15816 transformations
TRANSCRIPT
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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
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1
0100
0010
0001
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