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Extending my earlier question about linear transformations, what's the easiest way to test if there exists a projective linear transformation that takes one polygon to another (in $\mathbb{R}\mathbb{P}^2$)?

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up vote 2 down vote accepted

The answer Tim Gowers and Will Jagy gave you in response to your earlier question can be extended straightforwardly. A homogeneous linear transformation on R^3 has 3x3 - 1 = 4x2 degrees of freedom, so there is a unique projective-linear transformation between any given pair of non-degenerate quadrilaterals. Thus do with quadrilaterals what was done in the previous question with triangles.

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A hacky way would be, enumerate the points of the two polygons (with adjacent points adjacent in the enumerations), as in $P_1 = [p_{1,1}, p_{1,2}, \ldots, p_{1,n}]$ and $P_2 = [p_{2,1}, \ldots, P_{2,n}]$, build N systems of linear equations, in the form $\forall i, \mathbf{T} p_{1,i} = p_{2,i+j}$ (indexed by $j$). There will be only N such systems, and if any one of them can be satistied (that is, it there is a $\mathbf{T}$ that takes all points $p_{1,i}$ into $p_{2,i+j}$ for some $j$) you have an answer.

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