Quadric Surfaces

1. General equation of a quadric surface
   \(A{x^2} + B{y^2} + C{z^2} \) \(+\; 2Fyz + 2Gzx \) \(+\; 2Hxy + 2Px \) \(+\; 2Qy + 2Rz \) \(+\; D = 0,\)
   where \(x,\) \(y,\) \(z\) are the Cartesian coordinates of the points of the surface, \(A,\) \(B,\) \(C, \ldots\) are real numbers.
2. Classification of quadric surfaces
   This classification is based on invariants of the quadric surfaces. Invariants are special expressions composed of the coefficients of the general equation which do not change under parallel translation or rotation of the coordinate system. In total, there are \(17\) different (canonical) classes of the quadric surfaces.

Here the invariants are the ranks of the matrices \(e\) and \(E\), the determinant \(\Delta\) of the matrix \(E\), and signs of the eigenvalues \(k\) of the matrix \(e.\) The matrices \(e\) and \(E\) are given by
\(e = \left[ {\begin{array}{*{20}{c}}
A&H&G\\
H&B&F\\
G&F&C
\end{array}} \right],\;\) \(E = \left[ {\begin{array}{*{20}{c}}
A&H&Q&P\\
H&B&F&Q\\
G&F&C&R\\
P&Q&R&D
\end{array}} \right],\;\) \(\Delta = \det \left( E \right),\)

The roots \({k_1}\), \({k_2}\), \({k_3}\) are obtained from the solution of the equation
\( \left| {\begin{array}{*{20}{c}}
{A – k}&H&G\\
H&{B – k}&F\\
G&F&{C – k}
\end{array}} \right| \) \(= 0.\)

1. Real Ellipsoid (#\(1\))
   \({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} + {\large\frac{{{z^2}}}{{{c^2}}}\normalsize} \) \(= 1\)

1. Imaginary Ellipsoid (#\(2\))
   \({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} + {\large\frac{{{z^2}}}{{{c^2}}}\normalsize} \) \(= -1\)
2. Hyperboloid of One Sheet (#\(3\))
   \({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} – {\large\frac{{{z^2}}}{{{c^2}}}\normalsize} \) \(= 1\)

1. Hyperboloid of Two Sheets (#\(4\))
   \({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} – {\large\frac{{{z^2}}}{{{c^2}}}\normalsize} \) \(= -1\)

1. Real Quadric Cone (#\(5\))
   \({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} – {\large\frac{{{z^2}}}{{{c^2}}}\normalsize} \) \(= 0\)

1. Imaginary Quadric Cone (#\(6\))
   \({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} + {\large\frac{{{z^2}}}{{{c^2}}}\normalsize} \) \(= 0\)
2. Elliptic Paraboloid (#\(7\))
   \({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} – z \) \(= 0\)

1. Hyperbolic Paraboloid (#\(8\))
   \({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} – {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} – z \) \(= 0\)

1. Real Elliptic Cylinder (#\(9\))
   \({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} = 1\)

1. Imaginary Elliptic Cylinder (#\(10\))
   \({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} = -1\)
2. Hyperbolic Cylinder (#\(11\))
   \({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} – {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} = 1\)

1. Real Intersecting Planes (#\(12\))
   \({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} – {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} = 0\)
2. Imaginary Intersecting Planes \(\left({\text{#}13}\right)\)
   \({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} = 0\)
3. Parabolic Cylinder (#\(14\))
   \({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} – y = 0\)

1. Real Parallel Planes (#\(15\))
   \({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} = 1\)
2. Imaginary Parallel Planes (#\(16\))
   \({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} = -1\)
3. Coincident Planes (#\(17\))
   \({x^2} = 0\)
4. Equation of a sphere centered at the origin
   A sphere is a special case of an ellipsoid when the three semi-axes are the same and equal to the radius of the sphere. The equation of a sphere of radius \(R\) centered at the origin is given by
   \({x^2} + {y^2} + {z^2} \) \(= {R^2}.\)

1. Equation of a sphere centered at any point
   \({\left( {x – a} \right)^2} + {\left( {y – b} \right)^2} \) \(+\;{\left( {z – c} \right)^2} \) \(= {R^2},\)
   where \(\left( {a,b,c} \right)\) are the coordinates of the center of the sphere.
2. Diameter form of the equation of a sphere
   \(\left( {x – {x_1}} \right)\left( {x – {x_2}} \right) \) \(+\, \left( {y – {y_1}} \right)\left( {y – {y_2}} \right) \) \(+\, \left( {z – {z_1}} \right)\left( {z – {z_2}} \right) \) \(= 0,\)
   where \({P_1}\left( {{x_1},{y_1},{z_1}} \right),\) \({P_2}\left( {{x_2},{y_2},{z_2}} \right)\) are the endpoints of a diameter.
3. Four points form of the equation of a sphere
   \(\left| {\begin{array}{*{20}{l}}
   {{x^2} + {y^2} + {z^2}}&x&y&z&1\\
   {x_1^2 + y_1^2 + z_1^2}&{{x_1}}&{{y_1}}&{{z_1}}&1\\
   {x_2^2 + y_2^2 + z_2^2}&{{x_2}}&{{y_2}}&{{z_2}}&1\\
   {x_3^2 + y_3^2 + z_3^2}&{{x_3}}&{{y_3}}&{{z_3}}&1\\
   {x_4^2 + y_4^2 + z_4^2}&{{x_4}}&{{y_4}}&{{z_4}}&1
   \end{array}} \right| \) \(= 0.\)
   The points \({P_1}\left( {{x_1},{y_1},{z_1}} \right),\) \({P_2}\left( {{x_2},{y_2},{z_2}} \right),\) \({P_3}\left( {{x_3},{y_3},{z_3}} \right),\) \({P_4}\left( {{x_4},{y_4},{z_4}} \right)\) belong to the given sphere.
4. General equation of a sphere
   \(A{x^2} + A{y^2} + A{z^2} \) \(+\;Dx + Ey \) \(+\; Fz + M \) \(= 0,\;\) \(\left( {A \ne 0} \right)\)
   The center of the sphere has the coordinates \(\left( {a,b,c} \right)\) where
   \(a = – {\large\frac{D}{{2A}}\normalsize},\;\) \(b = – {\large\frac{E}{{2A}}\normalsize},\;\) \(c = – {\large\frac{F}{{2A}}\normalsize}.\)
   The radius of the sphere is given by
   \(R =\) \({\large\frac{{\sqrt {{D^2} + {E^2} + {F^2} – 4{A^2}M} }}{{2A}}\normalsize}\).