Geospatial Transformation

1. Spatial
Transformations
BY : EHSAN HAMZEI - 810392121

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.
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3. Linear Transformation
 A 2 x 2 linear transformation matrix allows:
 Scaling
 Rotation
 Reflection
 Shearing
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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. Affine Properties 4
 Preserves parallelism of lines, but not lengths and
angles.
 Lines are preserved.
 Proportional distances are preserved (Midpoints map
to midpoints).

6. An Affine Application 4
 Computer graphic

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. Projective Transformation 4
 Definition:

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. Cross Ratio 4

11. A Projective Application 4
 Robot Recognition

12. Review 4

13. Thanks 26