3D orientation - College of Computing

[Pages:45]3D orientation

? Rotation matrix ? Fixed angle and Euler angle ? Axis angle ? Quaternion ? Exponential map

Joints and rotations

Rotational DOFs are widely used in character animation 3 translational DOFs 48 rotational DOFs

Each joint can have up to 3 DOFs

1 DOF: knee

2 DOF: wrist

3 DOF: arm

Representation of orientation

? Homogeneous coordinates (review)

? 4X4 matrix used to represent translation,

scaling, and rotation

x

?

a point in the space is represented as

p=

y z

1

? Treat all transformations the same so that they

can be easily combined

Translation

x + tx

1

0

0

tx

x

y + ty

=

0

1

0

ty y

z + tz 0 0 1 tz z

1

000 1 1

new point

translation matrix

old point

Scaling

sxx

sx

0

0 0 x

syy

=

0

sy

0

0 y

szz 0 0 sz 0 z

1

0 0 01 1

new point scaling matrix old point

Rotation

x 1

y 0

z

= 0

1

0

0

cos sin

0

0

- sin cos

0

0 x

0 y

0

z

11

X axis

x cos 0 sin 0 x

y 0 1 0 0 y

=

z - sin 0 cos 0 z

1

0 001 1

Y axis

x cos - sin 0 0 x

y sin cos 0 0 y

z

=

0

0

1

0

z

1

0

0 01 1

Z axis

Composite transformations

h0 x0 y0 z0 000

1 1 1

h1

A series of transformations on an object can be applied as a series of

matrix multiplications

2

h2

p : position in the global coordinate

p

x : position in the local coordinate

3 3

h3

(h3, 0, 0)

p = T(x0, y0, z0)R(0)R(0)R(0)T(0, h0, 0)R(1)R(1)R(1)T(0, h1, 0)R(2)T(0, h2, 0)R(3)R(3)x

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