Hi Denis,
On 1/24/2011 3:41 PM, Denis Lila wrote:
Perhaps the problem is less with the word "boundary" than it
is with confusing our use of the word inside to describe the
concept of filling and containment with the topological concept
that a set has an interior in addition to (and mostly separate from)
its boundary?
That was exactly the problem. We classify every point as either inside
or outside, and I'm used to the interior, exterior, and boundary being
disjoint.
Right, my point was that the problem here isn't that we use the word
boundary, since when we use it we do acknowledge that there are points
"on" the boundary. The problem is that our definition of "inside" is
not similar to the topological definition of "interior".
Topology has interior, boundary, and exterior as disjoint sets.
Our definitions have inside and outside as disjoint sets, and we use the
word boundary only to describe how to determine which points are divided
into inside and outside - and I think our definition of boundary is
compatible with the topological concept. It's just that when we refer
to "inside" it may contain some points on the boundary, unlike the
topological "interior" which would not.
So, if you don't equate our "inside" term with topological "interior"
then there is no conflict with the fact that both fields use the term
boundary (and I think both use them compatibly).
Our "inside" is the topological "interior" unioned with the set of
points in/on the topological "boundary" that satisfy the "interior is
below or to the right" property.
Inside == Interior + ~half of Boundary
Outside == Exterior + ~half of Boundary
Interior+Boundary+Exterior == whole plane
No 2 of Interior, Boundary, or Exterior intersect
Inside+Outside == whole plane
Inside does not intersect Outside
Inside is a superset of Interior
Outside is a superset of Exterior
Both Inside and Outside intersect Boundary
...jim
