Spherical coordinates changing when the radius changes

Question

Assume i know the radius and some coordinates of a sphere. let say the initial radius is 20. When i change the radius of the sphere say 80, is there an equation to calculate the new coordinates using the initial coordinates ? thank you

Answer 1

Let's say we have the coordinates of the center as $(x_0,y_0,z_0)$, and the coordinates of some point as $(x_1,y_1,z_1)$, on the sphere.

If we dilate the sphere, by some positive constant $k>0$, this can correspond to a growth or shrink, depending on whether $k>1$ , by the way, then the center remains stationary, while the distances from the center change.

Let's rewrite that coordinate on the sphere as distances (they aren't actual distances, just relative position vectors.)

We have $(x_1,y_1,z_1)=(x_0+(x_1-x_0),y_0+(y_1-y_0),z_0+(z_1-z_0))$

When we scale by $r$, the distances between the center and the point are scaled by $r$ too.

Therefore, we have $\boxed{(x_1,y_1,z_1) \mapsto (x_0+k(x_1-x_0),y_0+k(y_1-y_0),z_0+k(z_1-z_0))}$.

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An example: We have a sphere with center $(2,2,3)$ and a point on it $(4,5,7)$. The sphere changes from radius $\sqrt{29}$ to $3\sqrt{29}$. This means our radius was tripled, or $k=3$.

Therefore, the differences in each coordinate triple. So the new point is $(8,11,15)$.