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Curvature (positive).
...When we drew our triangle, made of geodesic lines, on a plane, the sum of its angles at the vertices was π. A plane... is a flat, "non-curved," Euclidean surface. Therefore, the sum of the angles of this triangle is the Euclidean sum. In the previous experiment, we saw that if a triangle did not contain the apex of our cone, the sum remained Euclidean. However, when the triangle contains the apex S, then this sum exhibits an excess q, regardless of the triangle, as long as it contains this point. We will say that the apex of the cone is a point of concentrated curvature.
...We can now proceed to other experiments. After manufacturing two cones, with cuts q1 and q2, we can glue these two surface elements together.
...A simpler way to proceed is to make two cuts in a sheet of Bristol board and construct the following surface:
You can then draw as many geodesic triangles as you wish on this surface:
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Containing neither S1 nor S2: sum of angles: π
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Containing only S1: sum of angles: π + q1
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Containing only S2: sum of angles: π + q2
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Containing both points S1 and S2: sum of angles: π + q1 + q2
...It is easy to imagine that we could produce a large number of small cones with a small angle Δq and glue them together. We could even arrange it so that there is a constant curvature density per unit area, by equating this curvature to the sum of the Δq values associated with each vertex of these small cones.
...By making these small cones progressively smaller (as well as the elementary angle Δq associated with them), we can use this method to construct a portion of surface with constant curvature density.
The sphere is a surface with constant curvature density. We will simply say it has locally constant curvature.
An egg is a curved surface with variable curvature density. We will simply say it has locally variable curvature.
...General Relativity consists in identifying mass density ρ with local curvature. Of course, General Relativity does not deal with two-dimensional surfaces, nor even three-dimensional ones, but with four-dimensional hypersurfaces. Therefore, we should not expect too much from what precedes, and we should regard these figures only as didactic images, intended to clarify ideas. But they are not so bad after all.
2D didactic image of a celestial body.
A celestial body, like the Sun, is a concentration of matter, surrounded, if not by vacuum, at least by a near-vacuum (thus a region of very weak curvature). In two dimensions, the didactic image will be that of a blunted cone.
...A blunted cone is made from two parts: a spherical cap with constant curvature (or "constant curvature density") and a conical frustum. The conical frustum is "flat," its curvature density is zero. It is a Euclidean surface. This is the 2D didactic image of a celestial body with constant mass density ρ.
...Along the way, one might wonder how to perfectly join a conical frustum and a spherical cap so that the tangent plane remains continuous.
...It is simple. The conical frustum is made from a cone, which involves a cut of angle q. The spherical cap contains a certain "amount of curvature," which is also an angle—the sum of all the angles of the small cones that compose it. These two angles must be equal.
But how do we evaluate the amount of curvature contained in a given spherical cap?
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