Rotation representation and Interpolation

CSE 3541Matt Boggus

Rotation Matrices

1000

0100

00cossin

00sincos

1000

0cos0sin

0010

0sin0cos

1000

0cossin0

0sincos0

0001

Interpolating Orientations in 3D• Rotation matrices• Given rotation matrices Mi and time ti,

find M(t) such that M(ti)=Mi

x

y

z

M

uv

n

ux

vx

nx

uy

vy

ny

uz

vz

nz

Flawed Solution• Interpolate each entry independently• Example: M0 is identity and

M1 is 90o around x-axis

• Is the result a rotation matrix?

1 0 0 1 0 0 1 0 0

Interpolate ( 0 1 0 , 0 0 1 ) 0 0.5 0.5

0 0 1 0 1 0 0 0.5 0.5

1 0 0 1 0 0 1 0 0

Interpolate ( 0 1 0 , 0 0 1 ) 0 0.5 0.5

0 0 1 0 1 0 0 0.5 0.5

Orientation Representations

Rotation matrixFixed angles: rotate about global coordinate systemEuler angles: rotate about local coordinate systemAxis-angle: arbitrary axis and angleQuaternions: 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

zyx

A

Rotate about given axisEuler’s Rotation TheoremFairly easy to interpolate between orientationsDifficult to concatenate rotations

X

Y

Z

A Q

Axis-angle to rotation matrix

X

Y

Z

A Q Concatenate the following:Rotate A around z to x-z planeRotate A’ around y to x-axisRotate theta around xUndo rotation around y-axisUndo rotation around z-axis

Quaternions (intuitionally)

ARot A *)

2sin(

2cos

Same as axis-angle, but different formStill rotate about given axisMathematically convenient form

X

Y

Z

vs

qNote: in this form v is a scaled version of the given axis of rotation, A (which is a vector)

A Q

Quaternion

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)

q = s + xi + yj + zk where i, j, and k are imaginary numbers

Quaternion Arithmetic

Addition

Multiplication

Length

Inner Product

22112121 vsvsvvss

212112212121 vvvsvsvvssqq

212121 vvssqq

qqq

Quaternion Arithmetic

Inverse vsqq

2

11

000111 qqqq

111 pqpq

Unit quaternionq

qq ˆ

Identity

Quaternion Representation and Rotation

Vector

Rotation

v0

1)(' qvqvRotv q

Unit Quaternion Conversions

22

22

22

2212222

2222122

2222221

yxsxyzsyxz

sxyzzxszxy

syxzszxyzy

Rot zyxs

vvzyx

s

/),,(

)(cos2 1

Axis-Angle

QuaternionsWhat 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)

DescriptionApplies 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)

DescriptionRotates the transform around axis by angle degrees.

Rotations within Unity

transform.RotateAround(point : Vector3, axis : Vector3, angle : float)

DescriptionRotates 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)

DescriptionRotates 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

Global axes: x, y, zLocal axes: u, v, wP(s) = parametric curve function with s on [0,1]

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

Frenet Frame tangent & curvature vector

Frenet Frame tangent & curvature vector

MBUuP

MBUuP

UMBuP

'')(''

')('

)(

0026''

0123'

12

23

uU

uuU

uuuU

Frenet Framelocal coordinate system

Directly control orientation of object/camera

Use for direction and bank into turn, especially for ground-planar 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 point P(s+ds) for direction

Use auxiliary curve to define direction or up vector

Direction & Up vector

Direction vector

w

u=w × y-axis

v = u × wTo 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:1. Remember that2. Affects of scale are divided out by the inverse appearing in quaternion

rotation3. When interpolating quaternions, use UNIT quaternions – otherwise

magnitudes can interfere with spacing of results of interpolation

)()( vRotvRot kqq

Orientation interpolation

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?

Quaternions can be interpolated to produce in-between orientations:

21)1( kqqkq

Dual representation

Dual unit quaternion representations: q = –q

)()( vRotvRot kqq

For Interpolation between q1 and q2, compute cosine between q1 and q2 and between q1 and –q2; choose smallest angle

Interpolating quaternions

Linearly interpolating unit quaternions: not equally spaced

Unit quaternions form set of points on 4D sphere

Interpolating quaternions in great arc => equal spacing

Interpolating quaternions with equal spacing

2121 sin

sin

sin

)1sin(),,(slerp q

uq

uuqq

‘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

cos21 qqwhere