Quadric Surface Normalization

1. Quadric Surface Normalization refers to the Process of making suitable changes to the Implicit Coordinate Equation of the Quadric Surface by doing some Transformational and/or Algebraic Operations to the Coordinate Equation for the purpose of Identifying the Quadric Surface and/or Finding its Properties/Parameters.
   
   The Implicit Coordinate Equation obtained after the Process of Normalization is called the Normalized Equation of the Quadric Surface.
2. The Steps involved in Normalization and the Normalized Equations vary depending on the Type of the Quadric Surface. Following are the Different Kinds of Quadric Surface Normalization
   1. Hyperboloids, Ellipsoids, Elliptical and Hyperbolic Cylinder Normalization: This Normalization is done after the given Implicit Equation of Quadric Surface has been identified as such. Performing this kind of Normalization involves the following 2 steps in the order given below
      1. Translating the Implicit Equation of Quadric Surface so that its Center Lies in the Origin.
      2. Dividing the Implicit Equation of Quadric Surface with the Negative value of the Constant of Equation so that the Value of the Constant of Equation becomes -1.
   2. Sphere Normalization: This Normalization is done after the given Implicit Equation of Quadric Surface has been identified as a Sphere. Performing Sphere Normalization involves the following 2 steps in the order given below
      1. Translating the Implicit Equation of Quadric Surface so that its Center Lies in the Origin.
      2. Dividing the Implicit Equation of the Quadric Surface by Co-efficient of \(x^2\) or \(y^2\) or \(z^2\) (since all these are equal) so that the Value of their Co-efficients becomes 1.
   3. Plane Normalization: This Normalization is done after the given Implicit Equation of Quadric Surface has been identified as a Pair of Planes. Performing Plane Normalization further depends on the Value of the Co-efficients of the Square Terms (\(x^2, y^2, z^2, xy, xz\) and \(yz\)) 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 Quadric Surface with the Co-efficient of \(x^2\) term.
      2. If the Value of the Co-efficient of the \(x^2\) terms 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 Quadric Surface with the Co-efficient of \(y^2\) term.
      3. If the Value of the Co-efficients of the \(x^2\) and \(y^2\) terms are 0 and Value of the Co-efficient of the \(z^2\) term is Not 0, then the Normalized Equation is obtained by Dividing the Implicit Equation of Quadric Surface with the Co-efficient of \(z^2\) term.
      4. If the Value of the Co-efficients of the \(x^2\), \(y^2\) and \(z^2\) terms are 0 and Value of the Co-efficient of the \(xy\) term is Not 0, then the Normalized Equation is obtained by Dividing the Implicit Equation of Quadric Surface with the Co-efficient of \(xy\) term.
      5. If the Value of the Co-efficients of the \(x^2\), \(y^2\), \(z^2\) and \(xy\) terms are 0 and Value of the Co-efficient of the \(xz\) term is Not 0, then the Normalized Equation is obtained by Dividing the Implicit Equation of Quadric Surface with the Co-efficient of \(xz\) term.
      6. If the Value of the Co-efficients of the \(x^2\), \(y^2\), \(z^2\) , \(xy\) and \(xz\) terms are 0, then the Normalized Equation is obtained by Dividing the Implicit Equation of Quadric Surface with the Co-efficient of \(yz\) term.