Equation of a cone in $\mathbb{R}^n$ formed by taking all points within angle $\alpha$ of a line. What is the equation of a cone in $\mathbb{R}^n$ formed by taking all points within angle $\alpha$ of $e_1$ (the first standard normal vector)?
I would be interested in a parametric equation of its surface, but a closed equation would be ideal in order to compute certain integrals related to its volume.

The cone you described has equation $$x_1 = (\cot \alpha)\sqrt{x_2^2+x_3^2+\dots + x_n^2}$$
For double cone, the equation would be $$x_1^2 = (\cot^2\alpha)(x_2^2+x_3^2+\dots + x_n^2)$$
I like the disclaimer in the caption of a picture there: "A double cone (not shown infinitely extended)"
