How could I check to see if a point is inside a cone projected by another point?

Question

I use CesiumJS, a javascript library, to render entities in a 3-dimensional space. I need to detect if one entity is "viewing" another entity. My initial thought is to use a cone to represent the field of view of the entity. I need to be able to make the cone variable angle and height. What is the most efficient way to check if Entity 2 is inside the cone emitted by Entity 1? I am doing this calculation a bunch of times so I am trying to find a way to do it efficiently.

In Cesium, my entities are plotted using Lat/long/alt, which I figured I could think of as X,Y,Z. I also know the orientation of each point, so I know which way to make the cone emit. Cesium has some built-in functions for checking intersections but all seem to use rays or lines emitting from entity 1, which don't give me my desired output. Any help would be greatly appreciated.

Answer 1

Let's assume that your point where you check the viewing is $\vec P_0(x_0,y_0,z_0)$, you are looking in a direction $\vec d(d_x,d_y,d_z)$ in a cone with opening (angle between side and axis) $\alpha$, and you want to check if $\vec P_1(x_1,y_1,z_1)$ is inside the cone. Then you want the angle between the vector $\vec P_1-\vec P_0$ and $\vec d$ to be less than $\alpha$. You can write this using scalar product: $$\cos\left(\angle(\vec P_1-\vec P_0, \vec d)\right)=\frac{(\vec P_1-\vec P_0)\cdot \vec d}{|\vec P_1-\vec P_0||\vec d|}\\=\frac{(x_1-x_0)d_x+(y_1-y_0)d_y+(z_1-z_0)dz}{\sqrt{(x_1-x_0)^2+(y_1-y_0)^2+(z_1-z_0)^2}\sqrt{dx^2+dy^2+d_z^2}}$$ If the viewing angle is smaller than $\alpha$, then the last expression must be greater than $\cos\alpha$.