# Fundamental Theorem of Affine Transformations in 3D

Fundamental Theorem of Affine Transformations states that "Given two ordered sets of three non-collinear points each, there exists a unique affine transformation $f$ mapping one set onto the other."

There are several proof of the same but all of them consider collinear points in 2D. By that I mean that both the ordered sets of non-collinear points lies in the same plane.

I am wondering if things will change if the the ordered sets of non-collinear points lies in different planes Or the theorem is true for all pair of three non-collinear points.

• In 3D you need 4 non coplanar points. – Jens Schwaiger Aug 24 '18 at 6:32
• @JensSchwaiger Can you provide some reference? – Zahir Aug 24 '18 at 6:53

Let $V$ be a vector space of dimension $n$ and assume that $a_0,a_1,\ldots,a_n$ are affine Independent, i.e., the vectors $a_1-a_0,\ldots,a_n-a_0$ are linearly independent. Then given arbitrary points $b_0,b_1,\ldots,b_n\in V$ there is a unique linear mapping $g\colon V\to V$ such that $g(a_i-a_0)=b_i-b_0$ for all $1\leq i\leq n$. Then $f$ defined by $f(x):=g(x-a_0)+b_0=g(x)+b_0-g(a_0)$ is a (in fact the unique) affine mapping satisfying $f(a_i)=b_i$ for all $0\leq i\leq n$.