To begin to discuss Euclidean space we need a path-connected topological
space and a notion of
length for paths. If we continuously deform a
path, its length varies continuously, so lengths form a continuum.
The lengths of paths can be compared; furthermore, in the Euclidean world
(but not in Minkowski space, hence relativity's space-time) lengths are
positive, save only that a
constant path – i.e. a single point, the
path starting and ending without having gone anywhere – will have zero
length. This last being a degenerate case, I'm choosing to disallow the constant
path as a path – or, at least, allow length to not have to apply itself to
At least on a trajectory (or
open path – not ending where it
started), we can mark any two (distinct) points and examine the portion of the
path between these; this path is shorter than the original (provided at least
one of the chosen points isn't an end-point). Further for any path shorter than
the given trajectory, there is at least some portion of our trajectory which has
the same length as the given shorter path. In this sense, our continuum of
lengths is one-dimensional.
When one path ends where another begins, we can combine the two paths; the length of the composite must be equal to the sum of the lengths of the two parts. When a path loops round and ends where it started, we can combine it with itself arbitrarily often and infer that lengths may be multiplied by integers. This suffices to require that our one-dimensional continuum of lengths is a (one-sided) linear space over the positive scalars; in short, given some length to use as a unit, every length is simply a positive scalar multiple of this unit.
I chose to build up the discourse for a Euclidean space via stages which
can, as far as possible, be re-used for a
locally Euclidean space, such
as arises in discussion of smooth manifolds. Consequently, I shan't jump
straight from the linearity of lengths to the end-game, in which we express
(globally) Euclidean spaces as linear spaces over the positive scalars. The
road there is worth travelling.
To do geometry, we need paths that are
straight; they make the
discussion so much more straightforward: so let me try to characterize the
Chose two distinct points in our path-connected topological space and consider the lengths of all possible trajectory from one to the other. One can clearly pause part-way down any such trajectory, go round some loop back to the pause-point, then continue along down the trajectory: thus, for any trajectory between the points, there's a longer trajectory between them; so there's no such thing as a longest path between points. On the other hand, one may find a path than which no other is shorter (though, conceivably, there might be some other with exactly the same length): such a path is described as a shortest path between the points.
Chose any two distinct points on a shortest path between two given end-points and look at the portion of the given shortest path that connects the chosen points. If there were a shorter path between the chosen points, we could use it in place of the given portion of the given shortest path, thereby obtaining a shorter path between the original points, so the given portion must be a shortest path between the chosen points. Thus each portion of a shortest path is itself a shortest path.
The surface of a cylinder is a path-connected topological space. On it,
there are loops that are
locally shortest – that is, chose two
points on the loop and you'll find a shortest path between them which is a
portion of the loop. Chose two points on the loop and divide the loop into the
two portions whose end-points they are; if these two portions have equal
lengths, replace one of the points with an interior point of either portion and
start again; the resulting portions won't have equal length. Chose the longer
portion: it clearly isn't a shortest path between its end-points; yet all
sufficiently short portions of it are shortest paths between their
I'll describe a path as
locally short precisely if it is covered by
some list of portions each of which is a shortest path. Every shortest path is
When Obelix cuts a cake, he cuts a few dainty thin slices for other folk,
keeping the rest of the cake for himself. The interior of Obelix's slice is a
path-connected topological space. When Obelix cut the cake, he cut two flat
faces joining the centre to the outside; from any point on one face to any
point on the other, the paths within the cake have to go
round the hole
left when Obelix gave slices to other folk; the shortest path between two such
points goes to the center and back out, so it's made of straight pieces, but it
isn't straight. If we move one of the end-points of the bent path but keep it
in the plane formed by the two straight portions of the path, we can construct
the shortest path between its new position and the other, unmoved, end-point;
this path coincides with the original along one of the straight portions.
I'll describe a path, P, as
intrinsically angled if there's some
shortest path, R, sharing an end-point with P for which there is a shortest path
between the other ends of P and R which: doesn't have (R or) P as a portion and
isn't a portion of P but; shares a common portion with P.
I'll describe a path as a geodesic precisely if it is
locally short and not intrinsically angled. When a geodesic is also a shortest
path, I'll describe it as a
Now chose a point, K, in our path-connected topological space and look at
all the geodesics passing through it. To characterize
I'll look at some neighbourhood of K in terms of those geodesics.
One of the crucial properties of a locally Euclidean space is that for each
point, K, there is some length, d, for which there are no geodesic loops through
K of length less than 2.d. We can then, in effect, look at a
region around K of diameter d and be sure that the sphere doesn't intersect
I'll describe a path-connected topological space, U (typically an open sub-set of some larger p-c.t.s), with a positive path-metric as
star-shapedabout some point K
iff, for every X in U, there
is a unique geodesic within U connecting K to X. I'll describe K in U as
center of U precisely if U is star-shaped about K.
iff U is star-shaped about every K in U.
(A Euclidean space is convex; a locally Euclidean space can be stitched together out of overlapping star-shaped patches, which can be convex if one wishes.)
iff there is some (finite) length which is greater than
the length of any shortest geodesic in U. Such a length is described as
outer diameter of U; one will typically be interested in the least
iff there is some positive length, r, and some K in U for which:
(The second condition is there to enable me to take two points near K
and build a polygon
around K to show that, for instance, it's not on the
axis of Obelix's cake-slice.)
In such a case, I'll describe K as an
inner point of U and r as
inner radius for U.
iff U is voluminous and star-shaped about some inner point of U. I'll describe a center which is also an inner point as an inner center.
So, in a proto-Euclidean blob, chose an inner center and an inner
radius about that center. We can now look at the collection of points exactly
that far from our center: these we can try to characterize as a
sphere. Every point in our blob, aside from the center itself, is on
exactly one geodesic out of the center and this geodesic does cut the unit
sphere either on the way to the point or, if it is very near the center, after
passing through the point but before leaving the blob. We are thus able to
describe the each point in the blob in terms of a length (distance from some
chosen inner center) and a direction (point on the surface of the unit sphere).
I'll describe a proto-Euclidean blob, U, as locally Euclidean iff ...Valid CSS ? Valid HTML ? Written by Eddy.