| 9 |
There will exist, on the other hand,
genuine geometric singularities:
Etc....
..By the way, the fold is a particular region of the surface that
contains line curvature, negative on the left,
positive on the right. Deliberately, these two surfaces have been manufactured
with spherical caps. The final object has the topology of the sphere.
It therefore possesses a total curvature equal to 4p
. Suppose that the object on the left had been manufactured with
two times three quarters of a sphere (starting from spheres of the same
radius). These components therefore each correspond to a curvature of
3p. Total 6p . One therefore immediately knows
how much curvature (negative) the fold contains: - 2p
. This is uniformly distributed along the circular fold.
One therefore knows how to calculate the sum of the angles of triangle ABC. By measuring its
surface one first knows the quantity of curvature (angular)
that it contains. It is:
The quantity of curvature contained in the arc mn must be subtracted. It is:
The lens also has the
topology of the sphere. Therefore the fold contains a line curvature
positive and equal to 2p.
...One
can also calculate the sum of the angles of the bizarre triangle ABC,
which is constituted by three geodesic lines. The geodesics
blithely cross the folds. You only have to do the experiment
with your adhesive tape.
The arc mn contains the
line curvature: Suppose
that the kind of lens indicated above had been
manufactured with two identical quarter spheres. Each contains
a curvature equal to p . Therefore, in
the surface (without the fold) contains the curvature 2p
.
...By
accounting for the angular curvature contained in triangle ABC and in
the fold-arc one can know the deviation, positive, from the
Euclidean sum p .
One sees that one can quite
easily juggle with these problems of curvature, for surfaces.
...A
surface can comprise singular points or folds. In this case it is a matter
of genuinely geometric, intrinsic singularities,
and not resulting from a choice of coordinates.
...By
the way let us note that this line curvature could be distributed
over a portion of surface. For example, for the figure on the left, one would obtain:
...This is
the approach analogous to the one that had been presented
above, where the curvature concentrated at the apex of a cone
had been distributed over a spherical cap
(blunted cone). If
the two spherical caps composing the above surface represent
for example each two thirds of a sphere, i.e. a curvature
The shaded surface
will contain a negative curvature C, uniformly distributed,
such that:
Article
contents Science
contents Home
page
Back to
Science Guide Number of visits to this page since 1 July 2004
: