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Geodesics.
...In
the following we deal with 2d-surfaces. By the way, why is a surface a 2d
object ? Because you need two parameters to localize a point on it. Example
: longitude and latitude on the Earth.
...They
are things that you will better understand with yours hands than with your
head. Try to find some things before starting to read this text. You need
:
- Paper, both firm and
flexible. - Coloured sticky tape.
- Scissors
...Hereafter with this
kind of coloured sticky tape you can make constant width strips.
(6)
...Take
any surface, for example the coachwork of your car. You can glue your sticky
tape strip onto it, carefully avoiding creases. Then you get geodesics of
this surface. They are an infinity of it on the coachwork of your car.
You can do it on a flat
plate. The you get "straight lines of your plane".
"Straight lines" are
just peculiar geodesics, those on the plane.
(7)
With three geodesics
you make a triangle on this "euclidean surface". Then the sum of the three
angles is 180°.
Posicones.
...Now,
let us build a "posicone", some sort of a "positive cone". Just take a piece
of paper, a flat surface, and cut a sector with your scissors. Then put the
lines SA and SB together, as indicated on next figure.
(8)
...Now,
with our sticky tape, we draw three geodesic lines on that surface, as shown
on figure (9). We can measure the sum of the three angles and we find two
cases :
If the triangle does
not contain the summit S of this posicone, the sum is still 180°.
If it contains the summit,
whatever it is arranged around it, small or large, the sum is : 180° +
q .
(9)
To understand, you can
cut your posicone and put it flat. Then your q
angle appears clearly.
(10)
...You
can easily check that the two straight lines AH and we can therefore verify
easily the straight lines SH and SH' form and angle equal to q
.
... A
cone is a surface that can be put flat, after you cut along a line containing
the summit. In french we say that it is a "développable" surface, but
I don't find the english word in my dictionary. Same
thing for a cylinder.
(11)
...You
can draw geodesic lines on a cylinder, with your sticky tape strip. But, after
you manage your cut, and put your cylinder flat, your geodesic lines become...
straight lines.
You can also roll such
surfaces ( cone, cylinder ) on a plane. Next figure.
(12)
...You
can draw a geodesic on a cone or on a cylinder with ink, and then roll it
on a plane, using it as a printing matrix. Then, on your plan, you will print
straight lines.
...Conversely,
if you draw straight lines on your plan and roll the cone or the cylinder
on it, before the ink gets dry, you will print geodesics on it.
