How To Check From Certain Points A Length For Contact That Doesn't Intersect & Pass ?

[calculatingexistence]

The above is a visual demonstration of the first few steps I want to do.

To explain, the first step is the radius or length being created.
The second step is the length searching in all 2 dimensional directions from the two points (the left point and the right point of the length) for where it makes contact with another length's end.
The third step is recognizing those points.
The fourth step is searching in all 2 dimensional directions from the two points (the top and right point of the double triangle) for where it makes contact with another length's end.
The fifth step is recognizing those points.
The sixth is searching in all 2 dimensional directions from those FOUR points for where it makes contact with another length's end.

Mathematically, how do I make a visual program in java that does this and continues the simulation? Any help would be greatly appreciated.

For more info on why I want to do this, click http://calculatingexistence.webs.com...0existence.pdf

I guess the code would look something like this;

//frame 1
Create line segment 1 of length 1.
//frame 2
From each of line segment 1's end points (location of endpoints recorded) create a circle (circle number recorded) of radius length 1.
//frame 3
Analyze where circles (that were just recorded) intersect and make lines from line segment 1's end points to the points where circles (that were just recorded) intersect (record location of new endpoints).
//frame 4
At points (that were just recorded) create circles (circle number recorded) of radius length 1.
//frame 5
Analyze where circles (that were just recorded) intersect and make lines from (record of new endpoints) to the points where circles (that were just recorded) intersect (record location of new endpoints).
//frame 6
At points (that were just recorded) create circles (circle number recorded) of radius length 1.