Conic Section Normalization

1. Conic Section Normalization refers to the Process of making suitable changes to the Implicit Coordinate Equation of the Conic Section by doing some Transformational and/or Algebraic Operations to the Coordinate Equation for the purpose of Identifying the Conic Section and/or Finding its Properties/Parameters.
   
   The Implicit Coordinate Equation obtained after the Process of Normalization is called the Normalized Equation of the Conic Section.
2. The Steps involved in Normalization and the Normalized Equations vary depending on the Type of the Conic Section. Following are the Different Kinds of Conic Section Normalization
   1. Parabola Normalization: This Normalization is done after the given Implicit Equation of Conic Section has been identified as a Parabola. Performing Parabola Normalization involves Multiplying the Implicit Equation of the Conic Section by -1 if either the Co-efficient of \(x^2\) or Co-efficient of \(y^2\) or Both in the Equation is less than 0. Otherwise the Conic Section Equation is considered to be already Normalized.
   2. Circle Normalization: This Normalization is done after the given Implicit Equation of Conic Section has been identified as a Circle. Performing Circle Normalization involves the following 2 steps in the order given below
      1. Translating the Implicit Equation of Conic Section so that its Center Lies in the Origin.
      2. Dividing the Implicit Equation of the Conic Section by Co-efficient of \(x^2\) (or Co-efficient of \(y^2\) since both are equal) so that the Value of their Co-efficients becomes 1.
   3. Ellipse/Hyperbola Normalization: This Normalization is done after the given Implicit Equation of Conic Section has been identified as an Ellipse or a Hyperbola. Performing Ellipse/Hyperbola Normalization involves the following 2 steps in the order given below
      1. Translating the Implicit Equation of Conic Section so that its Center Lies in the Origin.
      2. Dividing the Implicit Equation of Conic Section with the Negative value of the Contant of Equation so that the Value of the Contant of Equation becomes -1.
   4. Line Normalization: This Normalization is done after the given Implicit Equation of Conic Section has been identified as a Pair of Lines. Performing Line Normalization further depends on the Value of the Co-efficients of the Square Terms (\(x^2, y^2\) and \(xy\)) as follows
      1. If the Value of the Co-efficient of the \(x^2\) term is Not 0, then the Normalized Equation is obtained by Dividing the Implicit Equation of Conic Section with the Co-efficient of \(x^2\) term.
      2. If the Value of the Co-efficient of the \(x^2\) term is 0 and Value of the Co-efficient of the \(y^2\) term is Not 0, then the Normalized Equation is obtained by Dividing the Implicit Equation of Conic Section with the Co-efficient of \(y^2\) term.
      3. If the Value of the Co-efficient of the \(x^2\) and \(y^2\) terms are 0, then the Normalized Equation is obtained by Dividing the Implicit Equation of Conic Section with the Co-efficient of \(xy\) term.