Topological Construction of Spherical Analogue of a Given Euclidean Pyramid

Given a regular Euclidean pyramid with square base, we use basic properties of great circle associated to it sides to prove the existence of its spherical counterpart. We also prove that its homeomorphic to its spherical counterpart.


Introduction
Euclidean geometry was believed to solve all the problems that could arise in society. Despite all these we came to discover that there were some questions Euclidean geometry could not answer. When we tried to find out, it led us to non-Euclidean Geometry. Euclid, a Greek mathematician who lived in approximately 300 B.C., is credited with collecting and organizing the postulates and theorems that are studied in geometry courses. The Parallel Postulate represents one of the most controversial assumptions made by Euclid. Over the years, Euclid's Parallel Postulate has troubled mathematicians. The thought that lines may intersect at possibly infinite. In 1795, the mathematician John Playfair devised an alternative formulation of Euclid's Parallel Postulate called Playfair's Axiom. Playfair's Axiom is more useful than Euclid's Parallel Postulate, as it answers the question, 'How many lines can be drawn through a point not on a line and, at the same time, parallel to that line?' In most geometry courses, Euclid's Parallel Postulate and Playfair's Axiom are used interchangeably. A development of geometry in which the Parallel Postulate or Playfair's Axiom do not hold is known as non-Euclidean geometry. This work is part of spherical geometry and one of it aims is to propose a method which can help us to construct spherical objet from given Euclidean one. It is well know that geometry aids in our perception of the world. We can use it to de-construct our view of objects into points, lines, circles, planes and spheres. For example, some properties of triangles that we know are that it consists of three straight lines and three angles that sum to π. Can we imagine other geometries that do not give these familiar results? The Euclidean geometry that we are familiar with depends on Euclid's parallel postulate. Since it is a postulate and not a theorem, it is assumed to be true without proof. If we alter that postulate, new geometries emerge. This project explores one model of that Non-Euclidean Geometry; the spherical geometry.
In this work we solve the following problem: is there a spherical pyramid homeomorphic to a prescribed Euclidean given pyramid? We also show how our construction can be use in engineering. Notably the existence of this homeomorphism can be use in the frame work of numerical metallurgy.
The main objective of this work is to prove the existence of spherical analogue of a given regular Euclidean pyramid with square basis. We prove their existence and we also prove that there are homeomorphic to their Euclidean counterpart. In other words we have proven that the two geometric entities are topologically equivalent. By compare their volume and area we show that those entity are not isometric [2]. More precisely we show that the volume and the area of spherical counterpart is bigger than it Euclidean analogue.
More explicitly, let SABCD be a given regular Euclidean pyramid with square basis ABCD. We denote ΔSAB, ΔSBC, ΔSCD and ΔSDA the lateral faces of SABCD. To construct the spherical analogue of SABCD we define and homeomorphism from IR 3 to IR 3 which push each sides of SABCD on its circumscribed sphere. We denote S SAB , S SBC , S SCD and S SDA respectively the image of ΔSAB, ΔSBC, ΔSCD and ΔSDA. Then, S SAB ∪ S SBC ∪ S SCD ∪ S SDA is the spherical analogue of our Euclidean pyramid.

On existence of Circumscribe Sphere to a Given Pyramid
In this subsection we will prove the existence of circumscribe sphere to a given regular pyramid with square basis.
Let E be a dimension 3 Euclidean space and SABCD five non coplanar points of E such that ABCD form a square. SABCD is a regular pyramid if the following triangle ΔSAD, ΔSDC, ΔSCB, ΔSBA are equilateral.  Any two great circles of the sphere bisect each other. The figure formed by the shorter arcs joining three points on the surface of a sphere, no two of which are diametrically opposite, is called a spherical triangle. The portion of a sphere included between two halves of great circles is called a lune. Two triangles, which have a common side and whose other sides belong to the same great circles, are called columnar, triangles.  have common perpendicular bisector plan P.
As matter of the fact, P is perpendicular to the line segment [AB] and passing through its midpoint and it is perpendicular to the line (AB) since line (AB) || (DC) and AB = DC, then it is also according to the Thales theorem, perpendicular to [AD] and it passing through its midpoint. Therefore, it is also its perpendicular bisector. Likewise ( ) Is direct orthonormal frame. Let e be the length basis's side and h the altitude of above pyramid.

On Associated Spherical Pyramid
We suppose that SABCD is a real Euclidean pyramid. Before we construct the spherical pyramid associated to SABCD, we shall first recall some analogy between the geometry of the sphere and the plane. In order to understand the analogy between plane and spherical geometry, it is necessary to observe that to right lines on the plane correspond on the sphere great circles, and the circles on the plane correspond circles on the sphere, which may be either great or small. We have proven that our Euclidean pyramid is inscribed in a sphere. Our pyramid has 8 sides and each side define with the centre Ω a unique Euclidean plan which intersect the sphere trough a great circle. It is well know that any two great circles of the sphere bisect each other. The sides of our spherical pyramid are defined by section of those great circles with the sphere. We recall that; when two arcs of circles intersect, the angle of the tangents at their points of intersection is called the angle of the arcs. It is also well know that; the angle of intersection of two great circles is equal to the inclination of their planes. We conclude that, the angle between two great circles is equal to the inclination of their planes. Therefore, we have a method to fine angle between two spherical line segments. Since the lateral surfaces of our spherical pyramid are deformed of their Euclidean counterpart. We need to define the spherical triangle. The special case of antipodal triangle we play a major role in this work. We recall its definition. Definition 2.2. Two triangles, whose corresponding vertices are diametrically opposite, are called antipodal triangle.

Great Circles Equations Associated to Euclidean
After defined spherical angle, we need to give the expression of some their trigonometric functions.

OD OD
; ; is the direct orthogonal frame image of the frame R by an isometry.

Topological Relation between the Two Pyramids
In this section, we suppose that we are in three dimensional Euclidean real space and we construct a homeomorphism between Euclidean pyramid and his spherical counterpart.

Cartesian Equation of SAD
is the equation of ∆SAD in the frame R.

Expression of Associates Spherical Pyramid Proposition 3.2.
( ) 2 2 Where: f = 2hp-cq; g = 2hq+cp; i = 1-c; and t = h-a is the expression of ∆ in the frame R'. proof: The first equation is just equation of plan PS,[AD] in the frame R ' . It remain the associate constraint on X, Y and Z. By replacing expressions of x, y and z as functions of X, Y and Z in the proof of lemma 3.1 we obtain: From in equations (6) + (7) we deduce that:  (9) and (8)    In the same manner, we prove that SSAD is close subset of IR 3 . Proposition 3.4. The map: is an homeomorphism. 3 To prove that M = M0, it is enough to justify that: √ 2 + 2 + 2 = � 0 2 + 0 2 + 0 2 . By subtracting the members of (E1) and (E2), we obtain: Extracting X − Y in (E4) and replacing it in (3), we obtain: Since M0 ∈ ∆SAD, then: But it follows from (3), that: Replacing (E6) in (E5), we obtain: With: f = 2hq + cp and t = a − h. This implies that: To conclude that f1 is surjective, we will show that f1(M) = M ' . M ∈ ∆SAD.
PSΩA, PSΩD and PDΩA are bounds plans of regions SSAD and ∆SAD therefore, for all M ' ∈SSAD, (ΩM ' ) ∩ ∆SAD is one point set. It follow bay construction that M ∈ ∆SAD. Denote: