Assume I have one spherical volume $s_0$ with origin $(x_0, y_0, z_0)$ and radius $r_0$ represented by $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 \leq r_0^2$.
There is another spherical volume $s_1$ with origin $(x_1, y_1, z_1)$ and radius $r_1$ represented by $(x - x_1)^2 + (y - y_1)^2 + (z - z_1)^2 \leq r_1^2$.
At every point within the spherical volume $s_0$ i.e. on all infinitely many spherical surfaces where $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 \leq r$ for $0 \leq r \leq r_0$, I have another set of infinite spherical sections with radii tangent to the spherical surface.
I would like to find the volume of intersection of these spherical sections on/in $s_0$ with $s_1$.
Illustrative partial picture here with $s_0$ in black, $s_1$ in purple, and the spherical sections with radii tangent to each level curve of $s_0$ depicted in red.
I started by trying to define the infinitely many gradients to each of the infinitely many surfaces, but quickly got lost.
How can I solve this problem?
Per questions in comments:
- The original sphere is placed at an arbitrary point to make the solution more general. Yes, centering the sphere at the origin makes the algebraic solution potentially easier / cleaner but will be less applicable.
- All of the small spherical sections that are on the contour lines of $s_0$ have radius $r_s$.
- The equation $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r$ defines a 3D spherical surface with an "empty" or "hollow" interior. To create a 3D spherical volume out of concentric spherical surfaces, I consider all spherical surfaces where $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r$ such that $0 \leq r \leq r_0$. This is akin to creating "level curves" or "contour lines" for each possible value of $r$
- So, for example, the spherical surface defined by $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r$ inside the spherical surface defined by $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r + \epsilon$. Matryoshka spheres.
- Each point on all spherical surfaces $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r$ for $0 \leq r \leq r_0$ has a defined tangent plane. Each point therefore has a small spherical section centered at that point but with a radius $r_s$ perpendicular to the tangent plane.
Funny you should mention Matryoshka dolls, because there seem to be so many layers of misunderstood terminology, excessive complications, and omissions of critical details, that I don't think we have time to peel them all away in a sequence of comments and edits. (It's worse actually than Matryoshka dolls, because rather than simply removing one layer of errors, you doubled the length of the question with attempted clarifications while introducing even more confusion in the first half.)
So far, I gather that you have a ball $s_0$ of radius $r_0$ with center $(x_0, y_0, z_0)$, consisting of all points $(x,y,z)$ satisfying the inequality $(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 \leq r_0^2$. (A sphere in $\mathbb R^3$ is a surface, all of whose points are at exactly the same distance from a point called the "center" of the sphere. Note that the center of a sphere is not actually part of the sphere; the word "of" signifies a relationship rather than membership. "Origin" in this context is usually a name for the point $(0,0,0)$.)
You also have a second ball $s_1$ of radius $r_1$ with center $(x_1, y_1, z_1)$, consisting of all points $(x,y,z)$ satisfying the inequality $(x - x_1)^2 + (y - y_1)^2 + (z - z_1)^2 \leq r_1^2$.
The third paragraph was the really confusing one, since "At every point within the spherical volume $s_0$" means something very different from "on all ... spherical surfaces where ...". The last part of the paragraph says something about a "set of infinite spheres" (surely you meant "infinite set of spheres", a very different kind of set); if $P$ is a point in $s_0$, is there such a set of spheres at $P$, or do you only get one of these sets of spheres if you look at an entire concentric sphere within $s_0$? And at that point you neglected to mention that all the spheres in these sets of spheres have the same radius.
(It's unclear why you replaced "spheres" with "spherical sections", since "section" has implications that don't appear to apply here. I'm going to assume you still actually mean spheres when you write "spherical section", since otherwise I have no idea what you mean.)
But now you seem to be saying that the red circles in your picture all represent spheres of radius $r_s$. So now to fully identify those spheres, we just need to determine where they are.
The part about a radius tangent to a level surface (there are no level curves described in the question) seems at first to indicate that the center of a red sphere is on one of the concentric spherical shells within $s_0$, or more simply, the center of the red sphere is in $s_0$. But did you mean instead that any point along any radius of the red sphere can be tangent to the concentric spherical shell? If so, then the center of a red sphere could be outside $s_0$.
You also mention the tangent plane at each point on each spherical shell. You mention that one of the "spherical sections" has a radius perpendicular to that plane. In fact, if you take any arbitrary plane $\pi_1$ and any sphere $s_2$ in Euclidean space, the sphere $s_2$has two radii perpendicular to the plane $\pi_1$. So it's not clear why you want us to know about the perpendicular radius; it seems to add no new information whatsoever. But the part where you write, "Each point therefore has a small spherical section centered at that point," does at least suggest that you meant the center of each red sphere to be inside or on the sphere $s_0$.
Looking at your drawing, however, it seems you have many red spheres whose centers are completely outside $s_0$. Some of them might even be tangent to $s_0$. (It's hard to tell because the picture is so crowded.)
Although it's still hard to mathematically identify exactly what spheres cannot be one of your red spheres, it appears that the centers of all of those spheres form a ball with center $(x_0, y_0, z_0)$. We can identify the set of spheres if we can say for sure that
In case 1, the union of all red spheres is a ball of radius $r_0 + r_s$ and center $(x_0, y_0, z_0)$; in case 2, the radius of the ball is $r_0 + 2r_s$. For the purpose of finding the union of these "spherical sections", it does not actually matter whether each red sphere is really a (hollow) sphere or is actually a (filled) ball.
I'm assuming you don't want to measure the volumes of intersections of the ball $s_1$ and the interiors of your red "spherical sections", and then add these volumes up over all sections, because the result is not a finite number. So my guess is that you were looking for the volume of the union of all those regions of intersection, which is the volume of the intersection of $s_1$ with the union of all of the red "spherical sections". In other words, it's just the volume of the intersection of two balls.
Now if you can just clearly and succinctly describe exactly which spheres or balls of radius $r_s$ are among your red "spherical sections", we can easily determine what ball to intersect with $s_1$.
If you're still having trouble coming up with such a description, you might try sharing how this problem came about in the first place.