Spherical Geometry

Spherical Geometry (Student Name

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TOC \o "1-3 " \h \z \u HYPERLINK \l "_Toc7 " I . Introduction PAGEREF _Toc7 \h 3

HYPERLINK \l "_Toc8 " II . The Sphere PAGEREF _Toc8 \h 4

HYPERLINK \l "_Toc9 " Lines on the Surface of the Sphere PAGEREF _Toc9 \h 5

HYPERLINK \l "_Toc0 " Great-Circle on the Surface of the Sphere PAGEREF _Toc0 \h 5

HYPERLINK \l "_Toc1 " Geodesic PAGEREF _Toc1 \h 6

HYPERLINK \l "_Toc2 " Antipodal on the Surface of the Sphere PAGEREF _Toc2

\h 7

HYPERLINK \l "_Toc3 " Triangles on the Surface of the Sphere PAGEREF _Toc3 \h 7

HYPERLINK \l "_Toc4 " III . Postulates of Euclidean (Flat Geometry and Spherical Geometry PAGEREF _Toc4 \h 8

HYPERLINK \l "_Toc5 " Postulates of Euclidean (Flat ) Geometry PAGEREF _Toc5 \h 8

HYPERLINK \l "_Toc6 " Postulates of Spherical Geometry PAGEREF _Toc6 \h 8

HYPERLINK \l "_Toc7 " IV . Spherical Distance PAGEREF _Toc7 \h 10

HYPERLINK \l "_Toc8 " Distance on the Surface of the Sphere PAGEREF _Toc8 \h 10

HYPERLINK \l "_Toc9 " Great-Circle Distance PAGEREF _Toc9 \h 10

HYPERLINK \l "_Toc0 " V . Lunes PAGEREF _Toc0 \h 12 HYPERLINK \l "_Toc1 " The Area of a Lune PAGEREF _Toc1 \h 12

HYPERLINK \l "_Toc2 " VI . Spherical Trigonometry PAGEREF _Toc2 \h 13

HYPERLINK \l "_Toc3 " The Area of a Spherical Triangle (Girard 's Theorem ) PAGEREF _Toc3 \h 14

HYPERLINK \l "_Toc4 " VII . Tracking of Hurricanes Using Spherical Geometry PAGEREF _Toc4 \h 16

HYPERLINK \l "_Toc5 " VIII . Reference PAGEREF _Toc5 \h 18 Introduction

There are so many objects of circular or spherical nature in our surroundings exists . Our Earth , Sun and other planets all are of spherical shape . Therefore it is important to understand the geometry of spherical objects . In case of plane geometry (Euclidean geometry ) the basic concepts are points and line . Spherical geometry is the geometry of the two-dimensional (2D ) surface of a sphere . In case of sphere points are defined in usual sense of straight line but in sense of the shortest paths between points on the surface of sphere that is also called as geodesic . Spherical geometry is an example of a non-Euclidean geometry . There are several applications of the principal of spherical geometry . Some of them are in space navigation and astronomy (Wikipedia Spherical Geometry

The Sphere

It is important to understand what is sphere before understanding spherical geometry . A sphere is a symmetrical object and set of points in three-dimensional space (such as round ball . All the points on sphere are equidistant from a point , which is called as center of the sphere (Polking , 2000 . This distance from center to the sphere is called as radius of the sphere that is generally represented by `r ' or `R

Figure 1 : Sphere and Three-Dimensional Co-Ordinate

Source (BBC : Spherical Geometry - Definitions

If we take three-dimensional co-ordinate geometry for defining any point in the given space . Than the three-variables are usually x , y , and z and any given point in the three dimensional space will be represented by (x , y , z ) for the relevant values of x...