A.8: Conic Sections and Quadric Surfaces

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A conic section is the curve of intersection of a cone and a plane that does not pass through the vertex of the cone. This is illustrated in the figures below.

$conePlaneCircle.svg$ $conePlaneEllipse.svg$ $conePlaneParabola.svg$ $conePlaneHyperbola.svg$

An equivalent¹ (and often used) definition is that a conic section is the set of all points in the $xy$-plane that obey $Q(x,y)=0$ with

\[ Q(x,y) = Ax^2 + By^2 + Cxy + Dx + Ey + F =0 \nonumber \]

being a polynomial of degree two². By rotating and translating our coordinate system the equation of the conic section can be brought into one of the forms³

This statement can be justified using a linear algebra eigenvalue/eigenvector analysis. It is beyond what we can cover here, but is not too difficult for a standard linear algeba course.

$\alpha x^2 + \beta y^2 =\gamma $ with $\alpha ,\be,\gamma \gt 0\text{,}$ which is an ellipse (or a circle),
$\alpha x^2 - \beta y^2 =\gamma $ with $\alpha ,\beta \gt 0\text{,}$ $\gamma \ne0\text{,}$ which is a hyperbola,
$x^2 = \delta y\text{,}$ with $\delta\ne 0$ which is a parabola.

The three dimensional analogs of conic sections, surfaces in three dimensions given by quadratic equations, are called quadrics. An example is the sphere $x^2+y^2+z^2=1\text{.}$

Here are some tables giving all of the quadric surfaces.

It is outside our scope to prove this equivalence. Technically, we should also require that the constants $A\text{,}$ $B\text{,}$ $C\text{,}$ $D\text{,}$ $E\text{,}$ $F\text{,}$ are real numbers, that $A\text{,}$ $B\text{,}$ $C$ are not all zero, that $Q(x,y)=0$ has more than one real solution, and that the polynomial can't be factored into the product of two polynomials of degree one.

name	elliptic cylinder	parabolic cylinder	hyperbolic cylinder	sphere
equation in standard form	\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\)	\(y=ax^2\)	\(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\)	\(x^2\!+\!y^2\!+\!z^2=r^2\)
\(x=\)constant cross-section	two lines	one line	two lines	circle
\(y=\)constant cross-section	two lines	two lines	two lines	circle
\(z=\)constant cross-section	ellipse	parabola	hyperbola	circle
sketch	$cylinder.svg$	$parabolic_cylinder.svg$	$hyperbolic_cylinder.svg$	$sphere.svg$

name	ellipsoid	elliptic paraboloid	elliptic cone
equation in standard form	\(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\)	\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z}{c}\)	\(\frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z^2}{c^2}\)
\(x=\) constant cross-section	ellipse	parabola	two lines if \(x=0\text{,}\) hyperbola if \(x\ne 0\)
\(y=\) constant cross-section	ellipse	parabola	two lines if \(y=0\text{,}\)hyperbola if \(y\ne 0\)
\(z=\) constant cross-section	ellipse	ellipse	ellipse
sketch	$ellipsoid.svg$	$elliptic_paraboloid.svg$	$cone.svg$

name	hyperboloid of one sheet	hyperboloid of two sheets	hyperbolic paraboloid
equation in standard form	\(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\)	\(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=-1\)	\(\frac{y^2}{b^2}-\frac{x^2}{a^2}=\frac{z}{c}\)
\(x=\) constant cross-section	hyperbola	hyperbola	parabola
\(y=\) constant cross-section	hyperbola	hyperbola	parabola
\(z=\) constant cross-section	ellipse	ellipse	two lines if \(z=0\text{,}\) hyperbola if \(z\ne 0\)
sketch	$hyperboloid1.svg$	$hyperboloid2.svg$	$hyperbolic_paraboloid.svg$

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