coordinate system - Real-world example of Affine Transform?

From the Wikipedia article:

In geometry, an affine transformation or affine map or an affinity (from the Latin, affinis, "connected with") between two vector spaces (strictly speaking, two affine spaces) consists of a linear transformation followed by a translation.

Can anyone give a real-world example of when and how this is used in GIS?

Answer

Routine uses of 2D and 3D affine transformations in GIS include

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  Map-to-display transformations

  
  



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  Registering images and rasters

  
  


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  Changing 3D viewpoints

  
  


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  Modifying features by rescaling, shifting, and rotation

  
  


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  Datum changes (3-point and 7-point formulas).

  
  

These are described in more detail and illustrated for the 2D case on this Web page, which is found when you search "affine transformation GIS". Other hits provide many more examples.

Affine transformations also provide some conceptual simplifications. For example, every regular grid of locations is affinely equivalent to the grid of points with integral coordinates and all ellipsoidal models of the earth are affinely equivalent to the unit sphere centered at the origin.

Finally, note that (at least since the late 1800's) Euclidean geometry is the study of the group of distance-preserving affine transformations. Because almost all GIS processing--spatial indexes, spatial relations, spatial queries, "geoprocessing," etc--uses algorithms based on the Euclidean geometry of the map, affine transformations are fundamental to GIS.