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Let's say we have some D dimensional Euclidean space.

Let me use the term S-Sphere to only indicate spheres that match the dimensionality of the space they reside in, while Circles are spheres with D-1 or lower dimensions (so a basketball in 3D space is a sphere, but it is a circle in 4D space.) I know this definition is extremely informal and crude, but it is just a change of words so I can explain easier.

Since an S-Sphere matches the dimensionality of the space it is in, it is parametrized as so:

S($c_s$, $r_s$)

where S indicates that we are talking about an S-Sphere, $c_s$ is the center, and $r_s$ is the radius.

Each circle is parametrized as so:

C($c_c$, $\vec{d}$, $r_c$, ${M}$)

where $c_c$ is the center, $\vec{d}$ is the vector orthogonal to the circle that shows what direction the center points in (or one possible direction if the difference between D and the dimension of the circle is greater than 2), $r_c$ is the radius, and ${M}$ is the matrix of column vectors that form a basis for the plane that the circle lies on, and are orthonormal to $\vec{d}$ and each other.

If all the parameters are defined, so we know the center and radius of any sphere we are working with, as well as the 4 parameters of any circle we are working with, how would I find the intersection of the edges? I have the S-Sphere & S-Sphere case down. For S-Sphere & Circle, I have the following:

${M}$*$\vec{a}$ + $\vec{c_c}$ = $\vec{c_s}$ + $\vec{b}$

where ||$\vec{a}$|| = $r_c$ and ||$\vec{b}$|| = $r_s$, and the centers have been vectorized so a point $(x,y)$ becomes the vector $\begin{bmatrix}x\\y\end{bmatrix}$.

I want to be able to solve for a vector field that satisfies the conditions that every point in it is on the edge of the S-Sphere as well as on the edge of the circle. So far, I can verify whether or not this is true for a vector $\vec{v}$ (by setting $\vec{v}$ = $\vec{c_s}$ + $\vec{b}$ and subtracting $\vec{c_c}$ and then manually solving), but I cannot find a way to create vectors from where I am at right now (I can check if a vector is in this space but I can't make one without checking a whole bunch of them and seeing if any stick, which is inefficient). Any help would be greatly appreciated, thank you.

Edit: For previous problems of this sort, I have just solved them by, instead of solving the equations, solving for a vector-generator (some equation set or rule set that produces vectors that satisfy the constraints) and a vector-checker (some equation set that allows me to validate whether a vector is in a set of interest, like the intersection of the vectors on the edge of the circle and on the edge of the S-Sphere.) Currently, the equations that I have work as a checker, as I can manually plug in some vectors to check if it is correct or not, but I cannot generate vectors from the intersection of those two sets.

Edit 2: I am making progress. I decided to use ${M}$ from the circle parameters to find the intersection between the hyperplane formed by ${M}$ and the S-Sphere. From there, it should reduce to the S-Sphere & S-Sphere solving problem, since the intersection will be of equal dimension as the circle (unless it is a point intersection, which can be easily checked for if the distance = the radius)

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  • $\begingroup$ Edit 2 looks like a really good idea. I think you don't need the vector $\vec d.$ $\endgroup$ – David K May 16 at 14:29

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