Spherical geometry – Wikipedia

The spherical geometry It is a non -Euclidean geometry conceived by the mathematician Bernhard Riemann. Spherical geometry has an immediate interpretation in Euclidean geometry. In fact, its model presents itself as “described” by the geometry of the surface of a sphere. It has practical applications in navigation and astronomy.

The spherical geometry was born from the denial of the postulate of Euclide, or equivalently by the IV.1 Postulate of Hilbert. However, so that it is a coherent axiomatic theory, it is also necessary to change the axioms of incidence and system of Euclidean geometry (in the case of elliptical geometry only the system) ^[first] . It is characterized by the absence of parallel straight lines.

Below we present first the axiomatic body of the plain spherical geometry and then we will analyze one of it. For a more intuitive understanding, you can, if you want, read the following paragraph before the axiomatic discussion: a spherical geometry model.

With reference to the axiomatic classification proposed by Hilbert for Euclidean geometry, we report below that relating to the plain spherical geometry.

The primitive concepts are the point, the couples of points called antipoal points, the straight line, and the plan. There are also two binary relationships and a primitive quaternary relationship:

• Contains : one point can be contained in a straight line or in a plane, and a straight line can be contained in a plane;
• To stop : the pair of AB points separates the pair of CD points, in symbols: S (AB | CD) (quaternary relationship);
• Congruence , indicated with the “≡” symbol: corners and segments can be congruent.

The segment between two points A It is B It is defined as the portion of the straight between the points A It is B (included A It is B ).

I – axioms of belonging [ change | Modifica Wikitesto ]

II – Sort axioms [ change | Modifica Wikitesto ]

III – Axioms of congruence [ change | Modifica Wikitesto ]

IV – Axiom of Riemann [ change | Modifica Wikitesto ]

V – continuity axiom (or dedekind) [ change | Modifica Wikitesto ]

As already mentioned previously a model of spherical geometry is the one built on a sphere as we will specify below.
In flat geometry the basic concepts are the point and the straight line. On a sphere, the points are defined in the usual sense. The straight lines are defined as maximum circles. Therefore, in spherical geometry the corners are defined between maximum circles, and a trigonometry derives in the spherical plane that differs from the Euclidean trigonometry in the plane (for example, the sum of the internal corners of a triangle is greater than a flat corner). Instead the spherical trigonometry in spherical space (but also in the elliptical one), if appropriate conventions are adopted on the measure of the sides and corners of the spherical triangles, coincides with the Euclidean and hyperbolic spherical trigonometry. That is, the spherical trigonometry belongs to the body of absolute geometry.
The distance between two points of the sphere is the minimum segment that unites them, geodesic.

piano	set of points of a spherical surface of the Euclidean space
point	point of the surface of the sphere
lift	Maximum circumference of the spherical surface (circumference of intersection of the spherical surface with a plan passing through the center of the sphere)
segment	part of a straight line delimited by two points of the straight line itself
belonging	usual belonging in an Euclidean sense
antipodal points	diametrically opposite points of the spherical surface
congruence between segments	congruence between the arches of maximum circumference in Euclidean geometry (defined by the congruence of the ropes or through the movements of the sphere)
corner between two straight lines	Dadrous corner between the two floors that cut the sphere according to the two straight lines, or angle that coincides with the corner of the two Euclidean lines tangent to the sphere at the point of intersection of the two spherical and lying straight lines in the plans they identified
congruence between corners	congruence between corners in an Euclidean sense

Based on this interpretation (model) all axioms and properties of spherical geometry are propositions in Euclidean geometry. In fact, for example, infinite straight lines pass for two antipodal points.

Theorem [ change | Modifica Wikitesto ]

1. two sides and the corner included;
2. two corners and the common side
3. I char to;
4. The three corners.

• Pythagorean theorem
  If ABC is a spherical triangle rectum in A and with hypotenuse A, and with B and c the lengths of its sides, then the cosine of the hypotenuse is the same as the product of the Coseni dei Cateti:
  ${displaystyle cos(a/k)=cos(b/k)cos(c/k)}$

  ^[3] By developing series in the second order of the trigonometric functions, the universally known expression of the Pythagoras theorem in Euclidean geometry is obtained:

  ${displaystyle a^{2}=b^{2}+c^{2}}$

  

• Area of ​​a spherical polygon
  The area of ​​a spherical polygon of n from the aware:
  
  ${displaystyle A=R^{2}(alpha _{1}+alpha _{2}+…+alpha _{n}-(n-2)pi )}$

  .
  Its demonstration is based on the possibility of decomposing a spherical polygon into triangles.

• Formula of Euler
  Given a convex spherical polyhedron with V leaders, s.
  V-S+F=2.
• All perpendicular to a straight line contribute in two points, antipodal points.
• Two antipodal points divide the straight line into two congruent parts.
• Two antipodal points divide into two congruent parts all the straight lines that pass through them.
• All the straight lines are congruent.
• Data four distinct points A, B, C, D of the same straight line, it applies to more one of the following reports: S (AB | CD), S (AC | BD), S (AD | BC).
• In a rectangle triangle, the opposite corner of one of the two sides of the right corner is acute, obtuse or rectum according to that this side is less, greater or congruent on the other side of the right corner.

In addition to the two -dimensional sphere, other spaces have a spherical geometry: these spaces are called spherical varieties. The spherical geometry is formally given by a structure of Riemannian varieties with a sectional curvature everywhere equal to 1.

The basic models of spherical varieties are the spheres

${displaystyle S^{n}}$

of arbitrary dimension (for example the three -dimensional sphere

${displaystyle S^{3}}$

). All other spherical varieties have the local structure of a sphere, but they can have a different global topology: among these there are the projective spaces, obtained by identifying the antipodal points of a sphere, which are not adjustable in size

${displaystyle n}$

even. In size

${displaystyle n=3}$

There are also lenticular spaces.

• Non -Euclidean geometries and the foundations of geometry by E. Agazzi, D. Palladino – Mondadori scientific editions and techniques.