I have a geometry preoblem obsession!?

Question

my mind always tries to calculate the shortest path from point A to B such that this par would consist of 1/4 of a circle and that they would form regular shapes for example http://www.geocities.com/hasha_2004/untitled.bmp
I mean really random stuff, like how long could the wall be after cutting through the A diagonal, please provide me with any theorems or advice because this thing has messed with my head long enough.

Answer 1

the best way to unite arcs to make the shortest path is to have the radius of the circle the smallest possible.
the sum of all of the arcs is directly proportional to the sum of the circumference's radius. (assuming a fixed arc of 90 degrees.)
If you will use other angles, the length of all the arcs shall be given by the some of the product of the diameters by the angles in radians.

Answer 2

In physics, things move according to the "principle of least action". For light rays, it's just the shortest distance between any two points on the ray. If you examine the figure given by your link, notice that the ray "reflects" from the curved surfaces exactly the way a ray of light would--at equal angles. On the other hand, if I have a collection of objects on the table, and I want to navigate through them from point A to point B, not bouncing from the objects as in your example, then, sorry, the solution tends to be much more complicated, and is in fact an NP-problem, similiar to the Travelling Salesman problem. There is no general mathematical solution for this sort of thing.

Answer 3

You might be the reincarnation of Archimedes. jk, It's normal for some ppl to think like that. In fact, who knows it might help some day.