Re: [Algorithms] Convexity [was Tangential Curvature of terrain]

[Ron L. <ro...@do...>]

Thatcher Ulrich wrote:

> From: Ron Levine <ro...@do...>
>>....
> > See your calculus book for the notion of "concave up" vs "concave
> > down" as applied to the graph of a function.  I would take issue with
> > Ulrich's use of the term "convex" where I use "concave down", because
> > it corresponds poorly with other important connotations in the usual
> > meaning of the term "convex".
>
> Fair enough, I abused the calculus terms.  In the context of a heightfield
> for GIS though, "concave up" and "concave down" are less intuitive than
> "concave" and "convex" in the geometric sense, since we're talking about a
> solid (the earth) with a well defined inside and outside.
>

Yes,  but that solid earth is not a convex body, not if it has any valleys.
True, you can slice off a solid convex cap under a region whose outer
surface is concave down, but without specifying that solid cap in its
entirety it is meaningless to use the term "convex".  If  it is not
meaningless to you, then you are certainly watering down a very powerful and
important term.

The point is that, in 3D,  convexity is a property of solids, not of
surfaces, certainly not a property of surface patches (except for convex
regions of PLANAR subspaces).  And it is a global property, not a local
property.  And it is an extremely important global property--many powerful
theorems or algorithms are true for convex solids in space (or convex
regions of a plane) but not true of non-convex bodies (*).  On the other
hand, the properties of "concave up"  and "concave down"  are local
properties, rather than global properties; they are properties of surface
patches, not of solids.

So by using "convex" in the way that you did, as a local property of surface
patches, you are muddying or watering down a very important nomenclature, a
nomenclature with powerful connotations, connotations which do not hold at
all for the sense in which you are using the term.  It is that abuse of an
important term with which I take issue, no matter how you may justify it to
yourself.

The words you use are important.  Mathematics is nothing more nor less than
the discipline of using language with precision. Elegant and useful
mathematics requires choosing definitions to have consequence.  Your use of
"convex" violates that principle.

(*) Just a couple of examples from collision detection:  GJK applies to all
convex solids, but to no non convex solids.    The Separating Axis Theorem
applies to all convex polyhedra, but to no non convex polyhedra.  There are
many many more examples in both pure and applied geometry.