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Assuming suppose we have 2 hyperplanes $H_1, H_2$ in $\mathbb{R}^n$ described parametrically by variables $T = [t_1 ... t_n]^T$ as:

$$H_1 = \mathcal{H}_1 T + T_{10} $$ $$H_2 = \mathcal{H}_2 T + T_{20} $$

I want to be able abstractly describe the "minimum" distance between $H_1, H_2$ and have some kind of method of computing it.

In the event that $H_1, H_2$ embed into two different parallel (n-1) dimensional hyperplanes $P_1, P_2$ techniques described in here apply.

But if you're in a situation of say $2$ skew lines, in $\mathbb{R}^3$ then those techniques are no longer relevant, and while the problem can be made into a convex optimization problem, I feel there should be some natural "algebraic" closed form here, that doesn't resort to modeling this is as a quadratic program.

NOTE: $H_1, H_2$ need not have the same dimensions. (i.e.) matrices $\mathcal H_1, \mathcal H_2$ can have null-space dimension anywhere from $0$ to $n$


Some work:

So I thought about this some more and come up with the following concrete answer:

Given points P,Q on the affine spaces they can be expressed as $\mathcal{H} T_1, \mathcal{H} T_2$ respectively.

$$ D(\mathcal{H}T_1,\mathcal HT_2)^2 = (\mathcal{H}T_1 - \mathcal HT_2)^T (\mathcal{H}T_1 - \mathcal HT_2) $$

It follows that the point minimizing this distance arises from looking at

$$ \frac{ \partial D}{ \partial t_i} = 0$$

w.r.t each of the $2n$ parameters.

This is a system of linear equations.

A concrete example below:

$H_1 = \begin{pmatrix} 2t \\ 3t\\ t \end{pmatrix} + \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix} $

$H_2 = \begin{pmatrix} 5s \\ 2s\\ 2s \end{pmatrix} + \begin{pmatrix} 3 \\ 5 \\ 1 \end{pmatrix} $

Our distance is given as $$ D' =D(t,s)^2 =(2t+1-5s-3)^2 + (3t+2 - 2s-5)^2 + (t+2-2s-1)^2$$

So it follows the solution to the system

$$ \frac{\partial D'}{\partial t}= 2(2)(2t+1-5s-3) + 2(3)(3t+2 - 2s-5) + 2(t+2-2s-1) = 0 $$

$$ \frac{\partial D'}{\partial s} = -2(5)(2t+1-5s-3) - 2(2)(3t+2 - 2s-5) - 2(2)(t+2-2s-1) = 0 $$

Is the unique closest point.

I suspect there is some "elegant" way to express this as an inversion of familiar matrices.

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  • $\begingroup$ For skew lines in $\mathbb R^3$, a geometric solution involves finding a line that intersects and is orthogonal to them both. You might be able to generalize this to a pair of flats. $\endgroup$
    – amd
    Feb 18, 2019 at 7:07
  • $\begingroup$ Hyperplane always means a $1$ codimensional (affine) subspace. It seems you only assume $H_1,H_2$ are affine subspaces. $\endgroup$
    – Berci
    Feb 18, 2019 at 7:39
  • $\begingroup$ that is correct, affine subspace is what i meant to say $\endgroup$ Feb 18, 2019 at 7:51

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