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Could you help me solve this: For projective coordinate system on the line $l$ are given points $A (2,1)$, $B (1,1)$, $C (0,1)$, $A_1 (0,1)$, $B_1 (1,5)$ and $C_1 (2,1 )$. Find a linear transformation where the points $A, B$ and $C$ go respectively at $A_1$, $B_1$, and $C_1$.

I tryed to solve it but I dont know what to do.

Here is what I tryed: \begin{equation} ρ_1 * \begin{pmatrix}0 \\ 1 \\ \end{pmatrix} = \begin{pmatrix}m & n \\ p & q \\ \end{pmatrix}* \begin{pmatrix} 2 \\ 1 \\ \end{pmatrix} \end{equation} \begin{equation} ρ_2 * \begin{pmatrix}1 \\ 5 \\ \end{pmatrix} = \begin{pmatrix}m & n \\ p & q \\ \end{pmatrix}* \begin{pmatrix} 1 \\ 1 \\ \end{pmatrix} \end{equation} \begin{equation} ρ_3 * \begin{pmatrix}2 \\ 1 \\ \end{pmatrix} = \begin{pmatrix}m & n \\ p & q \\ \end{pmatrix}* \begin{pmatrix} 0 \\ 1 \\ \end{pmatrix} \end{equation}

2m + n = 0
2p + q = $ρ_1$

m + n = $ρ_2$
p + q = 5*$ρ_2$

n = 2*$ρ_3$
q = $ρ_3$

Is that the right way ? I got some equations for m, n, p, q but I dont think they are right. What should I do now ?

Martin Hristov
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  • What equations did you get? If you don't write them down how will anyone know whether you go wrong and where? – DonAntonio Jun 07 '14 at 13:24
  • The factors $\rho_i$ indicate that you are in fact looking for a projective transformation, not a linear one. You can use the techniques from my answer here but reduce the dimension by one: instead of for points defining a transformation of the projective plane, you have three points defining a transformation of the projective line. – MvG Jun 10 '14 at 18:53

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