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

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


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


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.


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