Let's say we have some D-dimensional euclidean space, and we have some circles of dimension 0 to D-1 (circle dimensionality meaning the minimum number of vectors needed to fully define it, so a basketball would require 3 vectors and is a 3D circle, while a plate just needs 2 vectors, and is a 2D circle).
How would I find the vector equation for the intersection of these circles? I know the intersection is always 0 dimensional (a point) or takes on the dimensionality of the lower dimensional cirle minus 1, or has zero intersections total. Given two circles with the parameters (c, $\vec{d}$, r, and ${M}$), where c is the center, $\vec{d}$ is the direction the circle points in (or a single possible direction that it points in, depending on whether it is 2 dimensions or more lower than the embedded space), r is the radius, and ${M}$ is the matrix of column vectors that span the basis of the circle, and are orthonormal (this is so we can deal with tilted circles with as much ease as aligned circles), how do I formulate a solution to the circle formed by the intersection of the two circles? I am only looking for edge intersections, so the intersection of the set of points that are on each circle's edge. Thanks.