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# Spatial Transformation

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Geospatial Transformation

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### Spatial Transformation

1. 1. Spatial Transformations BY : EHSAN HAMZEI - 810392121
2. 2. 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. 1
3. 3. Linear Transformation  A 2 x 2 linear transformation matrix allows:  Scaling  Rotation  Reflection  Shearing 2
4. 4. Affine Transformation 3  Definition:  P(Px, Py) is transformed into Q(Qx , Qy ) as follows:  Qx = aPx + cPy + Tx  Qy = bPx + dPy + Ty
5. 5. Affine Properties 4  Preserves parallelism of lines, but not lengths and angles.  Lines are preserved.  Proportional distances are preserved (Midpoints map to midpoints).
6. 6. An Affine Application 4  Computer graphic
7. 7. 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
8. 8. Projective Transformation 4  Definition:
9. 9. 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
10. 10. Cross Ratio 4
11. 11. A Projective Application 4  Robot Recognition
12. 12. Review 4
13. 13. Thanks 26