Spatial Transformation
sBY : EHSAN HAMZEI - 810392121
Geometric transformations
Geometric transformations will map points in one space to points in another: (x',y',z') = f(x, y,
z).
These transformations can be very simple, such as scaling each coordinate, or complex, such as nonlinear twists and bends.
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Linear Transformation
A 2 x 2 linear transformation matrix allows: Scaling
Rotation
Reflection
Shearing
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Affine Transformation 3
Definition: P(Px, Py) is transformed into Q(Qx , Qy ) as follows: Qx = aPx + cPy + Tx
Qy = bPx + dPy + Ty
Affine Properties 4
Preserves parallelism of lines, but not lengths and angles.
Lines are preserved.
Proportional distances are preserved (Midpoints map to midpoints).
An Affine Application 4
Computer graphic
Projective Geometry 4
Euclidean geometry describes shapes “as they are”.
Projective geometry describes objects “as they appear”. (Ex: Railroad…)
Lengths, angles, parallelism become “distorted” when we look at objects
Projective Transformation 4
Definition:
Projective Properties 4
With projective geometry, two lines always meet in a single point, and two points always lie on a single line.
Mapping from points in plane to points in plane 3 aligned points are mapped to 3 aligned points Cross Ratio
Cross Ratio 4
A Projective Application 4
Robot Recognition
Review 4
Thanks 26