Is there an algorithm to generate a graph outline?

Question

I've searched the Internet and maybe I'm missing some correct keywords but I managed to find nothing like this. I only found the poly-lines (or just the lines) which are not exactly the graphs. I would like to generate a graph outline (of radius r) as seen in the picture. Is there something already available? I would like to avoid reinventing the wheel so to speak.

If anyone can at hint me at something or at least at some basic principle how to do it it would be great. Otherwise I'll "invent" one on my own of course.

Optimally in C#.

Update: I need to calculate outline polygon, not just visually draw it. The green points represents the resulting polygon. Also the "inner" holes are ignored completely. Only one outline polygon should be enough.

Update 2: Better picture to show some more extreme cases. Also the edges of graph never overlap so no need to accommodate for that.

Update 3: Picture updated yet again to reflect the bevel joins.

Answer 1

First, for every "line piece" from point A to B, generate the rectangle to it (all 4 points as "path", so to say). Then search two overlapping rectangles and merge them:

Merging is a bit complicated, the idea: Start with calculating the angle of all 8 lines (eg. if the rectangles are traversed clockwise). Then traverse one rectangle until the first line-line-intersection, check with the angles which direction is "outside", and move along the crossing line of the second rectangle ... until you arrive at the start point again => Now you traversed the shape of both together (and hopefully saved it somewhere).

Merge until only one large piece is left (or multiple non-overlapping pieces). In theory, starting from any point, you can traverse the whole shape, but there´s another problem: Holes are possible.
If one shape has two or more disjuct sets of points (where no point from set 2 is reachable from set 1 and vice-versa), all but one disjunct path is of a hole. An easy possibility to get the real outer border is to search for an extremum, ie. the point with the largest or smallest X or Y coordinate (only one of the 4 combinations in enough). This point surely is a part of the outer border.