The solid geometry consists in studying the objects defined in the plane geometry in a space in three dimensions and adding to it objects which are not contained in plans: closed surfaces (curved plans and surfaces) and volumes. It is thus about Géométrie in a space with three dimensions.
Euclidean geometry in space
One can adopt, in space with three dimensions, same the axioms as the Euclidean Géométrie.
When one studies the objects of the plane geometry, it is in general enough to be satisfied to imagine them in a plan. To solve a problem thus amounts considering various plans, and studying the properties of the objects contained in these plans. The solution comes in general owing to the fact that an object belongs to several plans at the same time.
The objects are known as “Coplanaire S” if they belong to same a plan. Let us note that:
- by two secant lines, it passes a plan and only one.
- by two not confused parallel straight lines, it passes a plan and only one.
- by three not aligned points, it passes a plan and only one.
- by a line and a point out of this line, it passes a plan and only one.
Example of nonplane objects
Open curved surfaces:
- Paraboloid S of revolution
- Hyperboloidal S of revolution
- cone S of revolution
Closed volumes:
- Polyhedral S
- cone S at polygonal base
- prism S
- Parallelepiped S
- Pyramid S
- volumes of revolution
- cone S of revolution
- Cylinder S
- Sphere S
- Ellipsoidal S
- Paraboloid S
Adaptation of concepts of plane geometry
Specific concepts
- Solid geometrical
Not-Euclidean geometry in space
One can apply the axioms of the not-Euclidean geometries (hyperbolic and elliptic geometry) in space.
The result is enough diverting for the common direction, but allowed the development of the theory of the General relativity, in particular by providing a geometrical model to gravity. One does not speak any more “line”, but of “geodetic”; thus, the trajectory of a satellite in space is geodetic, which makes it possible to predict for example the phenomenon in advance Périhélie; in the same way, the trajectory of a luminous ray between two star S corresponds to geodetic null length (what does not mean however that the two points of the space time are confused: let us recall that this one constitutes an not-Euclidean space).
By using a solid geometry Euclidean and the theory of the gravitation of Newton (force connecting the centers of the stars), one would obtain an elliptic trajectory without advance of the perihelion, contrary to what is noted in experiments (made abstraction of the advance of the perihelion due to the disturbances of another planets). One says sometimes, by joke, which the model of gravitation of Newton is completely valid only in one case: that where no massive body is there to disturb the model of it, which has obviously something of awkward.
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