) 2 θ 2 r × ) 3 θ 3 r × ) 4 θ 4 r × ) θ = I + ( ˆ r × ) θ + ( ˆ + ( ˆ + ( ˆ e ( ˆ + ··· 1! 2! 3! 4! r × ) 3 = − ( ˆ But notice that: ( ˆ r × ) r × ) 2 θ 2 r × ) θ 3 r × ) 2 θ 4 r × ) θ = I + ( ˆ r × ) θ + ( ˆ + − ( ˆ + − ( ˆ e ( ˆ + ··· 1! 2! 3! 4! 29 29 Exponential Maps r × ) 2 θ 2 r × ) θ 3 r × ) 2 θ 4 r × ) θ = I + ( ˆ r × ) θ + ( ˆ + − ( ˆ + − ( ˆ e ( ˆ + ··· 1! 2! 3! 4! ✓ θ 1! − θ 3 + θ 2 2! − θ 4 ◆ ✓ ◆ r × ) θ = ( ˆ e ( ˆ r × ) 2 + I +( ˆ r × ) 3! + ··· 4! + ··· r × ) θ = ( ˆ e ( ˆ r × ) 2 ( 1 − cos ( θ )) r × ) sin ( θ )+ I +( ˆ 30 30 Tuesday, September 17, 13

Quaternions • More popular than exponential maps e i θ = cos ( θ )+ i sin ( θ ) • Natural extension of • Due to Hamilton (1843) • Interesting history • Involves “hermaphroditic monsters” 31 31 Quaternions • Uber-Complex Numbers q = ( z 1 , z 2 , z 3 , s ) = ( z , s ) q = iz 1 + jz 2 + kz 3 + s i j = k ji = − k i 2 = j 2 = k 2 = − 1 jk = i k j = − i ki = j ik = − j 32 32 Tuesday, September 17, 13

Quaternions • Multiplication natural consequence of defn. q · p = ( z q s p + z p s q + z p × z q , s p s q − z p · z q ) • Conjugate q ∗ = ( − z , s ) • Magnitude || q || 2 = z · z + s 2 = q · q ∗ 33 33 Quaternions • Vectors as quaternions v = ( v , 0 ) • Rotations as quaternions r sin θ 2 , cos θ r = ( ˆ 2 ) • Rotating a vector x 0 = r · x · r ⇤ • Composing rotations Compare to Exp. Map r = r 1 · r 2 34 34 Tuesday, September 17, 13

Quaternions • No tumbling • No gimbal-lock • Orientations are “double unique” || r || = 1 • Surface of a 3-sphere in 4D • Nice for interpolation 35 35 Interpolation 36 36 Tuesday, September 17, 13

Rotation Matrices • Eigen system • One real eigenvalue • Real axis is axis of rotation • Imaginary values are 2D rotation as complex number • Logarithmic formula θ 2sin θ ( R − R T ) ( ˆ r × ) = ln ( R ) = ✓ Tr ( R ) − 1 ◆ θ = cos − 1 2 Similar formulae as for exponential... 37 37 Rotation Matrices • Consider: r r r 1 0 0 ⎡ ⎤ ⎡ ⎤ xx xy xz ⎢ ⎥ ⎢ ⎥ RI r r r 0 1 0 = ⎢ ⎥ yx yy yz ⎢ ⎥ ⎢ r r r ⎥ 0 0 1 ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ zx zy zz • Columns are coordinate axes after (true for general matrices) • Rows are original axes in original system (not true for general matrices) 38 38 Tuesday, September 17, 13