How can I calculate the transformation between two 3D triangles?

Question

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!

Answer 1

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.