Suppose I have two points, Point1 and Point2. At any given time, these points may be at different positions-- they are not necessarily static.

Point1 is located at some position at time t, and its position is defined by the continuous functions x1(t) and y1(t) giving the x and y coordinates at time t. These functions are not differentiable, they are constructed piecewise from line segments.

Point2 is the same, with x2(t) and y2(t), each function having the same properties.

The obstacles that might prevent visibility are simple (and immobile) polygons.

How can I find the boundary points for visibility?

i.e. there are two kinds of boundaries: where the points become visible, and become invisible.

For a become-visible boundary i, there exists some ?>0, such that for any real number a, a ? (i-?, i) , Point1 and Point2 are not visible (i.e. the line segment that connects (x1(a), y1(a)) to (x2(a), y2(x)) crosses some obstacles).

For b ? (i, i+?) they are visible.

And it is the other way around for becomes-invisible.

But can I find such a precise boundary, and if so, how?