Calculating rotation matrix of coordinate system from 2 known axis

Question

(https://i.stack.imgur.com/aG0QB.png)

In the image my main coordinate system is in the upper right corner. I measured $3$ points on a board and created a help coordinate system. V1 points directly to the origin of the help coordinate system. V2 and V3 lie on different axis of the help coordinate system.

I want to find the transformation matrix. So V1 is my translation, right? Subtracting V1 from V2 and V3 I can move the origin of the help coordinate system to the origin of the main coordinate system, right?

But how do I calculate the rotation?

score 0 · Accepted Answer · answered Jun 04 '13 at 10:50

0

You can construct a set of three orthogonal vectors from the following:

$$u_1 = v_2 - v_1$$ $$u_3 = u_1 \times (v_3 - v_1)$$ $$u_2 = u_3 \times u_1$$

Once you normalize all three of these vectors you can construct the rotation matrix as follows:

$$\mathbf R = [u_1\ u_2 \ u_3]$$

This is what I have referred to as a triad, although I am not sure how common that terminology is.

answered Jun 04 '13 at 10:50

Tpofofn

4,771

Thank you very much! That turned out a lot simpler than I thought... – komirdo Jun 04 '13 at 12:24
@Tpofofn could you please check my question here : https://math.stackexchange.com/questions/3599061/how-to-calculate-the-transformation-matrix-if-the-coordinate-system-translates-a/3601670?noredirect=1#comment7404018_3601670 thank you – FabioSpaghetti Mar 30 '20 at 18:12
@FabioSpaghetti, looks like it has already been answered. – Tpofofn Mar 31 '20 at 00:18
yes , in the meantime. Thank you ! – FabioSpaghetti Mar 31 '20 at 07:06

Calculating rotation matrix of coordinate system from 2 known axis

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