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Spherical Geometry is a type of Non – Euclidean Geometry because of the fact that in spherical geometry the fifth and the last postulate of the Euclidean Geometry does not hold true, in fact spherical geometry uses a completely different view of postulates as compared to Euclid’s Geometry. Thus, Spherical geometry is a geometry wherein the parallel postulate doesn’t hold true. The fifth postulate of Euclid’s known as the parallel postulate states that given any line and any point not lying on that given line, then there will exist exactly a line from that point, which will be parallel to that given line. Note that there will be one and only one single line parallel to the given line, according to Euclid’s.
Spherical Geometry is a type of Non Euclidean geometry, which is a flat surface in Cartesian coordinate system, we use a sphere as its plane and therefore, it is geometry on spheres.
This can be shown as below:
 Sphere in Flat Surface
That is, in spherical geometry, we study the geometry only on a sphere. Thus, all the terms which are related to Euclidean geometry like points, angles, straight lines, triangles and many more, are now applied on a sphere, thus making it a part of studying non Euclidean geometry.

A point in a plane represents the same meaning as in case of spherical geometry.
A line in a plane represents a great circle, in case of spherical geometry. Here, a great circle can be any circle inscribed on a sphere, whose centre is equal to the centre of the given sphere. Similarly, we can define a small circle as any line, in which the centre differs from the center of the given sphere.

The difference between great circle and small circle can be better understood by looking at the following diagram:

Sphere

We can now proceed to find out whether the first four postulates of Euclid’s holds true in case of spherical geometry or not, by considering the following points:

  • Euclid’s Postulate 1: There exists one and only one line, through any given two points, which is also true in case of spherical geometry.
  • Euclid’s Postulate 2: There exists one and only one plane, through any given three points, which are not lying on same line, this also holds true in case of spherical geometry.
  • Euclid’s Postulate 3: There exist at least two points, through any given line, which is also true in case of spherical geometry.
  • Euclid’s Postulate 4: All the angles which are right angle, that is of measure 90 degrees each, will always be congruent to each other, which is also true in case of spherical geometry.
Thus, all the first four postulates of Euclid’s holds true in spherical geometry except for the parallel postulate.


The need of spherical geometry raised because our Earth is a sphere, and the Cartesian coordinates system which is used in case of Euclidean geometry can not be applied in case of studying this non Euclidean geometry. Hence, the concept of longitude and latitudes were applied for studying different positions on our planet, Earth. 

In the 8th century B.C., Claudius Ptolemy wrote a book, named Almagest, wherein he explained in detail, about the work of earlier Egyptian and Babylonians, of various eclipses and various heavenly bodies like earth, stars etc, as Babylonians were the first who invented the concept of 360 degrees which completes a round circle. Moreover, Babylonians were the first, who developed a type of coordinate system, which is now known as spherical coordinate system.

In the 4th century B.C., the studies of Babylonians were further examined by Greeks, and Eudoxus developed a model for astronomy of stars and other heavenly bodies, which was known as The Two Sphere Model. Based on this model, in the same century, Autolycus wrote a book named, On the Rotating Spheres, in which he introduced another sphere of third type in which he introduced a pole and an equator, by which he showed that for an individual observer, there are some points, like stars, in the celestial sphere, which are categorized into either of the 3 categories mentioned below:
  • All the time visible points, or
  • All the time invisible points, or
  • Some rise and some set stars or points.

Around 300 B.C, Euclid’s also wrote a book on the concept of spherical geometry, named as Phaenomena, in which he solved some of the major problems relating to astronomy including the length of the day light at a particular longitude, on a particular date, etc. Euclid applied his definitions and postulates in the field of spherical geometry, in which the most famous of all propositions is stated below:
Let there are two circles which are parallel to the same given great circle, but they lie on the opposite sides of each other, then these two circles will be equal to each other in and only if, both of them cut equal length of arcs, from some other great circle, on any side of the given great circle.

Around the middle of the 1st century, Hipparchus also studied the concept of spheres of Babylonians and applied this to the above mentioned three spheres geometry, which helped in relating the coordinates of one sphere with the coordinates of the second sphere and so on. This change in the spherical geometry gave rise to an important part of spherical geometry known as Spherical Trigonometry, which is nowadays, very helpful in solving various astronomical problems.

Around 200 B.C., a most important book of spherical geometry was introduced by Theodosius, and the name of the book was Sphaerica. Around 100 A. D, another advanced version of spherical geometry was introduced by Menelaus, whose book’s name was On the Sphere, which is still being used by almost all the researches in the filed of spherical geometry.
The following is the list of some important spherical geometrical formulas:
1). We know that the parallel postulates of Euclid’s does not hold true in case of spherical geometry, which can be redefined in case of spherical geometry, and can be stated as follows:
Given any line and any point not lying on that same given line, then there would not exist a line from that point, which will be parallel to that given line.
2).In spherical geometry, any two lines will intersect not at one point, but they will intersect always at exactly two points, as shown below:
 
Intersecting Point in Sphere

3).Similarly, in spherical geometry, the angle sum property of a triangle does not hold true, that is the sum of all angles of a triangle inscribed in a sphere will not add up to 180 degrees.
4).Great Circle: is a circle in which, Radius of a sphere = Radius of a great circle.
 In other words, we can say that a great circle is a circle which passes through the center of the sphere and hence, a great circle always divides its sphere into two half equal hemispheres.
5).Small Circle: is a circle in which, radius of a sphere in not equal to the radius of a great circle
6).Antipodes: A great circle’s diameter’s end points are known as antipodes.
7). Axis: The diameter of any circle, perpendicular to the plane of that circle passing through its center is called as an axis.
8).Poles: An axis’s end points are known as poles.
9).Sides of any spherical triangle = arcs made out of the great circles.
10).Vertices of any spherical triangle = the points of intersection of three great circles.
11).Angles made from arcs = angles of the spherical triangle.
12).If s1, s2, s3 are the three sides of a spherical triangle, then either of the following inequality will hold true:
s1 + s2 > s3 or 
s1 + s3 > s2 or 
s3 + s2 > s1.
s1
13).If s1, s2, s3 are the three sides of a spherical triangle, then s1 + s2 + s3 < 360 degrees.
14).If triangle ABC is a spherical triangle, then the sum of the three angles would be less than 540 degrees but greater than 180 degrees, that is:
    180 degrees < angle A + angle B + angle C < 540 degrees.
    This can be better understood by looking at the following diagram:

Geometric Sphere

     
15).If a1, a2, and a3 are the three angles made by a spherical triangle, then the area of the triangle will be a1 + (a3 / a2) - $\Pi $ 
16).Area of a Diangle: If $\theta$ is the angle made by a diangle, then the area of the diangle will always be equal to 2$\theta$.
17).Parallel lines do not exist in the concept of spherical geometry; however they can be interpreted in a different way in non Euclidean geometry.
Spherical geometry is widely used for finding out the large distances of flights or driving or sailing etc. because of the concept of great circles in it. As we know, great circles are just the straight lines drawn on the sphere, provided that the centre of the great circle also lies at the center of the sphere itself. And because of the above definition of great circles, we can conclude that there do not have any concept of parallel lines in case of spherical geometry. 

However, the parallel postulate of Euclid’s has been redefined, with some modifications in case of spherical geometrical systems, which is known as the famous Elliptic Parallel Theorem as stated below:

Let p be any given point not lying on a given line l, then each and every line passing through the given point p will intersect with the given line l.

This statement of Elliptic Parallel Theorem can be better understood by looking at the following example:
Let the given line be the equator of the Earth, then, all the possible great circles, passing through any given point p on the Earth, will definitely intersect the equator of the Earth.

Similarly, we have another parallel postulate in spherical geometry which is known as the Hyperbolic Parallel Postulate, which is stated as below:

Let p be any given point not lying on a given line l, then there will exist at least two lines passing through the given point p, which will be parallel to the given line l. Note that in case of spherical geometry, the word parallel implies that the lines do not intersect each other. 

Hence, the above statement can be restated as follows:
Let p be any given point not lying on a given line l, then there will exist infinitely many lines passing through the given point p, which will not intersect with the given line l.
The Hyperbolic Parallel Postulate can be better understood by looking at the following diagram:

Hyperbolic Parallel Postulate

From the above, we can conclude that given a great circle or a line AB and a given point c, then by looking at the lines which are marked inside the green portion, we can find there are 3 lines, via, CD, CE and CF, which do not intersect the line AB and hence, will be parallel to AB, in the context of spherical geometry.
Also, all the lines or great circles that lie outside the limiting rays of the parallel lines are known as the limiting parallel. The limiting parallel lines are also known as ultra parallel or hyper parallel or super parallel.
We can draw the above section of diagram in a better way as follows:
 
Hyperbolic Parallel lines of Sphere

Here, we can see the line AB has two angles of parallelism, which are made by the corresponding two right and left parallel lines. These two left and right angles of parallelism would be always congruent to each other.
Spherical Geometry has a very special concept of triangles, which follows a different set of rules because of the spherical nature of its plane. This concept is explained in an easy way as following:

We know that, we can denote a unit sphere in a three dimensional space as S2, which is the set of all the points (x, y, z) belonging to R$^3$ such that x2 + y2 + z2 = 1. 

By the definition of great circles, we can say that the lines in the coordinate geometrical systems play the exact same role of great circles in case of spherical geometry. Since, great circles passes through the center of the sphere, therefore there can be infinitely many great circles passing through the center of the sphere.

Diangles: If two different great circles inside a sphere, intersect with each other, then they will intersect only at exactly two points, which will be the inverse of each other, then a region that will be bounded by these two great circles is known as a diangle.
Properties of Diangles:
  •     Both the angles made on the two vertices, of the diangles will be equal to each other.
  •     Both the diangles, made by the intersection of two great circles, will be congruent to each other.

A polygon or a spherical triangle on a sphere is known as a spherical polygon, in which each side of the given polygon is a part of the great circle. In a spherical triangle, then vertices of the triangle represent the points on the sphere and the sides of the triangle represent the arc segments of that great circle. For example, consider the following diagram of a spherical triangle:
 
This is a spherical triangle in which s1, s2 and s3 are the three sides and a1, a2 and a3 are the three angles of the spherical triangle.

A spherical geometry triangle uses the famous Girard’s Theorem which is used for finding out the area of these spherical triangles.
The statement of Girard’s Theorem is as follows:
If a1, a2, and a3 are the three angles made by a spherical triangle, then the area of the triangle will be a1 + (a3 / a2) - $\Pi $

We can generalize this result of area of spherical polygons having n sides, by the following statement:
Generalization: If a1, a2,  …, an, are the interior angles made by a spherical polygon having n vertices and n sides, 
the area of the spherical polygon = a1 + a2 + …+ an – (n - 2) $\Pi $
The proof of the above generalized theorem follows from the fact that if there is a polygon of n sides, where its sides, n $\geq$ 4, then the given polygon, can be divided into (n – 2) triangles within it, whose individual sum of the interior angles would be equal to the sum of interior angles of the given polygon.