Transcript ppt
Orientations • Goal: – Convenient representation of orientation of objects & characters in a scene • Applications to: – inverse kinematics – rigid body simulation – rag doll physics – etc. Euler Angles • …so far, used one angle per axis, ie. Euler angles. • Pros: – Simple! • Cons: – Singularities (“gimbal lock”) – Interpolation is tricky – Composing rotations is tricky Singularities • Singularities may arise when axes are rotated to coincide • Lose a degree of freedom • Consider an airplane’s roll-pitch-yaw – eg. (0,90,0) vs. (90,90,90) – identical position • IK problems: – Infinitely many solutions if coincident – Ill-posed when very close to coincident Interpolation • Vital for keyframe animation, blending mocap data, etc. • Linear interpolating each angle independently gives odd behaviour near singularities • Not “coordinate independent” – For 2 orientations, rotate-interpolate gives a different result than interpolate-rotate Composition • Given two rotations applied sequentially, find a single orientation that gives the same result. • Not at all obvious how to handle this 3x3 Orthogonal Matrices • Eliminate many issues of Euler angles, but… – Hard to work with: 9 components, 6 constraints – Highly redundant – Linear interpolation doesn’t work 3x3 Orthogonal Matrices • But: – Very effective for quickly transforming many points – Composition is straightforward – multiply • Still useful, but not as our primary representation Axis-angle • Unit axis vector and an angle indicating how much to rotate. • Very intuitive • 4 numbers, 1 constraint – less redundant • Comparison, interpolation, composition still difficult. 2D Rotations • Can use complex numbers to represent 2D rotations • View a point (x,y) as x+ i*y • Rotation about origin is multiplication by: – cos(theta) + i*sin(theta) • Equivalently, e^(i*theta) – ie. A unit magnitude complex number 2D Rotations • And composition is again multiplication • Isn’t this overkill? – In 2D, yes – But this idea extends elegantly to 3D Quaternions • • • • Generalizes complex numbers Two additional sqrt(-1) terms, j and k Form: a + bi + cj + dk Behave mostly like regular numbers: – But NOT commutative! a*b ≠ b*a Quaternions • Small set of simple multiplication rules: i*i = -1, j*j = -1, k*k = -1 i*j = k, j*k = i, k*i = j j*i = -k, k*j=-i, i*k = -j • When j & k components are 0, this reverts to basic complex numbers – ie. 2D rotations. Rotating points • Given complex number: q a • Conjugate is: q a ib 2 • Then: q 0 ix q 0 q ix ib • If q is unit magnitude, just gives back 0 + ix • Real part remains 0, imaginary part retains its length, x Rotating points • • • • Quaternion: q a ib jc kd Conjugate: q a ib jc kd Zero real-part quaternion: 0 ix jy Consider: q 0 ix jy kz q q 2 x 2 y kz 2 z 2 • Real part stays 0, imaginary part keeps length • For unit q, this is a rotation of the vector (x,y,z) in quaternion form Quaternion Rotation • So, to perform a rotation by unit quaternion q on vector v = (x,y,z) to get new vector v’ = (x’,y’,z’) just do: 0 ix ' jy ' kz ' q 0 ix jy kz q Quaternion Rotation • Unit-length quaternions can represent any orientation in 3D, without singularities • Relation to axis-angle form: – Real part is cos(theta/2) – Imaginary part is vector parallel to the rotation axis • Redundancy: q = -q – But since the two possible representations are maximally far apart, no harm done. Quaternion Interpolation • Interpolation is much more convenient – linear interpolating, then renormalizing q to unit-length works fairly well – For better results, ensure dot product of their imaginary parts is non-negative,by flipping sign, since q = -q is the same orientation. • We want the two quaternion’s “axes” to be pointing in the same direction, so that the interpolation path is as short as possible. • For improved interpolation of unit vectors, look into “slerp” Quaternion Composition • Similar to matrix form, just multiply • Consider 2 quaternion rotations applied to a point 0+ix+jy+kz q 1 q2 ix jy kz q 2 q1 q1 q 2 ix or q 3 ix jy kz q1 q 2 jy kz q3 for q 3 q 1 q2