# Various transformation in Homogeneous Coordinates

Today, I learned homogeneous coordinates which solve the notation problem of translation. By homogeneous coordinates, I can became to use translation as matrix notation.

Along this knowledge, there were lots of matrix form that show the various transformation.

1. Translation

$$\begin{pmatrix} 1&0&k\\ 0&1&h\\ 0&0&1 \end{pmatrix}$$

1. Rotation

$$\begin{pmatrix} cos\theta&-sin\theta&0\\ sin\theta&cos\theta&0\\ 0&0&1 \end{pmatrix}$$

1. Scalar Multiplication

$$\begin{pmatrix} s&0&0\\ 0&t&0\\ 0&0&1 \end{pmatrix}$$

ETC....

What I'm curious is above matrices are obtained by how? Is it just computation or other proof?

## 1 Answer

Presumably, since you know that homogeneous coordinates solve the problem of translation, you already have seen a proof that the matrix you have given does the job [If not, you should directly compute what this matrix does to $$(x,y,1)$$; it is a quick and illuminating exercise].

The other two matrices were obtained by just taking the ordinary rotation/scaling matrix, putting it in the upper-left corner, and then putting a $$1$$ in the lower-right. They clearly do what they say they do, because they do it to the first two coordinates, and leave the third ("homogenizing") coordinate unchanged.

For instance, starting with $$(x,y)$$ and converting to $$(x,y,1)$$, and applying the matrix, you compute that the algebraic mess you get is just $$(R_x, R_y, 1)$$, where $$R_x$$ and $$R_y$$ are just the coordinates of the point $$(x,y)$$ after it has been rotated through some angle. But $$(R_x,R_y,1)$$ is clearly the conversion of $$(R_x,R_y)$$, as desired.