Cone
Object type: Surface
Definition
In \mathbb{R}^3, a cone is a set of points (x,y,z) satisfying the equation z^2 = \left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 for some constants a, b > 0. Below the 'standard cone' z^2 = x^2 + y^2 is shown.
Each intersection with a plane z = c_0 \ne 0 yields an ellipse, and therefore the surface is also called an elliptic cone. If a = b, each such intersection is a circle, and the surface is called a circular cone. Intersection with planes containing the z axis yields a pair of lines through the origin; in particular, intersection with y = 0 yields the lines z = \pm x/a, ~ y = 0. In the case of a circular cone, the surface is the surface of revolution obtained as such a line is rotated about the z axis.
Parameterisation
The cone is the image \mathbf{r}(\mathbb{R}\times\left[0,2\pi\right[) where \mathbf{r} (u,v) = \underline{\mathbf{e}}\begin{pmatrix}au\cos v\\bu\sin v\\u\end{pmatrix}.
Properties
The following properties are with respect to the parameterisation given above.
Parameter-curve tangent vectors
The parameter-curve tangent vectors are \mathbf{r}_u(u,v) = \underline{\mathbf{e}}\begin {pmatrix}a\cos v\\b\sin v\\1\end{pmatrix}, \quad\quad \mathbf{r}_v(u,v) = \underline{\mathbf {e}}\begin{pmatrix}-au\sin v\\bu\cos v\\0\end{pmatrix}.
Standard unit normal
The standard unit normal vector field is \mathbf{\hat{N}}(u,v) = \frac{1}{\sqrt{b^2 u^2\cos^2 v + a^2 u^2 \sin^2 v + a^2 b^2 u^2 }}\underline{\mathbf{e}}\begin{pmatrix}-bu\cos v\\-au\sin v\\abu\end{pmatrix} which, in the case of a circular cone, reduces to \mathbf{\hat{N}}(u,v) = \frac{\text{sgn}u}{\sqrt{1+a^2}} \underline{\mathbf{e}}\begin{pmatrix}-\cos v\\-\sin v\\a \end{pmatrix}.
Area element
The area element is dA = \sqrt{b^2 u^2\cos^2 v+ a^2 u^2 \sin^2 v + a^2 b^2 u^2}~dudv which, in the case of a circular cone, reduces to dA = a|u|\sqrt{1+a^2}~dudv.
First fundamental form
The first fundamental form of the elliptic cone is \mathcal{F}(u,v) = \begin{pmatrix}a^2\cos^2 v +b^2\sin^2 v + 1&&(b^2-a^2)u\sin v\cos v\\(b^2-a^2)u\sin v\cos v&&a^2 u^2 \sin^2 v+b^2 u^2 \cos^2 v \end{pmatrix} which, in the case of a circular cone, reduces to \mathcal{F}(u,v) = \begin{pmatrix} a^2+1&&0\\0&&a^2 u^2\end{pmatrix}.
Second fundamental form
The second fundamental form is \mathcal{M}(u,v) = \frac{abu^2}{\sqrt{b^2 u^2\cos^2 v + a^2 u^2 \sin^2 v + a^2 b^2 u^2 }}\begin{pmatrix}0&&0\\0&&1\end{pmatrix}, which, in the case of a circular cone, reduces to \mathcal{M}(u,v) = \frac{a|u|}{\sqrt{1+a^2}}\begin{pmatrix}0&&0\\0&&1 \end{pmatrix}.
Christoffel symbols
In the case of a circular cone (a=b), the Christoffel symbols are \Gamma^1_{\alpha\beta} = \frac{-a^2u}{a^2+1}\begin{pmatrix}0&&0\\0&&1\end{pmatrix}, \quad\quad \Gamma^2_{\alpha\beta} = \frac {1}{u}\begin{pmatrix}0&&1\\1&&0\end{pmatrix}.
Curvatures
In the case of a circular cone (a=b), the principal curvatures are \kappa_1 = 0, \quad\quad \kappa_2 = \frac{1}{a|u|\sqrt{1+a^2}} with corresponding principal directions (1,0) and (0,1), respectively. Thus, the Gaussian and mean curvatures are K = 0, \quad\quad H = \frac{1}{a|u|\sqrt {1+a^2}}.