4. Affine maps

Affine maps

A function f\colon H\to M, where M and H are affine spaces is called an affine map, if for any p_0,\ldots, p_n\in H and a_0,\ldots, a_n\in K, such that a_0+\ldots +a_n=1,

    \[f(a_0p_0+\ldots +a_np_n)=a_0f(p_0)+\ldots + a_nf(p_n).\]

Equivalently, f is an affine map, if for a point p_0 and a linear mapping \varphi we get

    \[f(p)=f(p_0)+\varphi(\overrightarrow{p_0p}).\]

The map \varphi does not depend on p_0 and is called the derivative of f and denoted by f'.

It is worth noticing that

  • for any point p\in H and v\in T(H), f(p+v)=f(p)+f'(v),
  • for p,p'\in H, f'(p-p')=f(p)-f(p'),
  • if p_0,\ldots,p_n is a point basis of H and q_0,\ldots, q_n is a system of points in M, there exists exactly one affine map f, such that f(p_i)=q_i, for all 0\leq i\leq n,
  • if p_0;v_1\ldots,v_n is a basic system of H and q_0\in M and w_1,\ldots, w_n\in T(M), there exists exactly one affine map f, such that f(p_0)=q_0 and f'(v_i)=w_i, for 1\leq i\leq n.

Standard examples of affine maps H\to H

The map f(p)=p+v, for a given v\in T(H) is called the translation by v.

The map such that f(p_0)=p_0 for a givon p_0\in H and f'(v)=kv for a given k\in K is called a homothety with centre in p_0 and scale k.

If T(H)=V\oplus W, then a map f is called a projection onto p+V along q+W, if f(p)=p and f'(v)=v for v\in V and f'(w)=0 for w\in W.

If T(H)=V\oplus W, then a map f is called a reflection across p+V along q+W, if f(p)=p and f'(v)=v for v\in V and f'(w)=-w for w\in W.