III-1

III 3D TransformationHomogeneous Coordinates

• The three dimensional point (x, y, z) is represented by the homogeneous coordinate (x, y, z, 1)

• In general, the homogeneous coordinate (x, y, z, w) represents the three dimensional point (x/w, y/w, z/w)

1131

1333][

snmlrjigqfedpcba

T

• The generalized transformation matrix:

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Scaling• In general, this is done with the equations:

xn = sx * xyn = sy * yzn = sz * z

• This can also be done with the matrix multiplication:

wzyx

ss

s

wzyx

z

y

x

n

n

n

*

1000000000000

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• Scaling can be done relative to the object center with a composite transformation

• Scaling an object centered at (cx, cy, cz) is done with the matrix multiplication:

wzyx

ccc

ss

s

ccc

wzyx

z

y

x

z

y

x

z

y

x

n

n

n

*

1000100010001

*

1000000000000

*

1000100010001

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Shearing• Equivalent to pulling faces in opposite

directions•

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Rotation• Rotation can be done around any line or vector• Rotations are commonly specified around the x,

y, or z axis• A positive angle of rotation results in a

counterclockwise movement when looked at from the positive axis direction

• The matrix form for rotation– x axis

wzyx

wzyx

n

n *

10000cossin00sincos00001

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wzyx

wzyx

n

n

*

10000cos0sin00100sin0cos

wzyx

wzyx

n

n

*

1000010000cossin00sincos

– y axis

– z axis

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Reflection• Reflection through the xy-plane:

• Reflection through the yz-plane:

• Reflection through the xz-plane:

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Translations• The amount of the translation is added to or

subtracted from the x, y, and z coordinates• In general, this is done with the equations:

xn = x + tx

yn = y + ty

zn = z + tz • This can also be done with the matrix

multiplication:

wzyx

ttt

wzyx

z

y

x

n

n

n

*

1000100010001

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Combining Transformations

• Matrices can be multiplied together to accomplish multiple transformations with one matrix

• A matrix is built with successive transformations occurring from right to left

• A combination matrix is typically built from the identity matrix with each new transformation added by multiplying it on the left of the current combination

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Rotation about an Arbitrary Axis in Space

• Assume an arbitrary axis in space passing through the point with direction cosines and rotation about this axis by some angle

•

),,( 000 zyx),,( zyx ccc

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• Direction cosines:

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• The complete transformation is:

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Reflection through an Arbitrary Plane

•

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• The general transformation is:

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