Computer animation algorithms and techniques
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Computer Animation Algorithms and Techniques. Technical Background. Spaces and Transformations. Left-handed v. right handed Homogeneous coordinates: 4x4 transformation matrix (TM) Concatenating TMs Basic transformations (TMs) Display pipeline. Display Pipeline. Object Space Data.

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Computer Animation Algorithms and Techniques

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Computer animation algorithms and techniques

Computer AnimationAlgorithms and Techniques

Technical Background


Computer animation algorithms and techniques

Spaces and Transformations

Left-handed v. right handed

Homogeneous coordinates:

4x4 transformation matrix (TM)

Concatenating TMs

Basic transformations (TMs)

Display pipeline


Computer animation algorithms and techniques

Display Pipeline

Object Space Data

Object space to world space (TM)

World Space Data

World space to eye space (TM)

Eye Space Data

Perspective (TM)

Clipping

Perspective Divide

Image Space Data

Image space to display space (TM)

Display Space Data


Computer animation algorithms and techniques

Representing an orientation

Example: fixed angles - rotate around global axes

Y

Orientation:

X

Z


Computer animation algorithms and techniques

Y

Z

Y

X

X

Z

Working with fixed angles and Rotation Matrices (RMs)

Orthonormalizing a RM

Extracing fixed angles from an orientation

Extracing fixed angles from a RM

Making a RM from fixed angles

Making a RM from transformed unit coordinate system (TUCS)


Computer animation algorithms and techniques

Transformations in pipeline

object -> world: often rigid transforms

world -> eye: rigid transforms

perspective matrix: uses 4th component of homo. coords

perspective divide

image -> screen: 2D map to screen coordinates

Clipping: procedure that considers view frustum


Computer animation algorithms and techniques

Error considerations

Accumulated round-off error - transform data:

transform world data by delta RM

update RM by delta RM; apply to object data

update angle; form RM; apply to object data

orthonormalization

rotation matrix: orthogonal, unit-length columns

iterate update by taking cross product of 2 vectors

scale to unit length

considerations of scale

miles-to-inches can exeed single precision arithmetic


Computer animation algorithms and techniques

Orientation Representation

Rotation matrix

Fixed angles: rotate about global coordinate system

Euler angles: rotate about local coordinate system

Axis-angle: arbitrary axis and angle

Quaternions: mathematically handy axis-angle 4-tuple

Exponential map: 3-tuple version of quaternions


Computer animation algorithms and techniques

Transformation Matrix


Computer animation algorithms and techniques

Transformation Matrix


Computer animation algorithms and techniques

Rotation Matrices


Computer animation algorithms and techniques

Y

X

Z

Fixed Angles

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

Global axes


Computer animation algorithms and techniques

Y

Y

X

X

Z

Z

Gimbal Lock

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


Computer animation algorithms and techniques

Y

X

Z

Gimbal Lock

Fixed order of rotations: x, y, z

What do these epsilon rotations do?


Computer animation algorithms and techniques

Y

Y

X

X

Z

Z

Gimbal Lock

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


Computer animation algorithms and techniques

Y

X

Z

Euler Angles

Prescribed order: e.g., x, y, z or x, y, x

Rotate around (rotated) local axes

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


Computer animation algorithms and techniques

Y

X

Z

Axis-Angle

Q

A

Rotate about given axis

Euler’s Rotation Theorem

OpenGL

Fairly easy to interpolate between orientations

Difficult to concatenate rotations


Computer animation algorithms and techniques

Y

X

Z

Axis-angle to rotation matrix

Concantenate the following:

Rotate A around z to x-z plane

Rotate A’ around y to x-axis

Rotate theta around x

Undo rotation around y-axis

Undo rotation around z-axis

Q

A


Computer animation algorithms and techniques

Y

X

Z

Axis-angle to rotation matrix

A

P’

Q

P


Computer animation algorithms and techniques

Y

X

Z

Quaternion

Q

A

Same as axis-angle, but different form

Still rotate about given axis

Mathematically convenient form

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


Computer animation algorithms and techniques

Quaternion Arithmetic

Addition

Multiplication

Inner Product

Length


Computer animation algorithms and techniques

Quaternion Arithmetic

Inverse

Unit quaternion


Computer animation algorithms and techniques

Quaternion Represention

Vector

Transform


Computer animation algorithms and techniques

Quaternion Geometric Operations


Computer animation algorithms and techniques

Unit Quaternion Conversions

Axis-Angle


Computer animation algorithms and techniques

Quaternions

Avoid gimbal lock

Easy to rotate a point

Easy to convert to a rotation matrix

Easy to concatenate – quaternion multiply

Easy to interpolate – interpolate4-tuples

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


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