Skip to main content

Appendix B. Quadric Surface Types

Quadric surfaces are the three-dimensional analogues of conic sections. Their equations occur naturally in geometry, optimization, and multivariable integration. This appendix gives the standard axis-aligned forms and a practical method for recognizing them.

Definition B.0.1. Quadric Surface.

A quadric surface in \(\R^3\) is the zero set of a polynomial of degree at most two:
\begin{equation*} Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Iz+J=0, \end{equation*}
where the quadratic coefficients are not all zero.
We first consider nondegenerate quadrics with no mixed terms. Throughout the table, \(a,b,c\gt 0\text{.}\) Permuting \(x,y,z\) changes the orientation but not the type of surface.
Table B.0.2. Standard axis-aligned quadric surfaces.
Surface
Standard equation
Recognition feature
Ellipsoid
\(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1\)
Three positive squared terms
Elliptic cone
\(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}-\dfrac{z^2}{c^2}=0\)
Mixed signs and right side \(0\)
Elliptic cylinder
\(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\)
One variable is absent
Hyperboloid of one sheet
\(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}-\dfrac{z^2}{c^2}=1\)
One negative squared term
Hyperboloid of two sheets
\(-\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1\)
One positive squared term
Elliptic paraboloid
\(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=\dfrac{z}{c}\)
Like signs; one variable is linear
Hyperbolic paraboloid
\(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=\dfrac{z}{c}\)
Opposite signs; one variable is linear
The following gallery uses the \(z\)-axis as the distinguished axis. The equations and captions, rather than color alone, identify each surface.
Saddle surface curving upward along x and downward along y.
Figure B.0.4. The hyperbolic paraboloid \(z=x^2-y^2\text{.}\)
Several quick sign rules follow from the standard forms.
  • For a cone, the squared term with the different sign determines the axis. Both halves are present because that variable is squared.
  • For a hyperboloid of one sheet, the negative squared term determines the axis. For a hyperboloid of two sheets, the positive squared term determines the axis.
  • For either paraboloid, the variable that is not squared determines the axis. Reversing its sign reverses the opening direction.
  • If a variable is absent, the same planar curve is repeated parallel to that variable’s axis, producing a cylinder.
Coordinate traces provide a reliable check. Set one variable equal to a constant and identify the resulting curve. Ellipsoids have bounded elliptical traces; one-sheet hyperboloids have an elliptical waist; two-sheet hyperboloids have no points between their two vertices; and a hyperbolic paraboloid has parabolic traces opening in opposite directions.

Example B.0.5. Classification by Signs.

Classify \(x^2+4y^2-z^2=1\) and determine its axis.
Solution.
There are two positive squared terms, one negative squared term, and the right side is \(1\text{.}\) Hence the surface is a hyperboloid of one sheet. The negative term is the \(z^2\)-term, so its axis is the \(z\)-axis. At \(z=0\text{,}\) the trace \(x^2+4y^2=1\) is its narrowest ellipse.

Example B.0.6. Completing Squares.

Classify
\begin{equation*} x^2+4y^2+z^2-2x+8y-6=0. \end{equation*}
Solution.
Completing squares gives
\begin{equation*} (x-1)^2+4(y+1)^2+z^2=11. \end{equation*}
Dividing by \(11\) puts the equation in standard form:
\begin{equation*} \frac{(x-1)^2}{11} +\frac{(y+1)^2}{11/4} +\frac{z^2}{11} =1. \end{equation*}
Thus the surface is an ellipsoid centered at \(\langle 1,-1,0\rangle\text{,}\) with semiaxis lengths \(\sqrt{11}\text{,}\) \(\sqrt{11}/2\text{,}\) and \(\sqrt{11}\text{.}\)
Translations replace \(x,y,z\) by \(x-h,y-k,z-\ell\text{.}\) Mixed terms such as \(xy\) indicate that the principal axes may be rotated; an orthogonal change of coordinates removes these mixed terms. Some quadratic equations are degenerate: for example, \(x^2+y^2+z^2=0\) is one point, \(x^2+y^2+z^2=-1\) has no real points, and \(x^2-y^2=0\) is the union of two planes. Thus the table classifies the standard nondegenerate families, not every possible zero set of a quadratic polynomial.
For another introductory presentation of these standard forms, see [6].
PrevTopNext