Introduction to Coordinate Geometry and Coordinate Systems

1. In Coordinate Geometry we study How to perform Geometrical Measurments based on Spatial Reference Frames. These Spatial Reference Frames are called Coordinate Systems.
2. Coordinate Systems are used for following purposes
   1. Representing Geometric Objects like Curves and Surfaces.
   2. Representing Data Distributions (also known as Fields).
   3. Performing Calculations/Measurements related to Finding of Physical Quantities such as Positions / Distances / Length / Area / Volume / Angles etc.
3. Every Coordinate System is composed of a Set of Directions (Curved, Straight Line or a Combination of Both) called Coordinate Axes along which the measurements/calculations are done. Curves (or Straight Lines) along a Particular Coordinate Axis Intersect with the Curves (or Straight Lines) along other Coordinate Axes to form the Grid for a Coordinate System. Each such Point of Intersection specifies a Unique Location on the Grid known as a Coordinate Location or a Coordinate Point.
4. Coordinate Systems can have an Initial Location with repect to which all measurements/calculations are done. This location is called the Origin of the Coordinate System. Coordinate Systems without a well defined Origin are called Affine Coordinate Systems.
5. Based on how the Coordinate Axes (or Tangents to the Curves along the Coordinate Axes) of Coordinate Systems are aligned to each other at the Point of Intersection, Coordinate Systems can be of 2 types
   1. Orthogonal Coordinate Systems: In these Coordinate Systems the Coordinate Axes (or Tangents to the Curves along the Coordinate Axes) are All Mutually Perpendicular to each other at the Point of Intersection.
   2. Oblique Coordinate Systems: In these Coordinate Systems the Coordinate Axes (or Tangents to the Coordinate Axes) are Not All Mutually Perpendicular to each other at the Point of Intersection.
6. Based on the Kind of the Coordinate Axes, the Coordinate Systems can be of 2 types
   1. Cartesian Coordinate Systems: These are Orthogonal Coordinate Systems, having all the Coordinate Axes that are Straight Lines.
   2. Curvilinear Coordinate Systems: These are either Oblique Coordinate Systems or are Coordinate Systems in which atleast One of the Coordinate Axes is Not a Straight Line. Polar Coordinate System and Parabolic Coordinate System are some examples of 2D Curvilinear Coordinate Systems. Spherical Coordinate System, Polar Cylindrical Coordinate System and Parabolic Cylindrical Coordinate System are some examples of 3D Curvilinear Coordinate Systems.
7. The choice of a Coordinate System depends on the kind of Geometric Objects / Data Distribution that need to represented or the kind of measurements that need to be done. This is because some Coordinate Systems are much more suitable/intutive/easier for certain geometric/data measurements/representations than others.
   
   The representations/measurements can be done most intutively/easily using Orthogonal Coordinate Systems. Following are some commonly used Orthogonal Coordinate Systems
   1. 2D: Cartesian, Polar, Parabolic
   2. 3D: Cartesian, Spherical, Polar Cylindrical, Parabolic Cylindrical