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:
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:
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.