random unit vector in 2 dimension

Question

I wanted to generate a uniformly random unit vector in 2 dimensions and I have been using the following snippet in C++

double ranf(double x)
{
    return double(rand()) * x / (double(RAND_MAX));
}

double phi = ranf(2 * M_PI);

double ndx = cos(phi);
double ndy = sin(phi);

Is this correct?

Answer 1

You're generating a random angle phi uniformly between 0 and 2Pi, which is the "heading" of your vector. By using unscaled sin and cos, you're implicitly assuming that the length of the vector is 1 and decomposing it into its vertical and horizontal components. Looks basically correct to me.

A couple of minor details: 1) When you divide by RAND_MAX rather than RAND_MAX+1, you're allowing both zero and x to occur, in your case both 0 and 2Pi, and those two values are the same in angular terms. You may want to change the divisor accordingly. 2) I'd recommend multiplying by 2.0 rather than 2 to avoid the implicit up-conversion when you multiply it by M_PI, but this is probably taken care of by your compiler anyway.

There's another way to do this which generalizes to an arbitrary number of dimensions: Generate k-tuples (z₁, z₂,..., z_k) where each of the z's has a normal distribution. Calculate the norm N=sqrt(z₁² + z₂² + ... + z_k²), and use it to create a normalized vector X=(z₁/N, z₂/N,..., z_k/N). The result is that X has unit length (because of the normalization) and its direction is uniformly distributed over the k-dimensional unit hypersphere thanks to the distributional properties of the normals.