Determining angle and distance of points from single point source

Question

Suppose I have circle, with an arbitrary radius r. Additionally I also have a smaller set of random points within the circle, so they satisfy $x^2+y^2<r$ (Fig.A)

Is there a way to mathematically determine the angle and shortest distance to these points from one of the other points. In other words is there a function that maps an angle to the distance of the closest point from one point to a set of others. So in the case of Fig.B the function would be r for all angles except the angles of the blue lines, but would be different for Fig.C [stackexchange wont let me post images so here is a link: https://i.stack.imgur.com/Eimx4.png]

This is ultimately going to be used in a program, so it would be possible to iterate through the set of points but it is not ideal as the set could be incredibly large and it would also lead to redundantly checking points that lie on the same line. I could also iterate a line with an angle from $0$ to $2\pi$ using a small value, finding the shortest distance when it hits a point, but (I think) the value would have to be infinitely small to never miss any points. The former is probably the only option, but I was wondering if there was some mathematical magic that could reduce the amount of calculations needed.