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I am supposed to find the affine matrix, that transform the large square $P$ into $Q$

enter image description here

It is given, that the transformation consists of a scaling, rotation, and translation matrix. We also know 3 points: $(0.09,1.15)$ $(1,1)$ and $(1,1.67)$

I have come to this point.

$$ A= \left(\begin{matrix}a*cos(v)&-b*sin(v)\\a*sin(v)&b*cos(v)\end{matrix}\right) \left(\begin{matrix}x\\y\end{matrix}\right) +\left(\begin{matrix}0.09\\1.15\end{matrix}\right) $$

I just don't know how to determine $a$ and $b$ which are the coefficients of our scaling matrix.

Anyone know how I should continue?

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  • $\begingroup$ Use the $45 \deg$ angle and one of the two given points to determine two additional vertices of $Q$. Hint: what angle does the upper-right side of $Q$ make with the axes? This, together with one of its points, completely determines the line; similarly you can determine the equation of the upper-left side of $Q$ and then calculate their intersection. $\endgroup$
    – Alexander Geldhof
    Commented Nov 7, 2019 at 15:24
  • $\begingroup$ I'm not quite sure what you want me to do. "what angle does the upper-right side of Q make with the axes". That I don't really understand. $\endgroup$
    – Carl
    Commented Nov 7, 2019 at 15:36
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    $\begingroup$ The lines the sides of $Q$ lie on can be described by a (cartesian or parametric) equation. You'll need the direction and one point they run through to determine the equation; the indicated angle gives you all information you need for the directions. $\endgroup$
    – Alexander Geldhof
    Commented Nov 7, 2019 at 16:00
  • $\begingroup$ Ok, I can easily a vector since I know the angle, and I have a point, so I can make a parametric Line. But I need to find the length of the side, don’t I. And I don’t see how to find that. $\endgroup$
    – Carl
    Commented Nov 7, 2019 at 16:48
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    $\begingroup$ Determine the four lines. Their four intersections are the verteces of $Q$. You don't need the length directly. $\endgroup$
    – Alexander Geldhof
    Commented Nov 7, 2019 at 16:49

1 Answer 1

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You know one vertex of the transformed rectangle, $(x_0,y_0)=(0.09,1.15)$, and that its sides are rotated 45° relative to the coordinate axes. With this information you can use the point-normal form of equation of a line $n_xx+n_yy=n_xx_0+n_yy_0$ to develop equations for the extensions of the two rectangle sides that meet at this vertex. You also have a point on each of the sides opposite to these two, so use the point-line distance formula ${\lvert ax+by+c\rvert\over\sqrt{a^2+b^2}}$ to determine the width and height of the transformed rectangle, and from that, the appropriate scale factors.

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