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Concept of a geodesic.
...In everything that follows, we will only consider two-dimensional surfaces. It would be advisable for the reader to gather some materials before beginning this reading. Indeed, some things are best understood "by hand," through experimentation. Therefore, you will need:
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Bristol board or slightly thick paper.
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Adhesive tape, preferably colored.
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Scissors.
...You can find in stores "woven tape" (like duct tape) particularly convenient, which has the advantage of being able to be unrolled into strips of any width.
...Take a surface, for example the bodywork of your car. You can stick a strip onto it, arranging it so that it does not wrinkle and remains fully adhered 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 can draw a triangle on this plane, a "Euclidean surface," where the sum of the angles of the triangle equals 180°.
Posicônes.
Now let us construct a "posicône." To do this, make a cut along an angle q and rejoin the surface as indicated.
...Draw two geodesic triangles using the adhesive tape, one lying outside the vertex S of this posicône and the other containing this vertex S. Measure the sum of the angles of each triangle. You will observe that the sum of the angles of the triangle not containing the vertex remains equal to 180°, while the sum of the angles of the other triangle equals 180° + q.
To verify this, it would suffice to unfold this "posicône" and lay it flat. Then you would obtain this:
...You can then easily verify that the lines AH and H'B form an angle q. As one could experimentally observe, the geodesics of the cone, when laid flat, become straight lines in the plane, which simply means that the cone is a developable surface. The same applies to a cylinder.
...A developable surface is also a surface that can be rolled onto a plane (with certain precautions, for developable surfaces with negative curvature, as we shall see later).
...If you draw a geodesic on a cone or cylinder using greasy ink, you can use this "matrix" to "print" it onto a plane, resulting in a straight line. Conversely, if you draw a straight line using greasy ink on a plane and "print" it onto a cone or cylinder by rolling these objects over the plane, the resulting impression forms... geodesics.
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