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In the wiki-page, the explicit parametric equation for a right circular cone with apex at the origin and aperture $2\theta$ is only given for a $3$-dimensional cone whose axis coincides with the $z$-axis.

What would be the analogous formula for an $n$-dimensional cone with axis parallel to a vector $d$?

References and/or derivations are both very welcome. Thanks.

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$\newcommand{\Reals}{\mathbf{R}}\newcommand{\dd}{\partial}\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Basis}{\mathbf{e}}\newcommand{\Brak}[1]{\left\langle #1\right\rangle}\newcommand{\Vec}[1]{\mathbf{#1}}$If $\Vec{x}$ denotes a point of the unit sphere in the hyperplane orthogonal to $d$, the cone with vertex at the origin, having aperture $2\theta$ and opening along the ray spanned by $d$, can be parametrized by $$ X(t, \Vec{x}) = t \frac{d}{\|d\|} + t\tan\theta\, \Vec{x}. \tag{1} $$ One way to turn this into a useful specification is to fix:

  1. A parametrization $(f_{1}, \dots, f_{n-1})$ of the unit sphere in $\Reals^{n-1}$. (Stereographic coordinates cover the whole sphere with two conformal charts; generalized spherical coordinates might be preferable.)

  2. An orthonormal basis $(\Basis_{j})_{j=1}^{n-1}$ for the hyperplane orthogonal to $d$.

Then write $$ \Vec{x} = f_{1} \Basis_{1} + \cdots + f_{n-1} \Basis_{n-1} $$ in equation (1).

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    Thank you (once again!). I'm a bit confused as to what $x$ is exactly. How would it look if $d$ consisted of $d_1,d_{1+(n+1)},d_{1+2(n+1)},\dots,d_n=\frac{-1}{\sqrt{n}}$ and $0$ otherwise? – 2017-01-02
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    Depending on $d$, $\Vec{x}$ could be something of a mess; that's why I'd pick an orthonormal basis for the hyperplane $H$ orthogonal to $d$, then use that basis to define a Cartesian coordinate system in $H$. Finding that basis may be a pain, but it's algorithmic using [Gram-Schmidt](https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process). (Not sure I understand the components of the $d$ you mention; my browser's math fonts may be a bit wonky, but it looks like $d_{1} = d_{1 + (n+1)} = d_{1 + 2(n+1)} = \dots = d_{n} = -1/\sqrt{n}$...?) – 2017-01-02
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    Separately, [Parametric Equations for a Hypercone](http://math.stackexchange.com/questions/13555) looks more-or-less equivalent, which makes me realize I neglected to divide $d$ by its magnitude in (1). – 2017-01-02
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    All right. Yeah, the notation is ugly (but you got it right) because I'm actually working with matrices in $\mathbb{R}^{m\times m}$ and there $d$ is a "unit matrix" $-1/\sqrt{n}I_n$, where $I_n$ is the identity matrix - I simply "reshaped" $d$ (as a matrix) to be an $m^2=n$-dimensional vector. And thank you for the reference! – 2017-01-02
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    Just realized that my notation of the entries of $d$ in vector form doesn't make any sense, sorry about that! – 2017-01-02
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    As it happens, it's not too difficult to write down $m^{2} - 1$ matrices forming an orthonormal basis of the traceless matrices (i.e., orthogonal to $-1\sqrt{m} I_{m}$): You only need Gram-Schmidt for the diagonal matrices; the others can be taken to be symmetric or skew-symmetric with precisely two non-zero entries (or simply to have a single $1$ and the remaining entries $0$). – 2017-01-02
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    Btw, in the answer you referenced, shouldn't $\phi_1$ run from $0$ to $2\pi$ and the rest run from $0$ to $\pi$, instead of the other way around? Also, the "central angle" in that question is half the aperture from this question, right? – 2017-01-02
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    Regarding angle ranges, I think you're right: There's an "initial longitude" that runs over a whole circle, and "successive dimensions are added" by a half-turn. (Modulo indexing, which I haven't carefully checked, Wikipedia agrees with you.) Also yes, $\theta = 2\alpha$ in the notation of the linked question. – 2017-01-02