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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?

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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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