I have two spheres in three-dimensional space that touch at a single point, one of them controlled by a parameter $t$. I want to find the coordinates of that contact-point in terms of $t$.
The first is centered on the point $(0, 0, 1)$ with a radius of $1$, so it is represented by the equation $x^2 + y^2 + (z – 1)^2 = 1$.
The second is centered on the point $\left(t, \frac{\sqrt{2} – 1}{\sqrt{3}} t^2 – \frac{\sqrt{2} + 1}{4 \sqrt{3}}, 1 – \sqrt{\frac{3}{8}} + \frac{\sqrt{2} – 1}{\sqrt{6}} t^2 – \frac{\sqrt{2} + 1}{4 \sqrt{6}}\right)$ with a radius of $\frac{\sqrt{2} – 2}{2} \left(t^2 – \frac{1}{4}\right)$, where $-\frac{1}{2} < t < \frac{1}{2}$, so it is represented by the equation $(x – t)^2 + \left(y – \frac{\sqrt{2} – 1}{\sqrt{3}} t^2 + \frac{\sqrt{2} + 1}{4 \sqrt{3}}\right)^2 + \left(z – 1 + \sqrt{\frac{3}{8}} – \frac{\sqrt{2} – 1}{\sqrt{6}} t^2 + \frac{\sqrt{2} + 1}{4 \sqrt{6}}\right)^2 = \left(\frac{\sqrt{2} – 2}{2} \left(t^2 – \frac{1}{4}\right)\right)^2$.
For all valid values of $t$, the second sphere is located entirely inside the first except for a single point that is exactly on the surface of the first. I do also know that the contact-point lies on the line that passes through the center-points of both spheres.
I've tried using the techniques I would use to find the contact-point of two circles in a 2D plane, but they don't seem to translate into a 3D space.
Best Answer
Based on comments from @Intelligentipauca and @Cesareo, I figured out that the equation for the contact-point is $P = \frac{r_2 × C_1 - C_2}{r_2 - 1}$, where $C_1$ is the center of the first sphere, $C_2$ is the center of the second sphere, and $r_2$ is the radius of the second sphere.