This began as an idea to make a 3-Dimensional tunnel. The properties I wanted were as follows.
- The cross-section of the tunnel to be circular.
- The tunnel should be a set of circles, rather than a continuous surface.
- The Normals emerging from the circles’ centers should be parallel to the tunnel’s tangent (this tangent is along the direction of propagation) at every given point.
Firstly I have done trials in 2-Dimensions. The diagrams are presented below.
You can observe that the in all the above figures, the ellipses orient themselves tangentially or normally to the curve depending on their location.
It was a big leap from 2-D to 3-D. It wasn’t at all straightforward as it may seem. It required calculations of tangents’ orientations at various points along the tunnel and imparting that to the rotation of the circles. So for every circle there will be three rotation angles.
In summary, the program should be capable of plotting a 3-D curve and placing circles along the curve with their normals parallel to the tangent of the curve at that point.
I consider this one of the most difficult of the tasks I have undertaken. This took the longest time from the idea to result. I have also learnt a lot in the process and when the program worked I was exhilarated.
Since the method has been found, the same process can be used for any 3-D or 2-D curve that is differentiable everywhere along its length.