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CSCE 641: Computer Graphics Rotation Representation and Interpolation Jinxiang Chai
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Joints and Rotation Rotational dofs are widely used in character animation 3 translation dofs 48 rotational dofs 1 dof: knee2 dof: wrist3 dof: shoulder
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Orientation is described relative to some reference alignment A rotation changes object from one orientation to another Can represent orientation as a rotation from the reference alignment Orientation vs. Rotation
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Ideal Orientation Format Represent 3 degrees of freedom with minimum number of values Allow concatenations of rotations Math should be simple and efficient –concatenation –interpolation –rotation
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Outline Rotation matrix Fixed angle and Euler angle Axis angle Quaternion
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Matrices as Orientation Matrices just fine, right? No… –9 values to interpolate –don’t interpolate well
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Representation of orientation Homogeneous coordinates (review): –4X4 matrix used to represent translation, scaling, and rotation –a point in the space is represented as –Treat all transformations the same so that they can be easily combined
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Rotation New pointsrotation matrix old points
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Interpolation In order to “move things”, we need both translation and rotation Interpolating the translation is easy, but what about rotations?
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Interpolation of Orientation How about interpolating each entry of the rotation matrix? The interpolated matrix might no longer be orthonormal, leading to nonsense for the inbetween rotations
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Interpolation of Orientation Example: interpolate linearly from a positive 90 degree rotation about y axis to a negative 90 degree rotation about y Linearly interpolate each component and halfway between, you get this... Rotate about y-axis with 90 Rotate about y-axis with -90
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Properties of Rotation Matrix Easily composed? Interpolation? Compact representation?
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Properties of Rotation Matrix Easily composed? yes Interpolation? Compact representation?
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Properties of Rotation Matrix Easily composed? yes Interpolation? not good Compact representation?
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Properties of Rotation Matrix Easily composed? yes Interpolation? not good Compact representation? - 9 parameters (only needs 3 parameters)
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Outline Rotation matrix Fixed angle and Euler angle Axis angle Quaternion
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Fixed angles Angles are used to rotate about fixed axes Orientations are specified by a set of 3 ordered parameters that represent 3 ordered rotations about fixed axes Many possible orderings: x-y-z, x-y-x,y-x-z - as long as axis does immediately follow itself such as x-x-y
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E.g., ( z, y, x ) X Z Y Q = R x ( x ). R y ( y ). R z ( z ). P Ordered triple of rotations about global axes, any triple can be used that doesn’t immediately repeat an axis, e.g., x-y-z, is fine, so is x- y-x. But x-x-z is not. Fixed angles
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Euler Angles vs. Fixed Angles One point of clarification Euler angle - rotates around local axes Fixed angle - rotates around world axes Rotations are reversed - x-y-z Euler angles == z-y-x fixed angles
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Euler angle Interpolation Interpolating each component separately Might have singularity problem –Halfway between (0, 90, 0) & (90, 45, 90) –Interpolate directly, get (45, 67.5, 45) –Desired result is (90, 22.5, 90) (verify this!)
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Euler angle concatenation Can't just add or multiply components Best way: –Convert to matrices –Multiply matrices –Extract Euler angles from resulting matrix Not cheap
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Gimbal lock Euler/fixed angles not well-formed Different values can give same rotation Example with z-y-x fixed angles: –( 90, 90, 90 ) = ( 0, 90, 0 )
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Gimbal lock Euler/fixed angles not well-formed Different values can give same rotation Example with z-y-x fixed angles: –( 90, 90, 90 ) = ( 0, 90, 0 ) z x y
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Gimbal lock Euler/fixed angles not well-formed Different values can give same rotation Example with z-y-x fixed angles: –( 90, 90, 90 ) = ( 0, 90, 0 ) z x y z x y (90,0,0)
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Gimbal lock Euler/fixed angles not well-formed Different values can give same rotation Example with z-y-x fixed angles: –( 90, 90, 90 ) = ( 0, 90, 0 ) z x y z x y z x y (90,0,0)(90,90,0)
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Gimbal lock Euler/fixed angles not well-formed Different values can give same rotation Example with z-y-x Euler angles: –( 90, 90, 90 ) = ( 0, 90, 0 ) z y x z x y z x y z x y (90,0,0)(90,90,0)(90,90,90)
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Gimbal lock A Gimbal is a hardware implementation of Euler angles used for mounting gyroscopes or expensive globes Gimbal lock is a basic problem with representing 3D rotation using Euler angles or fixed angles
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Gimbal lock When two rotational axis of an object pointing in the same direction, the rotation ends up losing one degree of freedom
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Outline Rotation matrix Fixed angle and Euler angle Axis angle Quaternion
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(A x,A y,A z, ) Euler’s rotation theorem: An arbitrary rotation may be described by only three parameters. X Y Z Rotate an object by around A Axis angle
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Axis-angle rotation r r’ n Given r – Vector in space to rotate n – Unit-length axis in space about which to rotate – The amount about n to rotate Solve r’ – The rotated vector
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Axis-angle rotation r r’ r par Compute r par : the projection of r along the n direction r par = (n·r)n
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Axis-angle rotation r r’ r perp r par Compute r perp : r perp = r-r par
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Axis-angle rotation r r’ r perp v r par Compute v: a vector perpedicular to r par and r perp : v= r par xr perp
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Axis-angle rotation r r’ r perp v Use r par, r perp and v, θ to compute the new vector! r par Compute v: a vector perpedicular to r par and r perp : v= r par xr perp
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Axis-angle rotation r r’ n r par = (n·r) n r perp = r – (n·r) n V = n x (r – (n·r) n) = n x r r’ = r’ par + r’ perp = r’ par + (cos ) r perp + (sin ) V =(n·r) n + cos (r – (n·r)n) + (sin ) n x r = (cos )r + (1 – cos ) n (n·r) + (sin ) n x r
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Axis-angle rotation Can interpolate rotation well
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1. Interpolate axis from A1 to A2 Rotate axis about A1 x A2 to get A X Y Z 2. Interpolate angle from 1 to to get 3. Rotate the object by around A A1 x A2 Axis-angle interpolation
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Axis-angle rotation Can interpolate rotation well Compact representation Messy to concatenate - might need to convert to matrix form
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Outline Rotation matrix Fixed angle and Euler angle Axis angle Quaternion
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Remember complex numbers: a+ib, where i 2 =-1 Quaternions are a non-commutative extension of complex numbers Invented by Sir William Hamilton (1843) Quaternion: - Q = a + bi + cj + dk: where i 2 =j 2 =k 2 =ijk=-1,ij=k,jk=i,ki=j - Represented as: q = (w, v) = w + xi + yj + zk
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Quaternion 4 tuple of real numbers: w, x, y, z Same information as axis angles but in a more computational-friendly form
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Quaternion math Unit quaternion Multiplication Associative Non-commutative
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Quaternion math Conjugate Inverse
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Quaternion Example let then
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Quaternion Rotation
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Quaternion rotation
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Quaternion Example Rotate a point (1,0,0) about y-axis with -90 degrees z x y (1,0,0) (0,0,1)
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Quaternion Example Rotate a point (1,0,0) about y-axis with -90 degrees z x y (1,0,0) (0,0,1)
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Quaternion Example Rotate a point (1,0,0) about y-axis with 90 degrees z x y (1,0,0) (0,0,1) Point (1,0,0)
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Quaternion Example Rotate a point (1,0,0) about y-axis with 90 degrees z x y (1,0,0) (0,0,1) Point (1,0,0) Quaternion rotation
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Quaternion Example Rotate a point (1,0,0) about y-axis with 90 degrees z x y (1,0,0) (0,0,1)
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Quaternion composition Rotation by p then q is the same as rotation by qp
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Matrix Form For a 3D point (x 0,y 0,z 0 )
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Quaternion interpolation 1-angle rotation can be represented by a unit circle
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Quaternion interpolation 1-angle rotation can be represented by a unit circle 2-angle rotation can be represented by a unit sphere
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Quaternion interpolation Interpolation means moving on n-D sphere Now imagine a 4-D sphere for 3-angle rotation 1-angle rotation can be represented by a unit circle 2-angle rotation can be represented by a unit sphere
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Quaternion interpolation Moving between two points on the 4D unit Sphere: a unit quaternion at each step - another point on the 4D unit sphere move with constant angular velocity along the great circle between the two points on the 4D unit sphere
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Quaternion interpolation Direct linear interpolation does not work Linearly interpolated intermediate points are not uniformly spaced when projected onto the circle
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Quaternion interpolation Spherical linear interpolation (SLERP) Normalize to regain unit quaternion
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Quaternion interpolation Spherical linear interpolation (SLERP) Normalize to regain unit quaternion
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Quaternion interpolation Spherical linear interpolation (SLERP) Normalize to regain unit quaternion
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Quaternions Can interpolate rotation well Compact representation Also easy to concatenate
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