CSCE 441: Computer Graphics Rotation Representation and Interpolation Jinxiang Chai

Toy Example A 2D lamp with 6 degrees of freedom lower arm middle arm Upper arm base 2

Toy Example A 2D lamp with 6 degrees of freedom base

Toy Example A 2D lamp with 6 degrees of freedom Upper arm base

Toy Example A 2D lamp with 6 degrees of freedom middle arm Upper arm base

Toy Example A 2D lamp with 6 degrees of freedom lower arm middle arm Upper arm base

Joints and Rotation Rotational dofs are widely used in character animation 3 translation dofs 48 rotational dofs 1 dof: knee 2 dof: wrist 3 dof: shoulder

Orientation vs. Rotation 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 

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

Outline Rotation matrix Fixed angle and Euler angle Axis angle Quaternion

Matrices as Orientation Matrices just fine, right? No… 9 values to interpolate don’t interpolate well

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

Rotation New points rotation matrix old points

Interpolation In order to “move things”, we need both translation and rotation Interpolating the translation is easy, but what about rotations?

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

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

Properties of Rotation Matrix Easily composed? Interpolation? Compact representation?

Properties of Rotation Matrix Easily composed? yes Interpolation? Compact representation?

Properties of Rotation Matrix Easily composed? yes Interpolation? not good Compact representation?

Properties of Rotation Matrix Easily composed? yes Interpolation? not good Compact representation? - 9 parameters (only needs 3 parameters)

Outline Rotation matrix Fixed angle and Euler angle Axis angle Quaternion

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

Fixed Angles 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. X Z Y E.g., (qz, qy, qx) Q = Rx(qx). Ry(qy). Rz(qz). P

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

Euler Angles vs. Fixed Angles z-x-z Euler angles: (-60,30,45) z-x-z fixed angles: (45,30,-60)

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!)

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

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 )

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

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 (90,0,0) y x

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 (90,0,0) (90,90,0) y y x z x

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 (90,0,0) (90,90,0) (90,90,90) y y y z x z x x

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 (0,90,0) y z x

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

Gimbal Lock When two rotational axis of an object pointing in the same direction, the rotation ends up losing one degree of freedom

Outline Rotation matrix Fixed angle and Euler angle Axis angle Quaternion

Axis Angle Rotate an object by q around A (Ax,Ay,Az,q) A Y q Z X The axis is really only 2 DoFs - its length is irrelevant Z X Euler’s rotation theorem: An arbitrary rotation may be described by only three parameters.

Axis-angle Rotation Given r – Vector in space to rotate n – Unit-length axis in space about which to rotate q – The amount about n to rotate Solve r’ – The rotated vector r’ r n

Axis-angle Rotation Compute rpar: the projection of r along the n direction rpar = (n·r)n r’ rpar r

Axis-angle Rotation Compute rperp: rperp = r-rpar rperp rpar r’ r

Axis-angle Rotation Compute v: a vector perpedicular to rpar and rperp: v= rparxrperp v rperp rpar r’ r

Axis-angle Rotation Compute v: a vector perpedicular to rpar and rperp: v= rparxrperp v rperp rpar r’ Use rpar, rperp and v, θ to compute the new vector! r

Axis-angle Rotation rperp = r – (n·r) n q V = n x (r – (n·r) n) = n x r r’ rpar = (n·r) n r n r’ = r’par + r’perp = r’par + (cos q) rperp + (sin q) V =(n·r) n + cos q(r – (n·r)n) + (sin q) n x r = (cos q)r + (1 – cos q) n (n·r) + (sin q) n x r

Axis-angle Rotation Can interpolate rotation well

Axis-angle Interpolation 1. Interpolate axis from A1 to A2 Rotate axis about A1 x A2 to get A A1 q1 A Y q A2 A1 x A2 2. Interpolate angle from q1 to q2 to get q q2 Z X 3. Rotate the object by q around A

Axis-angle Rotation Can interpolate rotation well Compact representation Messy to concatenate - might need to convert to matrix form

Outline Rotation matrix Fixed angle and Euler angle Axis angle Quaternion

Quaternion Remember complex numbers: a+ib, where i2=-1 Quaternions are a non-commutative extension of complex numbers Invented by Sir William Hamilton (1843) Quaternion: - Q = a + bi + cj + dk: where i2=j2=k2=ijk=-1,ij=k,jk=i,ki=j - Represented as: q = (w, v) = w + xi + yj + zk

Quaternion 4 tuple of real numbers: w, x, y, z Same information as axis angles but in a more computational-friendly form

Quaternion Math Unit quaternion Multiplication Non-commutative Associative

Quaternion Math Conjugate Inverse

Quaternion Example let then

How to Represent Rotation? Axis-angle q r’ r n r’ = r’par + r’perp = r’par + (cos q) rperp + (sin q) V =(n·r) n + cos q(r – (n·r)n) + (sin q) n x r = (cos q)r + (1 – cos q) n (n·r) + (sin q) n x r

Quaternion Rotation

Quaternion Rotation

Quaternion Example Rotate a point (1,0,0) about y-axis with -90 degrees z (0,0,1) y x (1,0,0)

Quaternion Example Rotate a point (1,0,0) about y-axis with -90 degrees z (0,0,1) y x (1,0,0)

Quaternion Example Rotate a point (1,0,0) about y-axis with 90 degrees z (0,0,1) y x (1,0,0) Point (1,0,0)

Quaternion Example Rotate a point (1,0,0) about y-axis with 90 degrees z (0,0,1) y x (1,0,0) Point (1,0,0) Quaternion rotation

Quaternion Example Rotate a point (1,0,0) about y-axis with 90 degrees z (0,0,1) y x (1,0,0)

Quaternion Composition Rotation by p then q is the same as rotation by qp

Matrix Form For a 3D point (x0,y0,z0)

Quaternion Interpolation 1-angle rotation can be represented by a unit circle

Quaternion Interpolation 1-angle rotation can be represented by a unit circle 2-angle rotation can be represented by a unit sphere

Quaternion Interpolation 1-angle rotation can be represented by a unit circle 2-angle rotation can be represented by a unit sphere Interpolation means moving on n-D sphere Now imagine a 4-D sphere for 3-angle rotation

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

Quaternion Interpolation Direct linear interpolation does not work Linearly interpolated intermediate points are not uniformly spaced when projected onto the circle p0 p1

Quaternion Interpolation Spherical linear interpolation (SLERP) - Constant speed motion along a unit radius great circle arc, given the ends and an interpolation parameter between 0 and 1 - Normalize to regain unit quaternion p0 p1 p1(p1-1p0)1-t

Quaternion Interpolation Spherical linear interpolation (SLERP) Normalize to regain unit quaternion p0 p1 p1(p1-1p0)1-t

Quaternion Interpolation Spherical linear interpolation (SLERP) Normalize to regain unit quaternion p0 p1 p1(p1-1p0)1-t

Quaternions Can interpolate rotation well Compact representation Also easy to concatenate