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The notion of geodesics.
...In all that follows we shall deal only with two-dimensional surfaces. The reader would do well to equip himself with a little material before beginning this reading. There are indeed things that "are understood with the hands," by carrying out experiments. It is therefore necessary to have:
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Sheets of Bristol board, or somewhat stiff paper.
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Adhesive tape, preferably colored.
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Scissors.
...Particularly convenient "cloth-backed tape" is available commercially, which has the advantage of being able to be unrolled in strips of any width.
...Take a surface, for example the bodywork of your car. You can stick a strip onto it, arranging things so that it does not wrinkle, so that it always adheres completely to its support. In doing so you will trace geodesics of the "car-surface."
...If you perform this operation on a flat surface, you will obtain.. straight lines. Straight lines are the geodesics of the plane.
...With three geodesics you will be able to trace a triangle on this plane, a "Euclidean surface," the sum of the angles of this plane being equal to 180°.
The posicones.
Let us now make a "posicone." To do this, cut out an angle q and glue the surface back together as shown.
...Let us trace two geodesic triangles, with the adhesive tape, one being outside the vertex S of this posicone and the other containing this vertex S. Let us measure the sum of the angles of the triangle. We then observe that the sum of the angles of the triangle that does not contain the vertex remains equal to 180°, whereas that of the other triangle equals 180°+q.
To verify this it would suffice to reopen this "posicone" and lay it flat. Then one would obtain this:
...One can then easily verify that the lines AH and H'B make an angle q. As one could observe "experimentally," the geodesics of the cone, after being laid flat, become straight lines of the plane, which simply means that the cone is a developable surface. The same thing for a cylinder.
...A developable surface is also a surface that can be rolled onto a plane (subject to certain precautions, for developable surfaces with negative curvature, as will be seen further on).
...If one traces a geodesic on a cone or a cylinder, with greasy ink, one can use this "matrix" to "print" it on a plane, in which case one obtains a straight line. And, conversely, if one inscribes a straight line in greasy ink on a plane and "prints" it on a cone or a cylinder, by rolling these objects on the plane, the impression gives... geodesics.
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