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I am given the $3$-D coordinates of two triangles. For example:

  • for $\triangle ABC$, the coordinates are: $A=(0, 0, 0)$, $B= (3.37576, 0, 0)$, $C=(5.14131, -2.47202, 0)$

and

  • for $\triangle DEF$ the coordinates are: $D=(0, 0, 0)$, $E(3.73345, 0, 0)$, $F=(7.06825, -3.44094, 0)$.

How to calculate the transformation matrix between two triangles?

Any help will be largely appreciated!

  • 2
    Do you see that both triangles lie in the x-y coordinate plane? (i.e., both triangles are, essentially, in 2-dimensions), Each with a vertex at the origin (A, D), and each with one side lying along the x-axis, overlapping (side DE a bit longer than side AB?), and each lying in the same quadrant?2011-05-31

1 Answers 1

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This particular case is rather simple. Note:

$\frac{3.73345}{3.37576} \approx 1.105958,$

$\frac{7.06825 - 5.14131\times 1.105958}{-2.47202} \approx -0.559128, \text{ and}$

$\frac{-3.44094}{-2.47202} \approx 1.391958.$

so giving a solution of

$ \left( \begin{array}{ccc} 1.105958 & -0.559128 & c \\ 0 & 1.391958 & f \\ 0 & 0 & i \end{array} \right)$

where $c$, $f$ and $i$ can take any values because, as amWhy noted, you only have information about the x-y plane. In general it will be slightly more complicated.

  • 0
    You could try [Wikipedia on matrices and linear transformation](http://en.wikipedia.org/wiki/Matrix_%28mathematics%29#Linear_transformations) and links from that article2011-06-02