I want to know how thickness is defined.
Let me start with the simplest shape due to its symmetry. This is a circle. Assume a circle with radius of 1. If we draw a circle with the same center and radius 2, we will have created a ring with thickness 1.
How can we extend this to different shapes? For example let's take a rectangle with length of sides a=1 and b=2. I can think of 2 ways to create a ring of thickness 1.
- create a rectangle with length of sizes 2 and 3.
- create a shape like previous rectangle, but the corners would be rounded. That'a because the thickness in the corners would be $\sqrt2$, not 1.
So my definition of thickness something like
- The minimum distance between any point of the inner shape and any point of the outer shape should be 1.
- there should be no point (A) in inner shape where there is no point (B) in outer circle such as AB=1.
- there should be no point (A) in outer shape where there is no point (B) in inner circle such as AB=1.
Can this be correct? Assuming that above is correct:
If inner shape is circle $(x-a)^2+(y-b)^2=r^2$, then the outer shape should be $(x-a)^2+(y-b)^2=(r+1)^2$.
How about if the inner shape is an ellipsis? Or a square?
Best Answer
You will always get the set of all points outsude of the inner shape exactly distance 1 from the inner shape. For a circle or ellipse, you get anityer circle or ellipse with the radius or semiaxes increased in length by 1.
For a square, you add a rectangle of thickness 1 to each edge and then add quarter circles at the corner to fill in the gaps.
By the way, have you heard of extremal length of an annulus?https://en.m.wikipedia.org/wiki/Extremal_length
It is really cool. It assigns a number to each ring that tells how 'thick' it is using complex analysis. It is hard to compute but useful.