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Rotation representation and Interpolation CSE 3541 Matt Boggus 0 0 1 0 cos sin 0 sin cos 0 0 0 Rotation Matrices cos 0 sin 0 0 0 0 1 0 sin 0 1 0 0 0 cos 0 0 0 1 cos sin sin cos 0 0 0 0 0 0 0 0 1 0 0 1 Interpolating Orientations in 3D • Rotation matrices • Given rotation matrices Mi and time ti, find M(t) such that M(ti)=Mi y v u x z n ux uy uz M vx vy vz nx ny nz Flawed Solution • Interpolate each entry independently • Example: M0 is identity and M1 is 90o around x-axis 0 0 1 0 0 1 0 0 1 Interpolate ( 0 1 0 , 0 0 1 ) 0 0.5 0.5 0 0 1 0 1 0 0 0.5 0.5 • Is the result a rotation matrix? Orientation Representations Rotation matrix Fixed angles: rotate about global coordinate system Euler angles: rotate about local coordinate system Axis-angle: arbitrary axis and angle Quaternions: mathematically handy axis-angle 4-tuple Fixed Angles • Any orientation can be described by composing three rotations, one around each global coordinate axis • Roll, pitch and yaw (perfect for flight simulation) http://www.fho-emden.de/~hoffmann/gimbal09082002.pdf Gimbal Lock • Two or more axes align resulting in a loss of rotation degrees of freedom. http://www.fho-emden.de/~hoffmann/gimbal09082002.pdf Fixed Angles in the Real World • Apollo inertial measurement unit • To “prevent” lock, they added a fourth Gimbal http://www.hq.nasa.gov/office/pao/History/alsj/gimbals.html Axis-Angle A x Y Q A y z X Z Rotate about given axis Euler’s Rotation Theorem Fairly easy to interpolate between orientations Difficult to concatenate rotations Axis-angle to rotation matrix Y Q A X Z Concatenate the following: Rotate A around z to x-z plane Rotate A’ around y to x-axis Rotate theta around x Undo rotation around y-axis Undo rotation around z-axis Quaternions (intuitionally) Rot Y sin( ) * A A cos 2 2 Q A X Same as axis-angle, but different form Still rotate about given axis Mathematically convenient form s q v Z Note: in this form v is a scaled version of the given axis of rotation, A (which is a vector) Quaternion q = s + xi + yj + zk where i, j, and k are imaginary numbers Rotation Quaternions Identity (no rotation) 1, -1 180 degrees about x-axis i, -i 180 degrees about y-axis j, -j 180 degrees about z-axis k, -k angle θ, axis (unit vector) Quaternion Arithmetic Addition s1 s2 v1 v2 s1 v1 s2 Multiplication q1q2 s1s2 v1 v2 Inner Product Length s2v1 s1v2 v1 v2 q1 q2 s1s2 v1 v2 q qq v2 Quaternion Arithmetic Identity 1 1 q Inverse q 2 s qq 1 q 1q 1 pq 1 Unit quaternion 1 v 0 q p qˆ q q 1 0 0 Quaternion Representation and Rotation Vector Rotation 0 v v' Rot q (v) qvq1 Unit Quaternion Conversions Rot s x y 1 2 y 2 2 z 2 2 xy 2 sz 2 xz 2sy 2 2 1 2x 2z 2 yz 2 sx z 2 xy 2 sz 2 2 2 xz 2 sy 2 yz 2sx 1 2 x 2 y Axis-Angle 2 cos 1 ( s) ( x, y , z ) v / v Quaternions What an animator needs to know Avoid gimbal lock Easy to rotate a point Easy to convert to a rotation matrix Easy to concatenate – quaternion multiply Easy to interpolate – interpolate 4-tuples Rotations within Unity transform.Rotate(eulerAngles : Vector3, relativeTo : Space = Space.Self) Description Applies a rotation of eulerAngles.z degrees around the z axis, eulerAngles.x degrees around the x axis, and eulerAngles.y degrees around the y axis (in that order). Rotations within Unity transform.Rotate(axis : Vector3, angle : float, relativeTo : Space = Space.Self) Description Rotates the transform around axis by angle degrees. Rotations within Unity transform.RotateAround(point : Vector3, axis : Vector3, angle : float) Description Rotates the transform about axis passing through point in world coordinates by angle degrees. This modifies both the position and the rotation of the transform. Rotations within Unity Vector3.RotateTowards(current : Vector3, target : Vector3, maxRadiansDelta : float, maxMagnitudeDelta : float) Description Rotates a vector current towards target. Orientation along a curve Orientation on a curve in 2D Orientation on a curve in 2D Orientation on a curve in 2D q1 and q2 are orthogonal to P’(u) i.e. q1 = Rotate(90) * P’(u) q2 = Rotate(-90) * P’(u) Frenet Frame a moving (coordinate) frame that provides a coordinate system at each point of the curve that is "best adapted" to the curve near that point Global axes: x, y, z Local axes: u, v, w P(s) = parametric curve function with s on [0,1] Frenet Frame tangent & curvature vector Frenet Frame tangent & curvature vector P (u ) UMB P ' (u ) U ' MB P ' ' (u ) U ' ' MB U ' 3u Uu 3 u 2 2 u 1 2u 1 0 U ' ' 6u 2 0 0 Frenet Frame local coordinate system Directly control orientation of object/camera Use for direction and bank into turn, especially for groundplanar curves (e.g. roads) w = P’(s) u = P’’(s) × P’(s) v=w×u Frenet Frame – sometimes undefined Frenet Frame – discontinuity Other ways to control orientation Use auxiliary curve to define direction or up vector Use point P(s+ds) for direction Direction & Up vector v=u×w w u=w × y-axis Direction vector To keep ‘head up’, use y-axis to compute over and up vectors perpendicular to direction vector If up vector supplied, use that instead of y-axis Orientation interpolation Preliminary note: Rot q (v) Rot kq (v) 1. Remember that 2. Affects of scale are divided out by the inverse appearing in quaternion rotation 3. When interpolating quaternions, use UNIT quaternions – otherwise magnitudes can interfere with spacing of results of interpolation Orientation interpolation Quaternions can be interpolated to produce in-between orientations: q (1 k )q1 kq2 2 problems analogous to issues when interpolating positions: 1. How to take equidistant steps along orientation path? 2. How to pass through orientations smoothly (1st order continuous) 3. And another particular to quaternions: with dual unit quaternion representations, which to use? Dual representation Rot q (v) Rot kq (v) Dual unit quaternion representations: q = –q For Interpolation between q1 and q2, compute cosine between q1 and q2 and between q1 and –q2; choose smallest angle Interpolating quaternions Unit quaternions form set of points on 4D sphere Linearly interpolating unit quaternions: not equally spaced Interpolating quaternions in great arc => equal spacing Interpolating quaternions with equal spacing sin( 1 u ) sin u slerp (q1 , q2 , u ) q1 q2 sin sin where q1 q2 cos ‘slerp’, sphereical linear interpolation is a function of • the beginning quaternion orientation, q1 • the ending quaternion orientation, q2 • the interpolant, u Still a linear order interpolation