3
$\begingroup$

This post continues 'Constrained morphing' of planar convex regions

If an $m$-gon $P_m$ is to be morphed (altered continuously) into an $n$-gon $P_n$ with same area and perimeter, can one maintain the area and perimeter constant throughout the morphing? What if a further constraint is applied: if m and n are different, the morphing should cause the number of edges to change monotonically? Further, if the two polygons and all intermediate ones need to be convex, what happens?

Note: As far as I know, two triangles with same area and perimeter can be morphed into one another, keeping area and perimeter constant throughout. I am not sure about two convex quadrilaterals with equal area and perimeter - the guess is that the morphing can happen via the rectangle with same area and perimeter as the two quads.

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ Whenever $4\pi A\le P^2$, there is a canonical figure with area $A$ and perimeter $P$, namely a circle cut off by a secant. So if the edges can vary arbitrarily, it would be enough to morph any shape into the canonical shape of its perimeter and area. If the edges can’t vary arbitrarily, there is still a similarly canonical $n$-gon with those characteristics, that can similarly be used to simplify the problem. But actually writing out the details seems tedious. $\endgroup$
    – user44143
    Commented Jan 1, 2023 at 2:57
  • $\begingroup$ Guess what you mean is that the 'canonical figure' of same area and perimeter can be used as an intermediate stage in the morphing of one planar convex region into another of same area and perimeter. That might need smoothening out the polygon followed by a roughing up with number of edges growing infinitely in between. And as you too seem to suggest, I don't readily see if the transition to and from the canonical figure can be done keeping area and perimeter equal throughout. $\endgroup$
    – Nandakumar R
    Commented Jan 1, 2023 at 8:04
  • $\begingroup$ It is known that any two parallel orthogonal polygons have a morph (Spriggs, Michael. "Morphing parallel graph drawings." Ph.D. thesis, Univ. Waterloo (2007).) It would be quite interesting if they could morph while retaining area constant. $\endgroup$
    – Joseph O'Rourke
    Commented Jan 1, 2023 at 14:32
  • $\begingroup$ It appears that two rectilinear polygons with same area can be morphed into each other with area maintained constant throughout - the process goes via a rectangle of the same area and the number of edges keeps altering. The same approach seems to work for morphing two rectilinear polygons of same perimeter into each other. A limitation of the method is that simultaneously 2 unconnected portions of the rectilinear polygon will need to be altered. So insisting that either the entire polygon or a connected portion thereof needs to be continuously altered could be a more difficult challenge. $\endgroup$
    – Nandakumar R
    Commented Jan 12, 2023 at 7:12

1 Answer 1

Reset to default
1
$\begingroup$

I am going to answer a question you did not ask, but is in the same intellectual neighborhood: morphing with constant perimeter.

Q. Given a polygon $P$ of $n$ edges, not necessarily convex, can it be continuously morphed to a convex $n$-gon $C$, such that (a) all intermediate shapes are simple polygons (not self-crossing), and (b) all edge lengths remain the same throughout (and so the perimeter is constant)?

The answer is Yes, as described in this paper:

Jason Cantarella, Erik D. Demaine, Hayley Iben, and James O'Brien, “An Energy-Driven Approach to Linkage Unfolding”, in Proceedings of the 20th Annual ACM Symposium on Computational Geometry (SoCG 2004), Brooklyn, New York, June 9–11, 2004, pages 134–143. PDF download.

Energy

Fig. 3: To maximize visibility, the animation zooms as time proceeds; in fact, all edge lengths remain constant.

Clearly area is not preserved! But perimeter and $n$ and simplicity is preserved.

Improve this answer
$\endgroup$
1
  • 1
    $\begingroup$ Thanks for pointing this out - a different aspect of the problem; one that I was unaware of. This morphing was conceived as a generalization from the study of linkages - in the latter, the links are rigid and of fixed length whereas in the morphing I was thinking about, one can change the lengths (and sometimes, the number) of links freely. $\endgroup$
    – Nandakumar R
    Commented Jan 1, 2023 at 8:07

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .