Point coordinates of the quadric surfaces: \(x,\) \(y,\) \(z,\) \({x_1},\) \({y_1},\) \({z_1},\) \(\ldots\)
Real numbers: \(A,\) \(B,\) \(C,\) \(\ldots,\) \(a,\) \(b,\) \(c,\) \({k_1},\) \({k_2},\) \({k_3}\)
Invariants: \(e\), \(E\), \(\Delta\)
Real numbers: \(A,\) \(B,\) \(C,\) \(\ldots,\) \(a,\) \(b,\) \(c,\) \({k_1},\) \({k_2},\) \({k_3}\)
Invariants: \(e\), \(E\), \(\Delta\)
Radius of a sphere: \(R\)
Center of a sphere: \(\left( {a,b,c} \right)\)
Center of a sphere: \(\left( {a,b,c} \right)\)
- 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. - 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. - Real Ellipsoid (#\(1\))
\({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} + {\large\frac{{{z^2}}}{{{c^2}}}\normalsize} \) \(= 1\) - Imaginary Ellipsoid (#\(2\))
\({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} + {\large\frac{{{z^2}}}{{{c^2}}}\normalsize} \) \(= -1\) - Hyperboloid of One Sheet (#\(3\))
\({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} – {\large\frac{{{z^2}}}{{{c^2}}}\normalsize} \) \(= 1\) - Hyperboloid of Two Sheets (#\(4\))
\({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} – {\large\frac{{{z^2}}}{{{c^2}}}\normalsize} \) \(= -1\) - Real Quadric Cone (#\(5\))
\({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} – {\large\frac{{{z^2}}}{{{c^2}}}\normalsize} \) \(= 0\) - Imaginary Quadric Cone (#\(6\))
\({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} + {\large\frac{{{z^2}}}{{{c^2}}}\normalsize} \) \(= 0\) - Elliptic Paraboloid (#\(7\))
\({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} – z \) \(= 0\) - Hyperbolic Paraboloid (#\(8\))
\({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} – {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} – z \) \(= 0\) - Real Elliptic Cylinder (#\(9\))
\({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} = 1\) - Imaginary Elliptic Cylinder (#\(10\))
\({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} = -1\) - Hyperbolic Cylinder (#\(11\))
\({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} – {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} = 1\) - Real Intersecting Planes (#\(12\))
\({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} – {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} = 0\) - Imaginary Intersecting Planes \(\left({\text{#}13}\right)\)
\({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} + {\large\frac{{{y^2}}}{{{b^2}}}\normalsize} = 0\) - Parabolic Cylinder (#\(14\))
\({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} – y = 0\) - Real Parallel Planes (#\(15\))
\({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} = 1\) - Imaginary Parallel Planes (#\(16\))
\({\large\frac{{{x^2}}}{{{a^2}}}\normalsize} = -1\) - Coincident Planes (#\(17\))
\({x^2} = 0\) - 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
\({x^2} + {y^2} + {z^2} \) \(= {R^2}.\) - 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. - 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. - 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. - 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
\(R =\) \({\large\frac{{\sqrt {{D^2} + {E^2} + {F^2} – 4{A^2}M} }}{{2A}}\normalsize}\).
\(e = \left[ {\begin{array}{*{20}{c}} A&H&G\\ H&B&F\\ G&F&C \end{array}} \right],\;\) \(E = \left[ {\begin{array}{*{20}{c}} A&H&Q&P\\ H&B&F&Q\\ G&F&C&R\\ P&Q&R&D \end{array}} \right],\;\) \(\Delta = \det \left( E \right),\)
The roots \({k_1}\), \({k_2}\), \({k_3}\) are obtained from the solution of the equation
\( \left| {\begin{array}{*{20}{c}} {A – k}&H&G\\ H&{B – k}&F\\ G&F&{C – k} \end{array}} \right| \) \(= 0.\)
