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Given two general position point sets in $\mathbb{R}^2$ of the same size and order type, is it possible to continuously move the points of one set until they coincide with those of the other set in such a way that order type is preserved (equivalently, no three points become collinear) throughout the operation?

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It would be nice to have a definition of "order type" here in this question. – Jeff Strom May 29 '12 at 15:01
According to the paper that Nima Hoda links to, the order type of a (finite) set $P$ of points in general position (no 3 are collinear) is a mapping that assigns to each triple $(p,q,r)$ of distinct points in $P$ the orientation of this triple (clockwise, or counterclockwise). (It should be false for infinite sets.) – Goldstern May 29 '12 at 16:26
Yes, Jeff, I should have included the definition for convenience. Thanks for filling that in, Goldstern. – Nima Hoda May 29 '12 at 20:31
up vote 6 down vote accepted

I believe the answer is (in general) No for the following reason. A configuration of points in $\mathbb{P}^2$ is projectively dual to an arrangement of lines in $\mathbb{P}^2$. The question you ask, translated to arrangements of lines, is whether an arrangement can always be continuously moved to any isomorphic arrangement, all the while remaining isomorphic. This was asked by Ringle in 1956 (the isotopy conjecture), and answered negatively by Mnev's Universality Theorem in 1985: not only can the space of line arrangments isomorphic to a given arrangement be disconnected, it can have the homotopy type of any algebraic variety.

Of course it is not difficult to change from $\mathbb{P}^2$ to $\mathbb{R}^2$. I find this explicit claim in a 1988 paper by Suvorov (Springer link):
Here, "nondegenerated" means, I believe, what we would call today "simple": no three lines share a point. A "rigid" isotopy keeps the lines straight (in a "flexible" isotopy, the objects are pseuodolines).

Suvorov's example has 14 lines in $\mathbb{P}^2$. Later (1996) Richter-Gebert found another 14-line/point example, presented in his paper "Two Interesting Oriented Matroids" (CiteSeer link):
      Jurgen R-G Fig 1
He shows that

[its] realization space [...] is an open interval from which one point has been deleted.

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Thanks a lot for the answer! – Nima Hoda May 29 '12 at 21:46

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