Projective transformation between polygons. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T23:40:50Z http://mathoverflow.net/feeds/question/33432 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/33432/projective-transformation-between-polygons Projective transformation between polygons. Adeel 2010-07-26T19:21:30Z 2010-07-27T01:36:40Z <p>Extending my <a href="http://mathoverflow.net/questions/33303/linear-transformation-takes-a-polygon-to-another-one" rel="nofollow">earlier question about linear transformations</a>, what's the easiest way to test if there exists a projective linear transformation that takes one polygon to another (in $\mathbb{R}\mathbb{P}^2$)?</p> http://mathoverflow.net/questions/33432/projective-transformation-between-polygons/33450#33450 Answer by Alexandre Passos for Projective transformation between polygons. Alexandre Passos 2010-07-26T22:25:07Z 2010-07-26T22:25:07Z <p>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.</p> http://mathoverflow.net/questions/33432/projective-transformation-between-polygons/33459#33459 Answer by Per Vognsen for Projective transformation between polygons. Per Vognsen 2010-07-27T01:36:40Z 2010-07-27T01:36:40Z <p>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.</p>