rick parent - cis681 orientation use quaternions interpolating rotations is difficult

Rick Parent - CIS681

ORIENTATION

Use Quaternions

Interpolating rotations is difficult


Object Representation

Desired operationsInterpolation between transformationsConcatenation of one transformation after another

Define object in world spaceObject space data ScaleRotation Translation

Handle scale, rotation, translation, independently

Rotation deserves special attention!


Repeated Rotations: Error Management

Task: Rotate an object some every frame

Issue: Avoiding accumulated roundoff error


Repeated Rotations: Error Management

Method 2D = create_rotation_matrix()M create_rotation_matrix()M = D MObject = apply M to object

Method 1M = create_rotation_matrix()Object = apply M to Object

Method 3 = + M = create_rotation_matrix()Object = apply M to object

<= repeat

<= repeat

<= repeat


Orientation Representation

orientation


Interpolation

O1

O2

O 1.5


Concatenation

O1

O2


Orientation Representation

Rotation Matrix

Fixed Angles

Euler Angles

Axis-Angle

Quaternion


Rotation Matrices

0

0

0

1

a

d

g

b c

e f

h i

0 0 0

9 values but 3 degrees of freedom

Euler’s rotation theorem: An arbitrary rotation may be described by only three parameters.


Rotation Matrices

00

0

1

01

0

-1 00 0

0 1

0 0 0

000

1

0-10

1 00 00 1

0 0 0

Can’t interpolate rotation matrices

90o z-axis -90o z-axis0

0

0

1

0

0

0

0 0

0 0

0 1

0 0 0

??


Fixed Angles

E.g., (z, y, x)

XZ

Y

Q = Rx(x). Ry(y). Rz(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

X

Z

Y

X

Z

Y

Note: left-hand coordinate system

Original orientationOrientation represented by

(0,90,0)

Using order z-y-x


Fixed Angles

(45,90,0)

X

Z

Y

X

Z

Y

Original

Using order z-y-x


Gimbal Lock

From (0,90,0), how can the object change its orientation?

X

Z

Y

a) (,90,0)

b) (,90+,0)

c) (,90,)

What do these do?

Using order z-y-x


Fixed Angles

(-45,90,0)(0,90,0)

X

Z

Y

Z

X

Y

Is same asX-axisrotation

(0,90,45)(0,90,0)

Changing Z-axis parameter


Fixed Angle Interpolation

(0,90,0) to (90,0,90)(0,0,0)

(90,0,90)(0,90,0)


Euler Angles

Ordered triple of rotations about local axes, As with fixed angles, 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

Y

Z

x

y

z


Euler Angles

X

Y

Z

x

y

z

Use (z,y,x)

P Rz(1)P

P Rz(1)Ry (2)Rz ( 1) P

P Rz(1)Ry (2)Rz ( 1)Rz (1)P

P Rz(1)Ry (2)P

P Rz(1)Ry (2)Rx (3)Ry ( 2)Rz ( 1) P

P Rz(1)Ry (2)Rx (3)Ry ( 2)Rz ( 1)Rz (1)Ry (2)P

P Rz(1)Ry (2)Rx (3)P

Show that Euler angle ordering is equivalent to reverse ordering in fixed angles

…and so has the same problems


Axis-Angle

(Ax,Ay,Az,)

Euler’s rotation theorem: An arbitrary rotation may be described by only three parameters.

X

Y

Z

Rotate object by around A

?


Axis-Angle Interpolation

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 object by around A

A1 x A2


Quaternions

(cos(/2),sin(/2)*A)

q =[s,v]=[s,x,y,z]

Has the same information as axis-angle but in a more computational-friendly form


Quaternions

s1 v1 s2 v2 s1 s2 v1 v2 s1 v1 s2 v2 s1s2 v1 v2 s1v2 s2v1 v1 v2 q s2 x 2 y 2 z2

q 1 0 0 0 q

q 1 s v

q2

qq 1 1 0 0 0

Basic math operations


Quaternions - rotate a point

v = (x,y,z) => [0,v]

Rotq (v) v q 0 v q 1


Composite transformations

Rotq (Rotp (v)) Rotq ( p 0 v p 1)

qp 0 v p 1q 1

qp 0 v (qp) 1

Rotqp (v)

Rotation by p then by q is the same as rotation by qp(where qp is quaternion q multiplied by quaternion p)


Quaternion Rotation

q

||q||Unit quaternion =>

Rot s v Rotk s v Rot ks kv

Rot s v Rot s v


Quaternion Interpolation

Fixed angles

(90,0,90)(0,90,0)

quaternions

[0.7,0.0,0.7,0.0] [0.5,0.5,0.5,0.5]


Quaternion Interpolation

Linearly interpolating fixed angles from (0,90,0) to

(90,0,90)

Interpolating quaternions from (0.5,0.0,1.0,0.0) to (0.5,0.5,0.5,0.5) using sphereical linear interpolation

QuickTime™ and aVideo decompressor

are needed to see this picture.

QuickTime™ and aVideo decompressor

are needed to see this picture.

rick parent - cis681 orientation use quaternions interpolating rotations is difficult

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