In this paper, we introduce a variation of the well-studied Yao graphs. Given a set of points $Ssubset mathbb{R}^2$ and an angle $0 < theta leq 2pi$, we define the continuous Yao graph $cY(theta)$ with vertex set $S$ and angle $theta$ as follows. For
each $p,qin S$, we add an edge from $p$ to $q$ in $cY(theta)$ if there exists a cone with apex $p$ and aperture $theta$ such that $q$ is the closest point to $p$ inside this cone. We study the spanning ratio of $cY(theta)$ for different values of $theta$. Using a new algebraic technique, we show that $cY(theta)$ is a spanner when $theta leq 2pi /3$. We believe that this technique may be of independent interest. We also show that $cY(pi)$ is not a spanner, and that $cY(theta)$ may be disconnected for $theta > pi$.
In this paper, we show that the $theta$-graph with three cones is connected. We also provide an alternative proof of the connectivity of the Yao graph with three cones.
