Is there a equation specifying distance of a point lying outside sphere to every point lying inside volume of sphere?

Question

Suppose we have a sphere of radius $r$ and a point $m$ lying $d$ away from the sphere's center is there an equation that specifies distance of every point inside the sphere to that point $m(0,0,0)$. Preferably in spherical coordinates.

Answer 1

If you want an answer in terms of spherical coordinates, define your coordinates so that the poles are collinear with your external point (it makes sense to take advantage of symmetry wherever you can). You can then use the dot product to calculate the distance. Defining $M, P, O$ as the point outside the sphere, the point inside the sphere, and the centre of the sphere respectively:

$$\overrightarrow{MP} = \overrightarrow{MO} + \overrightarrow{OP}$$

You know the magnitude of the vectors $\overrightarrow{MO}$ and $\overrightarrow{OP}$, and you know the angle between them.