Finite Cone

Definition¹

Let $v$ be a unit vector² in $\mathbb{R}^n$. For each non-zero $x\in \mathbb{R}^n$, let $\angle(x,v)$ be the angle between two vectors $v,x$. Then, for given $v$, $\rho \gt 0$, $0 \lt \kappa \le \pi$, the set $C$ is called a finite cone of height $\rho$, axis direction $v$, and aperture angle $\kappa$ with the vertex at the origin. $$ C= \left\{ x \in \mathbb{R}^n \ \ \big| \ \ x=0\ \mathrm{or}\ 0<|x|\le \rho,\ \angle (x,v)\le \kappa/2 \right\} $$

$x+C=\left\{ x+y\ \ \big| \ \ y\in C \right\}$ is the translation of cone $C$, with the vertex changed from the origin to $x$.

Explanation

Although the cone is translated as a ‘cone’ in geometry, its actual 3D shape is not a cone. Hence, it’s more appropriate to read it as ‘cone’. In 2D, the cone becomes a sector.

Simply put, a cone is a collection of straight lines that extend from a certain point ($x$ in the diagram below) based on conditions of angle and size. Thinking of it as a collection of straight lines makes sense when considering the segment condition, weak cone condition, etc. It is used assuming that the given domain is not sufficiently good. For example, even if it’s impossible to determine an open ball centered at some point, a finite cone with that point as its vertex can exist. This is because a finite cone in $n$ dimensions is part of a $n$-dimensional ball.

The cone in 3D looks like the diagram above. It seems similar to a cone but is not. Specifically, it’s more like a cone ice cream. The diagram illustrates all vectors whose angle difference is less than $\kappa/2$ with vector $v$ as the base and size is less than $\rho$ based on $x$ as the viewpoint.

In 2D, it resembles a sector.