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

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

I  wrote:

>
>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.
>
>.....
>
>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. 
 
To be a little more explicit, for the non-mathematicians who may not
be familiar with it, the standard mathematical definition of "convex"
is as follows:
 
" A set (i.e., any set of points) in a Euclidean space (of any
dimension) is convex if for any two points in the set the line segment
joining them lines entirely within the set".

This is the definition with consequence, the definition that leads to
all the nice properties of convex sets that we know and love.  The
concave down surface patches that Thatcher was discussing in the post
in question, considered as point sets,  cannot be understood to
satisfy this definition in any sense.  They have none of the nice
properties that are normally connoted by the term.