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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. 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 which is given by a Unique Ordered Set of as many Real Number Values as the Number of Coordinate Axes.
   
   The First Value of the Ordered Set gives the Value of First Axis for the Location.
   
   The Second Value of the Ordered Set gives the Value of Second Axis for the Location.
   
   The Third Value of the Ordered Set gives the Value of Third Axis for the Location and so on.
   
   The Variables that are used to denote the Value of Any Axis for a Location are called Coordinate Variables.
3. Coordinate Systems are classified based on
   1. Presence or Absence of an Origin: Coordinate Systems May or May Not have an Initial Location with repect to which all measurements/calculations are done. If present, this location is called the Origin of the Coordinate System.
   2. Based on the Number of the Coordinate Axes:
      
      The Number of the Coordinate Axes present in a Coordinate System gives the Dimension of the Coordinate System. An \(N\)-Dimensional Coordinate System has \(N\) Number of Coordinate Axes.
   3. Based on the Nature of the Coordinate Axes:
      
      The Coordinate Systems in which All Coordinate Axes are Straight Lines are called Straight Line Coordinate Systems.
      
      The Coordinate Systems in which Atleast One of the Coordinate Axes is a Non Straight Line Curve are called Curvilinear Coordinate Systems.
   4. Based on the Nature of the Intersection of Coordinate Axes:
      
      The Coordinate Systems in which All the Coordinate Axes (or All the Tangents to the Curves along the Coordinate Axes) are Mutually Perpendicular to each other at the Point of Intersection are called Orthogonal Coordinate Systems.
      
      The Coordinate Systems in which Any One Pair of Coordinate Axes (or Any One Pair of Tangents to the Curves along the Coordinate Axes) are Not Perpendicular to each other at the Point of Intersection are called Oblique Coordinate Systems.
4. Following are some of the Most Commonly used Coordinate Systems
   1. Cartesian Coordinate Systems: These are Orthogonal Coordinate Systems, having all the Coordinate Axes that are Straight Lines.
   2. Polar Coordinate System: It is an Orthogonal, 2D Curvilinear Coordinate System.
   3. Parabolic Coordinate System: It is also an Orthogonal, 2D Curvilinear Coordinate System.
   4. Spherical Coordinate System: It is an Orthogonal, 3D Curvilinear Coordinate System.
   5. Polar Cylindrical Coordinate System: It is also an Orthogonal, 3D Curvilinear Coordinate System.
5. 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.
   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.
6. Coordinate Systems are used for mathematically representing Geometric Objects like Curves and Surfaces through the following
   1. Scalar Coordinate Equations: These are some Scalar Function Equations that a particular Geometric Object satisfies. The Variables in these Scalar Function Equations are the Coordinate Variables of the Coordinate System in which the Geometric Object is represented.
   2. Parametric Equations / Position Vector Functions: In this representation the Coordinate Variables of the Coordinate Systems are themselves given in terms of Scalar Functions of 1 or More Parameter Variables. The value of the Coordinate Variables thus obtained Form the Components of Position Vector of Points on the Geometric Object.
   3. Vector Equations: These are some Vector Equations that a particular Geometric Object satisfies.
   Each of the above representation can have Various Types Depending on the Actual Geometric Object.
7. Coordinate Systems are used for mathematically representing Data Distributions / Fields through the following
   1. Scalar Coordinate Expressions: These are Scalar Function Expressions that represent Fields. The Variables in these Scalar Function Expressions are Coordinate Variables of the Coordinate System in which the Field is represented.
   2. Vector Functions: These are Vector Function Expressions that represent Fields. Each Component of the Vector Function is a Scalar Function Expression. The Variables in these Scalar Function Expressions are Coordinate Variables of the Coordinate System in which the Field is represented.