# Random Rotation matrix

I am asked to generate a random matrix that satisfy the requirements for rotation matrices. Since I don't really know how to do that, I thought of creating a random quaternion(hopefully, there is no error there. If there is, please let me know). I start by randomizing an angle between 0 and 2$\pi$ and a vector $t$ (axis) . then I get the unit vector of t. Now I want to convert the quaternion to a rotation matrix. So is there a sample of code I could use to do so? Anyhow, Thanks!

    rg=2*pi*[0,1];
theta=diff(rg)*rand(1,1)+rg(1);

t=rand(3,1)*2-1;

q = [cos(theta/2),t(1)* sin(theta/2),t(2)* sin(theta/2),t(3)* sin(theta/2)];
R = zeros(3);

• A 3D rotation matrix, right? – Parcly Taxel May 13 '17 at 15:43
• @ParclyTaxel Yes – Abu Bakr May 13 '17 at 16:43

Here's one way:

Choose $X,Y,Z$ independently with a standard normal distribution (i.e. $\dfrac 1 {\sqrt{2\pi}} e^{-x^2/2} \, dx$).

Then replace $(X_1,Y_1,Z_1)$ with $\dfrac{(X_1,Y_1,Z_1)}{\sqrt{X_1^2+Y_1^2 + Z_1^2}}.$

Then choose $(X_2,Y_2,Z_2)$ independently from the standard normal distribution. First replace $(X_2,Y_2,Z_2)$ with $(X_2,Y_2,Z_2)$ minus the orthogonal projection of $(X_2,Y_2,Z_2)$ onto $(X_1,Y_1,Z_1),$ and finally replace this new $(X_2,Y_2,Z_2)$ with $\dfrac{(X_2,Y_2,Z_2)}{\sqrt{X_2^2+Y_2^2+Z_2^2}}.$

Then choose $(X_3,Y_3,Z_3)$ independently from the standard normal distribution. First replace $(X_3,Y_3,Z_3)$ with $(X_3,Y_3,Z_3)$ minus the sum of the orthogonal projections of $(X_3,Y_3,Z_3)$ onto $(X_1,Y_1,Z_1)$ and $(X_2,Y_2,Z_2),$ and finally replace this new $(X_3,Y_3,Z_3)$ with $\dfrac{(X_3,Y_3,Z_3)}{\sqrt{X_3^2+Y_3^2+Z_3^2}}.$

Now you have a randomly chosen orthonormal basis of $\mathbb R^3$. Let its members be the three columns of your matrix. The determinant of this matrix will be either $1$ or $-1$. If it is $-1$ then interchange two of its columns. The result will be a randomly chosen rotation matrix.

• Thanks a lot!! this really helped and the code now works! – Abu Bakr May 16 '17 at 11:43
• @AbuBakr : I'm glad it helps. – Michael Hardy May 16 '17 at 17:21