What is the intersection between $d$-dimensional sphere and $d-1$ dimensional subspace?

Question

I'm trying to find out the intersection between a $d$ dimensional sphere and a $d-1$ dimensional subspace (generalization of a sphere and plane intersection). I'm pretty sure that the intersection is a $d-1$ dimensional sphere but I don't know how to show it for the general subspace.

I can show that the intersection is a $d-1$ dimensional sphere only for a very specific subspace and not the general subspace:

If a general point of the subspace if of the following form: $(x_1,x_2,\cdots,x_{d-1},0)$ (generalization of $XY$ plane in $3D$)

and a sphere is of the form: $(x_1-c_1)^2+(x_2-c_2)^2+\cdots+(x_{d-1}-c_{d-1})^2+(x_d-c_d)^2=R^2$

$(c_1,c_2,\cdots,c_d)$ is the center of the sphere and $R$ is its radius.

Then it's easy to see that the intersection between the sphere and the subspace can be calculated by substituting $x_d=0$ in the sphere equation so we get that the intersection is: $(x_1-c_1)^2+(x_2-c_2)^2+\cdots+(x_{d-1}-c_{d-1})^2=R^2-c_d^2$ which is a $d-1$ dimensional sphere.

But how can I show it for any subspace? Is the intersection is still a $d-1$ dimensional sphere for the general case as well? I will appreciate any help or link that explains about it if exists (didn't find any link).

Thanks in advance.

Answer 1

Here’s an outline of how to do this using homogeneous coordinates. The sphere is represented by a matrix of the form $$Q = \left[\begin{array}{c|c}I & -\tilde{\mathbf C} \\ \hline -\tilde{\mathbf C}^T & \tilde{\mathbf C}^T\tilde{\mathbf C}-R^2\end{array}\right],$$ where $\tilde{\mathbf C}$ is the inhomogeneous Cartesian coordinate vector of the sphere’s center. You can parameterize the subspace as $\mathbf X = M\mathbf x$, where $M$ is a full-rank $n\times(n+1)$ matrix. The columns of $M$ form a basis for this space. Points in the intersection of the sphere and subspace then satisfy $$(M\mathbf x)^TQ(M\mathbf x)= 0,$$ i.e., in the coordinate system defined by $M$, the intersection is the quadric with matrix $M^TQM$. For a suitable and natural choice for $M$, you can then show that this matrix represents a $(d-1)$-sphere.

Answer 2

Let the hyperplane be given by $H=a+E$ where $E$ is the $d-1$-dimensional subspace generated by the orthonormal basis $e_1,e_2,\ldots,e_{d-1}$. Then there is some unit vector $e_d$ orthogonal to all $e_i$, $i<d$. Thus $e_1,e_2,\ldots,e_{d}$ is a basis of $\mathbb{R}^d$. Then we may assume without loss of generality that $a=\alpha e_d$. Let $m=m_E+\beta e_d$ with $m_E\in E$. Consider the sphere $( x-m)^2=R^2$ and let $x=a+z=\alpha e_d+z$ with $z\in E$. Since $e_d$ is orthogonal to $z$ the equation for the intersection of the hyperplane and the sphere $( x-m)^2=R^2$ reads as $(z-m_E)^2+(\alpha-\beta)^2=R^2$ which represents a $d-1$-dimensional sphere if $R^2- (\alpha-\beta)^2\geq0$ (and the empty set otherwise).