Quadratic Surface

There are 17 standard-form quadratic surfaces. The general quadratic is written

$$ax^2+by^2+cz^2+2fyz+2gzx+2hxy+2px+2qy+2rz+d=0.$$ (1)

Define

$$e = \left[\begin{array}{ccc}a & h & g\\ h & b & f\\ g & f & c\end{array}\right]$$ (2)

$$E = \left[\begin{array}{cccc}a & h & g & p\\ h & b & f & q\\ g & f & c & r\\ p & q & r & d\end{array}\right]$$ (3)

$$\rho_3 = {\rm rank\ }e$$ (4)

$$\rho_4 = {\rm rank\ }E$$ (5)

$$\Delta = {\rm det\ }E,$$ (6)

and $k_1$, $k_2$, as $k_3$ are the roots of

$$\left\vert\matrix{a-x & h & g\cr h & b-x & f\cr g & f & c-x}\right\vert=0.$$ (7)

Also define

$$k\equiv\cases{ 1 & if the signs of nonzero $k$s are the same\cr 0 & otherwise.\cr}$$ (8)

Surface	Equation	$\rho_3$	$\rho_4$	$\mathop{\rm sgn}\nolimits (\Delta)$	$k$
Coincident Planes	$x^2 = 0$	1	1
Ellipsoid (Imaginary)	${x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2} =-1$	3	4	$+$	1
Ellipsoid (Real)	${x^2\over a^2}+{y^2\over b^2}+{z^2\over c^2} = 1$	3	4	$-$	1
Elliptic Cone (Imaginary)	${x^2\over a^2}+{y^2\over b^2}-{z^2\over c^2} = 0$	3	3		1
Elliptic Cone (Real)	$z^2 = {x^2\over a^2}+{y^2\over b^2}$	3	3		0
Elliptic Cylinder (Imaginary)	${x^2\over a^2}+{y^2\over b^2} =-1$	2	3		1
Elliptic Cylinder (Real)	${x^2\over a^2}+{y^2\over b^2} = 1$	2	3		1
Elliptic Paraboloid	$z = {x^2\over a^2}+{y^2\over b^2}$	2	4	$-$	1
Hyperbolic Cylinder	${x^2\over a^2}-{y^2\over b^2} =-1$	2	3		0
Hyperbolic Paraboloid	$z = {y^2\over b^2}-{x^2\over a^2}$	2	4	$+$	0
Hyperboloid of one Sheet	${x^2\over a^2}+{y^2\over b^2}-{z^2\over c^2} = 1$	3	4	$+$	0
Hyperboloid of two Sheets	${x^2\over a^2}+{y^2\over b^2}-{z^2\over c^2} =-1$	3	4	$-$	0
Intersecting Planes (Imaginary)	${x^2\over a^2}+{y^2\over b^2} = 0$	2	2		1
Intersecting Planes (Real)	${x^2\over a^2}-{y^2\over b^2} = 0$	2	2		0
Parabolic Cylinder	$x^2+2rz = 0$	1	3
Parallel Planes (Imaginary)	$x^2 = -a^2$	1	2
Parallel Planes (Real)	$x^2 = a^2$	1	2

See also Cubic Surface, Ellipsoid, Elliptic Cone, Elliptic Cylinder, Elliptic Paraboloid, Hyperbolic Cylinder, Hyperbolic Paraboloid, Hyperboloid, Plane, Quartic Surface, Surface

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 210-211, 1987.