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I have an interesting problem but I don't know if it can be solved using probability theory and geometry. The average distance from the origin to the unit circle is 1. What would be the average distance if I translate the unit circle by some arbitrary vector in terms of that vector’s magnitude? What about if I scale (increase radius of) the unit circle before translating it, how will the scaling term affect the result? What is the distance distribution?

Here is what I know. Through sampling and averaging, the expected distance when I move the circle one unit away from the origin is about 1.27. The farther away you go, the closer the expected distance value converges to the offset distance. 10 units away gives me 10.027, and 100.005 for 100 units away. Regardless of offset, the expected distance is only equal to the radius of the circle when the circle is at the origin, and the value only increases the further out the circle is. If no (simple) solution can be found, what is an elegant equation to model these observations? Can accuracy be controlled by modifying non-input parameters of this equation?

Application: This is more than a random fun problem. In digital signal processing (DSP) and digital music, interactions between different sine waves of the same frequency but different phases create interference. This interference can increase of decrease the amplitude of the output signal as well as change its phases. Amplitudes and phases of signals with the same frequency can be represented with vectors where the magnitude represents the amplitude of the wave and the angle represents the phase. The sum of the two vectors models interference. If the amplitude of two input signals are known but their phases (random with uniform distribution) are not, what is the expected amplitude of the signal when added together? Using a circle to represent all possible phase differences between a two waves, with the offset representing the amplitude of the first wave, and an radius represented as the amplitude of the second wave. Adding the offset vector to another one at random orientation representing radius will get you an offset circle, and the expected distance to the resulting point on that circle should model the expected amplitude of the combined wave. From here, you can see where the actual question comes from. Please correct me if this model is wrong.

Extra Credit I am much more interested in the Mathematics solution but if you do have a model, or a solution, I would also like to know by proof if it satisfies these three properties. Let me know if I am asking too much, I don't know if this site is strictly Q and A or if discussion is allowed.

Property 1: f(a,b) = f(b,a) approximately for a model

Property 2: f(sa,sb) = s*f(a,b)

Property 3: f(a,f(b,c)) = f(f(a,b),c)

This is in addition to the convergence, and minimum value properties I mentioned in paragraph 2.

f(,) Is the model or solution taking in the parameters for radius and offset, (offset vector orientation shouldn't matter right?). a,b,c are the paramaters, non-negative real numbers. s is a real number scalar

Thank You.

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  • $\begingroup$ Without going through all of the calculations at the moment, you might want to write your equation for the circle in parametric form, use those coordinates in the standard norm function, then apply the Mean Value Theorem by integrating wrt your parameter over your interval (presumably $[0,2\pi]$ and dividing by the length of the interval. $\endgroup$ – rnrstopstraffic Sep 29 '16 at 5:48
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Partly answer your toy model: Let $x > 0$ be the magnitude and $r > 0$ be the radius of the circle. WLOG assume the centre of the circle always lie on the positive $x$-axis. I assume you mean that you are picking a point uniformly on the circumference. So the coordinate of the random point can be expressed as $$ (x+r\cos\Theta,r\sin\Theta)$$ where $\Theta \sim \text{Uniform}(0, 2\pi)$. By symmetry we can just consider the upper half circle in this problem, i.e. restricting $\Theta \sim \text{Uniform}(0, \pi)$. The average distance is then given by

$$ E\left[\sqrt{(x + r\cos\Theta)^2 + r^2\sin^2\Theta}\right] = \frac {1} {\pi}\int_0^{\pi}\sqrt{(x + r\cos\theta)^2 + r^2\sin^2\theta} d\theta$$

which should be an elliptic integral (without elementary function as the solution).

The remaining application part left to other experts.

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A circle can be modeled with the vector $\vec{u}=<r_0\cos{t}, r_0\sin{t}>$. Displacement can be modeled with an additional vector, $\vec{d}=<x_0,y_0>$

So the displaced circle is $\vec{r}=\vec{u}+\vec{d}$

$r^2=u^2+d^2+2\vec{u}\cdot \vec{d}$

$r=\sqrt{u^2+d^2+2\vec{u}\cdot \vec{d}}$

$r_{avg}=\frac{1}{\pi}\int_0^{\pi}\sqrt{r_0^2+d^2+2\vec{u}\cdot \vec{d}}\ dt$

This can be replaced with :

$r_{avg}=\frac{1}{\pi}\int_0^\pi r_0\sqrt{1+\frac{d^2}{r_0^2}+2d/r_0\cos{t}}$

This has no closed form solutions, but by a sign change you can restate it in terms of Legendre Polynomials which could facilitate signal processing.

This set up does have some physical applications. It's the electric field at the origin froma displaced ring of charge for example. Some tweaks and it represnets a current loop.

Often when you are dealign with displacements, you can do an anlaogue to Fourier Analysis using Lgendre' Polynomials.

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