# Math Help - Help finding 90 degree angle in sphere

1. ## Help finding 90 degree angle in sphere

Hello,

Struggling artist here, please excuse my very basic understanding / lack of technical terms. I've included a quick diagram to make it easier to explain my question.

Is it possible, assuming distance "a" is known, to determine what distance "b" will be? I'm trying to better understand how far apart 90 degree slices are when rotating a spherical object, but obviously the distance changes depending on where the two points are on the rotating axis relative to the viewpoint. Does that make sense?

(also, in the diagram,, by "b" i mean distance on the centre line).

Thank you!

2. ## Re: Help finding 90 degree angle in sphere

Are you saying that 6 is "a" and "b" is the distance moved, i.e. 9?

3. ## Re: Help finding 90 degree angle in sphere

Interesting geometry problem. I'm assuming that you mean that $a$ and $b$ are lengths on a two-dimensional photograph of the sphere (what we in math jargon would call the projection of the sphere onto a plane). In that case...

Let's define $\theta$ as the rotation angle of the sphere. Let $\theta=0^{\circ}$ correspond to the situation where $a$ equals the sphere's radius and $b$ is zero. If the rotation angle $\theta$ is increased, then $a$ decreases (and $b$ increases) according to:

$a=R\sin(90^{\circ}-\theta)$
$b=R\sin(\theta)$
in which $R$ is the radius of the sphere.

So if you know distance $a$ then you calculate the rotation angle $\theta$ which you can use in turn to calculate $b$. A quick rewrite of the formulas given for this:
$\theta=90^{\circ}-\arcsin\left(\frac{a}{R}\right)$
$b=R\sin(\theta)$

Let us know if this gives you the results that you were expecting.

4. ## Re: Help finding 90 degree angle in sphere

Perfect, thank you corsica.