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General Quadratic Equations in 3 Variables and Quadric Surfaces

1. Any equation given in the forms of the following are General Quadratic Equations in 3 Variables
   
   \(Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + K=0\)   ...(1)
   
   \(Ax^2 + By^2 + Cz^2 + 2Dxy + 2Exz + 2Fyz + 2Gx + 2Hy + 2Iz + K=0\)  ...(2)
   
   Please note that in such equations Atleast 1 of Co-efficients \(A\) (i.e. Co-efficient of \(x^2\)), \(B\) (i.e. Co-efficient of \(y^2\)), \(C\) (i.e. Co-efficient of \(z^2\)), \(D\) (i.e. Co-efficient of \(xy\)), \(E\) (i.e. Co-efficient of \(xz\)) or \(F\) (i.e. Co-efficient of \(yz\)) Must be Non-Zero.
   
   Also note that both equation (1) and equation (2) are just different representation of the same equation.
2. The General Quadratic Equations in 3 Variables geometrically represent a Set of Surfaces known as Quadric Surfaces.
3. Determining the Type of Surface and its Related Parameters/Properties that any given General Quadratic Equations in 3 Variables represents is called Quadric Surface Analysis.
4. For analysing General Quadratic Equations in 3 Variables , a certain set of Matrices, Determinants and Related Algebraic Entities are used. Representation of these Matrices, Determinants and Related Algebraic Entities differ depending on whether Any Given Quadratic Equation is Analysed in form of equation (1) or equation (2).


Following are the Matrices, Determinants and Related Algebraic Entities used to analyse the General Quadratic Equations in 3 Variables

\(E=\begin{bmatrix} A & \frac{1}{2}D & \frac{1}{2}E & \frac{1}{2}G \\ \frac{1}{2}D & B & \frac{1}{2}F & \frac{1}{2}H \\ \frac{1}{2}E & \frac{1}{2}F & C & \frac{1}{2}I \\ \frac{1}{2}G & \frac{1}{2}H & \frac{1}{2}I & K\end{bmatrix}\) (when Analysed in form of equation (1)) OR \(E =\begin{bmatrix} A & D & E & G \\ D & B & F & H \\ E & F & C & I \\ G & H & I & K\end{bmatrix}\) (when Analysed in form of equation (2))

\(e=\begin{bmatrix} A & \frac{1}{2}D & \frac{1}{2}E \\ \frac{1}{2}D & B & \frac{1}{2}F \\ \frac{1}{2}E & \frac{1}{2}F & C \end{bmatrix}\) (when Analysed in form of equation (1)) OR \(e =\begin{bmatrix} A & D & E \\ D & B & F \\ E & F & C \end{bmatrix}\) (when Analysed in form of equation (2))

\(\Delta=|E|=\begin{vmatrix} A & \frac{1}{2}D & \frac{1}{2}E & \frac{1}{2}G \\ \frac{1}{2}D & B & \frac{1}{2}F & \frac{1}{2}H \\ \frac{1}{2}E & \frac{1}{2}F & C & \frac{1}{2}I \\ \frac{1}{2}G & \frac{1}{2}H & \frac{1}{2}I & K\end{vmatrix}\) (when Analysed in form of equation (1)) OR \(\Delta=|E|=\begin{vmatrix} A & D & E & G \\ D & B & F & H \\ E & F & C & I \\ G & H & I & K\end{vmatrix}\) (when Analysed in form of equation (2))

\(R_E = R_4 =\) Rank of \(E\) Matrix

\(R_e = R_3 =\) Rank of \(e\) Matrix

\(K =true\) If All Non-Zero Eigen Values of Matrix \(e\) Have Same Sign otherwise \(K=false\)

\(M=true\) if \(A=B=C\) and \(D=E=F=0\) otherwise \(M=false\) (Applicable only for Ellipsoid/Sphere)

\(I=true\) if All Nomalized Non-Zero Eigen Values of Matrix \(e\) are \( >0 \) otherwise \(I=false\) (Applicable only for Real and Imaginary Elliptical Cylinders)


5. Any General Quadratic Equation in 3 Variables can represent any 1 of the following 19 Geometric Entities based on the values of the calculated Matrices, Determinants and Algebraic Entities
   
   

   S.No.	Geometric Entity	\(\Delta\)	\(R_E\) / \(R_4\)	\(R_e\) / \(R_3\)	K	M	I
   Hyperboloids
   1.	Hyperboloid of 1 Sheet	> 0	4	3	false	NA	NA
   2.	Hyperboloid of 2 Sheets	< 0	4	3	false	NA	NA
   Ellipsoids
   3.	Real Ellipsoid	< 0	4	3	true	false	NA
   4.	Imaginary Ellipsoid	> 0	4	3	true	false	NA
   Spheres
   5.	Real Sphere	< 0	4	3	true	true	NA
   6.	Imaginary Sphere	> 0	4	3	true	true	NA
   Paraboloids
   7.	Elliptical Paraboloid	< 0	4	2	true	NA	NA
   8.	Hyperbolic Paraboloid	> 0	4	2	false	NA	NA
   Cones
   9.	Real Cone	NA	3	3	false	NA	NA
   10.	Imaginary Cone	NA	3	3	true	NA	NA
   Cylinders
   11.	Real Elliptic Cylinder	NA	3	2	true	NA	true
   12.	Imaginary Elliptic Cylinder	NA	3	2	true	NA	false
   13.	Hyperbolic Cylinder	NA	3	2	false	NA	NA
   14.	Parabolic Cylinder	NA	3	1	NA	NA	NA
   Pair of Planes
   15.	2 Intersecting Real Planes	NA	2	2	false	NA	NA
   16.	2 Intersecting Imaginary Planes	NA	2	2	true	NA	NA
   17.	2 Parallel Real Planes	NA	2	1	NA	NA	NA
   18.	2 Parallel Imaginary Planes	NA	2	1	NA	NA	NA
   19.	2 Co-Incident Real Planes	NA	1	1	NA	NA	NA

   
   Whether a General Quadratic Equation in 3 Variables represents 2 Parallel Real Planes or 2 Parallel Imaginary Planes can only be found out after Factoring the Quadratic Equation.