Computer AnimationRick Parent

Computer AnimationAlgorithms and Techniques

Technical Background

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Spaces and Transformations

Left-handed v. right handedHomogeneous coordinates: 4x4 transformation matrix (TM) Concatenating TMsBasic transformations (TMs)Display pipeline

1zyx

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Display Pipeline

Object Space Data

World Space Data

Eye Space Data

Image Space Data

Display Space Data

Object space to world space (TM)

World space to eye space (TM)

Perspective (TM)ClippingPerspective Divide

Image space to display space (TM)

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Representing an orientation

Example: fixed angles - rotate around global axes

X

Y

Z

Orientation:

PRRRP xyz )()()('

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Working with fixed angles and Rotation Matrices (RMs)

Extracing fixed angles from an orientation

Extracing fixed angles from a RM

Making a RM from fixed angles

Orthonormalizing a RM

Making a RM from transformed unit coordinate system (TUCS)

X

Y

Z

X

YZ

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Transformations in pipeline

object -> world: often rigid transformsworld -> eye: rigid transformsperspective matrix: uses 4th component of homo. coordsperspective divideimage -> screen: 2D map to screen coordinatesClipping: procedure that considers view frustum

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Error considerations

Accumulated round-off error - transform data:transform world data by delta RMupdate RM by delta RM; apply to object dataupdate angle; form RM; apply to object data

orthonormalizationrotation matrix: orthogonal, unit-length columnsiterate update by taking cross product of 2 vectorsscale to unit length

considerations of scalemiles-to-inches can exeed single precision arithmetic

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Orientation Representation

Rotation matrixFixed angles: rotate about global coordinate systemEuler angles: rotate about local coordinate systemAxis-angle: arbitrary axis and angleQuaternions: mathematically handy axis-angle 4-tupleExponential map: 3-tuple version of quaternions

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Transformation Matrix

ponm

lkji

hgfe

dcba

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Transformation Matrix

1000z

y

x

tkji

tgfe

tcba

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Rotation Matrices

1000

0100

00cossin

00sincos

1000

0cos0sin

0010

0sin0cos

1000

0cossin0

0sincos0

0001

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Fixed Angles

Fixed order: e.g., x, y, z; also could be x, y, xGlobal axes

X

Y

Z

PRRRP xyz )()()('

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Gimbal Lock

0900

X

Y

Z

Fixed angle: e.g., x, y, z

000

X

Y

Z

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Gimbal Lock

0900Fixed order of rotations: x, y, z

X

Y

Z

0900

What do these epsilon rotations do?

0900

0900

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Gimbal Lock

0900

X

Y

Z

90090

X

Y

Z

Interpolating FA representations does not produce intuitive rotation because of gimbal lock

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Euler Angles

Prescribed order: e.g., x, y, z or x, y, xRotate around (rotated) local axes

X

Y

Z

Note: fixed angles are same as Euler angles in reverse order and vice versa

PRRRP zyx )()()('

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Axis-Angle

zyx

A

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

X

Y

Z

A Q

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Axis-angle to rotation matrix

X

Y

Z

A Q Concantenate 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

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Axis-angle to rotation matrix

*

)(

*

)sin()ˆ)(cos(ˆ

0

0

0

ˆ

AAIARot

aa

aa

aa

A

azaaaaa

aaaaaa

aaaaaa

A

zyx

xy

xz

yz

zyzxz

zyyyxy

zxyxxx

X

Y

Z

AQ

P

P’

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Quaternion

ARot A *)

2sin(

2cos

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

X

Y

Z

vs

q

Note: in this form v is a scaled version of the given axis of rotation, A

A Q

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Quaternion Arithmetic

Addition

Multiplication

Length

Inner Product

22112121 vsvsvvss

212112212121 vvvsvsvvssqq

212121 vvssqq

qqq

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Quaternion Arithmetic

Inverse vsqq

2

11

000111 qqqq

111 pqpq

Unit quaternionq

qq ˆ

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Quaternion Represention

Vector

Transform

v0

1)(' qvqvRotv q

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Quaternion Geometric Operations

)())(( vRotvRotRotv qppq

)()( vRotvRot qq

vqqvqqvRotRotv qq

)())(( 111

)()( vRotvRot kqq

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Unit Quaternion Conversions

22

22

22

2212222

2222122

2222221

yxsxyzsyxz

sxyzzxszxy

syxzszxyzy

Rot zyxs

vvzyx

s

/),,(

)(cos2 1

Axis-Angle

Computer AnimationRick Parent

Quaternions

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

How about smoothly (in both space and time) interpolate?