Affine Map

up:: Affine Space

An affine transformation is a map which preserves affine structures ─ i.e. it preserves, in particular, “distances” between points in the set (with respect to the associated group action).

A map between affine spaces $f:(A_M,V_M,{+}_M)\to (A_N,V_N,{+}_N)$ is said to be an affine transformation if there is a Linear Transformation between their respective vector spaces $L:V_M\to V_N$ (which is called the underlying linear map of $f$) such that

$$\forall A\in A_M,\forall v\in V_M:f(A+v)=f(A)+L(v)$$

Equivalently, one can say that, for any pair of points $A,B\in V_M$, we have that

$$\overrightarrow{f(A)f(B)}=L\left(\overrightarrow{AB}\right)$$

Properties

• The underlying linear map of an affine map is unique
• An affine map which is bijective is an Affine Isomorphism

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References

• Affine transformation - Wikipedia