# affine geometry

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## affine geometry:

see geometry**geometry**

[Gr.,=earth measuring], branch of mathematics concerned with the properties of and relationships between points, lines, planes, and figures and with generalizations of these concepts.

**.....**Click the link for more information. .

*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Affine Geometry

a branch of geometry that studies those properties of figures in a plane or in space that are preserved under any affine transformation of the plane or space. An example of such a transformation is the transformation of similitude. The properties of a geometric figure which are preserved under any affine transformation are naturally called the affine invariants of this figure. A basic affine invariant is the simple ratio of three points *M*_{1}, *M*_{2}, and *M*_{3} which lie on a straight line. If *x*_{1}*x*_{2}, and *x*_{3} are the respective abscissas of these points, then the simple ratio is equal to (*x*_{2} − *x*_{1}) / (*x*_{3} − *x*_{2}). The affine invariants of any system consisting of *n* points (*n* ≥ 4) can be expressed in terms of simple ratios. Hence, in particular, it follows that the center of gravity of a geometric figure is preserved in affine transformations. In arbitrary affine transformations, parallel straight lines remain parallel. The methods and facts of affine geometry are widely used in different branches of natural science—mechanics, theoretical physics, and astronomy. For example, small deformations of a continuous medium, elastic in the first approximation, can be studied by the methods of affine geometry.

### REFERENCES

Aleksandrov, P. S.*Lektsii po analiticheskoi geometrii*. Moscow, 1968.

Efimov, N. V.

*Vysshaia geometriia*, 4th ed. Moscow, 1961.