fom
Posts:
1,969
Registered:
12/4/12


Re: Calculating the area of a closed 3D path or ring
Posted:
Mar 16, 2013 1:30 PM


On 3/16/2013 10:33 AM, Math Guy wrote: > fom used improper usenet message composition style by unnecessarily > fullquoting: > >> When I had been trying to visualize the stated problem, >> I assumed that a "ring" would be a convex star. > > I clearly stated in the first post that the ring resembled the perimeter > of a horse saddle, or a potato chip. > > How does either of those resemble a star? >
The sense of a star in this case is the idea that there is some point in the region which can form a line segment with any given point of the region different from itself such that every point of that line segment is a point of the region.
Of course, with curved differentiable surfaces, one would have to interpret the sense of my statement in terms of geodesics.
As for your description, you asked for numerical methods and suggested that the data available would be some finite number of points on the boundary.
Assuming a Euclidean interpretation of coordinates, the closest approximation to a ring you could have is a polygonal line closing upon itself. Hence corners.
Given the barycenter and triangular regions formed with each consecutive pair of points on the boundary, every point of the region different from the barycenter forms a line segment in the surface.
Sadly, my choice of words is often not as precise as I would like. But, it can usually be reasonably clarified. I hope I have done so.

