  For each pair $(Q_i,Q_j)$ of reference points and each real number $r$ there
is a unique hyperplane $h \perp Q_iQ_j$ such that $d(P,Q_i)^2 - d(P,Q_j)^2 = r$
for points $P$ in $h$. Take $n$ reference points in $d$-space and for each pair
$(Q_i,Q_j)$ a finite set of real numbers. The corresponding perpendiculars form
an arrangement of hyperplanes. We explore the structure of the semilattice of
intersections of the hyperplanes for generic reference points. The main theorem
is that there is a real, additive gain graph (this is a graph with an additive
real number associated invertibly to each edge) whose set of balanced flats has
the same structure as the intersection semilattice. We examine the requirements
for genericity, which are related to behavior at infinity but remain
mysterious; also, variations in the construction rules for perpendiculars. We
investigate several particular arrangements with a view to finding the exact
numbers of faces of each dimension. The prototype, the arrangement of all
perpendicular bisectors, was studied by Good and Tideman, motivated by a
geometric voting theory. Most of our particular examples are suggested by
extensions of that theory in which voters exercise finer discrimination.
Throughout, we propose many research problems.
