# Finding the linear mapping between homogeneous coordinates of affine camera

If I have an affine camera with a projection relationship governed by:

$$$$\begin{bmatrix} x & y \end{bmatrix}^T = A \begin{bmatrix} X & Y & Z \end{bmatrix}^T + b$$$$ where A is a 2x3 matrix and b is a 2x1 vector. How can I form a matrix representing the linear mapping between the world point $$(X,Y,Z)$$ and image point $$(x,y)$$ if they are represented by homogeneous vectors?

## 1 Answer

Let $$V = [X,Y,Z]^T$$, $$A\in\mathbb{R}^{2\times 3}$$, $$b\in\mathbb{R}^{2}$$. Then: \begin{align} Mh= \begin{bmatrix}A & b \\ 0 & 1\end{bmatrix}\begin{bmatrix} V \\ 1 \end{bmatrix} &= \begin{bmatrix} a_{11} & a_{12} & a_{13} & b_1 \\ a_{21} & a_{22} & a_{23} & b_2 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} \begin{bmatrix} X \\ Y \\ Z \\ 1 \end{bmatrix} = \begin{bmatrix}AV +b \\ 1 \end{bmatrix} = \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} = v \end{align} Let me know if you want me clarify anything.