[metapost] envelope proposal 4

[Larry Siebenmann]

Hi Jacko,

I have been talking too fast about envellopes, sliding over
several difficulties.  Here are a couple of 
of my sins.


Sin (1).

<< On the other hand the (possibly disconnected) 
"perimeter" ("frontier") of the inked region does vary 
continuously with the path, and that is all that meets 
the eye of the reader. >>

I should have said "semicontinuously" as a subset of the
plane. Only the area of the inked  region clearly varies
continuously in the ordinary sense. Also one should forget
parametrization here. "Semicontinuous" means roughly that
although things can *vanish* instantaneously, they cannot
get big instantanously. Consider the inner envellope of a
varying rectangle for a fixed pen. It can disaappear
instanty, in case at a given path deformation parameter it is a
mere segment.  


Sin (2).

<< ...define continous envelope paths for proposal 4 in all
cases SO AS TO VARY CONTINUOUSLY AS THE PATH CONTROL POINTS
VARY CONTINUOUSLY. >>

This notion of continuous variation only makes sense if 
we reparametrize proportional to arc length, keeping the 
same interval of definition.  Then it seems to work 
except for a finite list of exceptional phenomena such 
as cusps. Inflexions that run parallel to a polygonal 
pen edge are another exceptional phenomenon.

Fortunately, most of the horrors one meets are ultimately
drowned in ink and forgotten.  Still, I reckon that the
polygonal pen business was not an easy wicket for those who
first worked it out.  There is surely a literature on it;
has anyone seen a good reference? 

I am not keen to get seriously involved with polygonal
pens...

LS