A.8: Conic Sections and Quadric Surfaces
- Page ID
- 92256
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.
An equivalent 1 (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 two 2. By rotating and translating our coordinate system the equation of the conic section can be brought into one of the forms 3
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 \(\alpha ,\beta,\gamma \gt 0\text{,}\) 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
\[x^2+y^2+z^2=1. \nonumber\]
Here are some tables giving all of the quadric surfaces.
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 |
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 |
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 |
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.