prof. RNDr. Pavel Pech, CSc.
University of South Bohemia
Over 120 years have elapsed since K. Petr, the professor of Charles’s university in Prague, established a theorem which is called after him Petr’s theorem. So, it is a good occasion to remind some results which are connected with this theorem.
First, plane dosed polygons are harmonically analyzed, i.e. they are expressed in the form of the sum of fundamental k-regular polygons. Using this harmonic analysis the Petr’s theorem is studied.
Then we will investigate plane linear polygon transformations. We will study two kinds of transformations – S-transformation which is described by means of complex numbers and preserves similarity and A-transformation which is defined by real numbers and is invariant under any affinity.
In the next part, we use A-transformations to generalization of linear polygon transformations into a space which leads to a space analog of harmonic analysis of polygons.