but shortest distance and perpendicular are the same thing!
To an infinitely long line, yes, but in my previous wording there was the danger that someone might consider each edge of the polygon to be a finitely long line segment. The shortest distance to a line segment, if you're somewhere beyond one end of it, is nothing like the perpendicular distance to the containing line.
also... how is it irregular if convex and all the lengths are the same...
The angles are allowed to vary. For example, imagine a square of side 1, and an equilateral triangle of side 1. Join them together along one side, so that you get a pentagon looking a little like the end view of a house. This is convex, and all its side lengths are 1, but by no stretch of the imagination is it a regular pentagon.
It does work on irregular polygons, provided all their sides are the same length and they're convex. For example, it does work for the house-shaped pentagon I described above: for any point within the house shape, the sum of the perpendicular distances from all five sides is a constant.
... *thinks* i was thinking of the dist between the point and the base... not the sides oops. still not 100% convinced... need to get a piece of paper! is it the sum of ALL the sides then? that would help :)
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To an infinitely long line, yes, but in my previous wording there was the danger that someone might consider each edge of the polygon to be a finitely long line segment. The shortest distance to a line segment, if you're somewhere beyond one end of it, is nothing like the perpendicular distance to the containing line.
also... how is it irregular if convex and all the lengths are the same...
The angles are allowed to vary. For example, imagine a square of side 1, and an equilateral triangle of side 1. Join them together along one side, so that you get a pentagon looking a little like the end view of a house. This is convex, and all its side lengths are 1, but by no stretch of the imagination is it a regular pentagon.
i've forgotten the rule now...
it can't work on irregular ones can it?
or was that the point?....
not woken up yet...
oops.
still not 100% convinced... need to get a piece of paper!
is it the sum of ALL the sides then?
that would help :)