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For a ray from the origin with a given angle in $R^n$, I am trying to find the radius at which that ray intersects the frontier of the unit n-cube. In two dimensions, the picture is this: enter image description here

Given $\theta$, find the distance r to the edge of the unit square. Or, in $n$ dimensions, given angles $\theta_1,...,\theta_{n-1}$, find the distance $r$ to the edge of the unit hypercube.

If somebody could provide a good source of information on doing trigonometry in arbitrary dimensions, that may be enough. As it is I don't know how to approach this problem.

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2 Answers 2

In 2D it is min$(1/\cos\theta, 1/\sin\theta)$ Where are you measuring your angles? In 3D, is $\theta_1$ the angle in the plane as you show and $\theta_2$ the angle out of the plane?

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In higher dimensions it is usually best to use a directional vector from the sphere $S^{n-1}$ instead of an angle $\theta$. For convex bodies (or even star bodies) the radial function. $\rho$: $S^{n-1}$$\rightarrow R_{\ge0}$ gives the radius of the body in a particular direction. In your case, you are working with the $l_\infty$ ball. The radial function, when restricted to $S^{n-1}$ is simply the receprical of the minkowski functional:

http://en.wikipedia.org/wiki/Minkowski_functional

In short, you need a "nice" formula for the radial function $\rho$ of the $l_\infty$ ball. You may find such a formula in a convex geometry book. In particular, Koldobsky has a book on Fourier analysis and convex geometry which contains extensive results on $l_p$ balls and radial functions.

I'll edit my answer if I think of more specific references.

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