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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.