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