15816 transformations

8/2/2019 15816 Transformations

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TRANSFORMATIONS


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INTRODUCTION

Geometric transformations play a central role in geometric modeling and

viewing.

They are used in modeling to express the locations of entities relative to

others and to move them around in the modeling space.

They are used in viewing to generate different views of a model for

visualizations and drafting purposes.

Typical CAD operations to translate, rotate, zoom and mirror entities are

all based on Geometric transformations.


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TWO DIMENSIONAL TRANSFORMATIONS


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Transformation of straight line


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ROTATION


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Continued


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SCALING


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THREE DIMENSIONAL TRANSFORMATIONS

A point in three dimensional space [x y z] is represented by a four

dimensional position vector

[x y z h] = [x y z 1] [T]

The transformation from homogeneous coordinates to ordinary

coordinates is given by[x* y* z* 1] = [x/h y/h z/h 1]

The homogeneous transformation matrix (4x4) is the combination

of 3x3 sub matrix in the form of rotation, scaling, shearing and

reflection. The 1x3 lower left submatrix produces translation, and

3x1 submatrix produces a perspective transformation. The final

lower right hand 1x1 submatrix produces overall scaling.


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THREE DIMENSIONAL SCALING

The diagonal terms of the general 4x4 transformation produces local and

overall scaling.

Consider

[X] [T] = [x y z 1]

= [ax ey jz 1] = [x* y* z* 1]

which shows the local scaling effect.

1000

000

000

000

j

e

a


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THREE DIMENSIONAL ROTATION

For rotation about the x-axis, the x coordinates of the position vector do

not change. In effect, the rotation occurs in plane perpendicular to the x-

axis. Similarly, rotation about the y-axis and z- axis occurs in plane

perpendicular to the y and z axis, respectively.

For rotation about the x-axis, the x coordinates of the transformed

position vector does not change.

[T] =

1000

0cossin0

0sincos0

0001


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1000

0cos0sin

0010

0sin0cos

1000

0100

00cossin

00sincos


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THREE DIMENSIONAL REFLECTION

Some orientations of a three dimensional object cannot be obtained using

pure rotation, they require reflection. In three dimension, reflection occur

through plane. For a pure reflection, the determinant of the reflection

matrix is identically -1.

Transformation matrix for a reflection through the xy plane is

[T] =

1000

0100

0010

0001


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[X] =

[X*]= [X] [T]

=

1211

1212

1202

1201

1111

1112

1102

1101

1211

1212

1202

1201

1111

1112

1102

1101

1000

0100

0010

0001


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The block A*B*C*B*E*F*G*H* shows new transformed position vectors

[X*] =

1211

1212

1202

1201

1111

1112

1102

1101


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THREE DIMENSIONAL TRANSLATION

[T] =

The translated homogeneous coordinates are obtained by writing

[x y z h] = [x y z 1]

[x y z h] = [(x+l) (y+m) (z+n) 1 ]

1

0100

0010

0001

nml

1

0100

0010

0001

nml

15816 transformations

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