{"id":800,"date":"2026-05-04T14:49:07","date_gmt":"2026-05-04T14:49:07","guid":{"rendered":"https:\/\/home.pf.jcu.cz\/~csgg2026\/?p=800"},"modified":"2026-05-04T14:51:33","modified_gmt":"2026-05-04T14:51:33","slug":"petrova-veta-a-harmonicka-analyza-mnohouhelniku","status":"publish","type":"post","link":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/cs\/petrova-veta-a-harmonicka-analyza-mnohouhelniku\/","title":{"rendered":"Petrova v\u011bta a harmonick\u00e1 anal\u00fdza mnoho\u00faheln\u00edk\u016f"},"content":{"rendered":"\n

prof. RNDr. Pavel Pech, CSc.<\/strong><\/p>\n\n\n\n

Jiho\u010desk\u00e1 univerzita v \u010cesk\u00fdch Bud\u011bjovic\u00edch<\/em><\/p>\n\n\n\n

Uplynulo v\u00edce ne\u017e 120 let od doby, kdy Karel Petr, profesor Univerzity Karlovy v Praze, formuloval v\u011btu, kter\u00e1 se dnes naz\u00fdv\u00e1 Petrova v\u011bta. Je proto vhodn\u00e1 p\u0159\u00edle\u017eitost p\u0159ipomenout n\u011bkter\u00e9 v\u00fdsledky s touto v\u011btou souvisej\u00edc\u00ed.<\/p>\n\n\n\n

Nejprve se zam\u011b\u0159\u00edme na rovinn\u00e9 uzav\u0159en\u00e9 mnoho\u00faheln\u00edky a podrob\u00edme je harmonick\u00e9 anal\u00fdze, tj. vyj\u00e1d\u0159\u00edme je ve form\u011b sou\u010dtu z\u00e1kladn\u00edch k-pravideln\u00fdch mnoho\u00faheln\u00edk\u016f. Tuto harmonickou anal\u00fdzu n\u00e1sledn\u011b vyu\u017eijeme ke studiu Petrovy v\u011bty.<\/p>\n\n\n\n

D\u00e1le se budeme zab\u00fdvat line\u00e1rn\u00edmi transformacemi rovinn\u00fdch mnoho\u00faheln\u00edk\u016f. Budeme zkoumat dva druhy transformac\u00ed, S-transformaci, kter\u00e1 je pops\u00e1na pomoc\u00ed komplexn\u00edch \u010d\u00edsel a zachov\u00e1v\u00e1 podobnost, a A-transformaci, kter\u00e1 je definov\u00e1na pomoc\u00ed re\u00e1ln\u00fdch \u010d\u00edsel a je invariantn\u00ed v\u016f\u010di libovoln\u00e9 afinit\u011b.<\/p>\n\n\n\n

Nakonec pou\u017eijeme A-transformace k zobecn\u011bn\u00ed line\u00e1rn\u00edch transformac\u00ed mnoho\u00faheln\u00edk\u016f do prostoru, co\u017e vede k prostorov\u00e9 analogii harmonick\u00e9 anal\u00fdzy mnoho\u00faheln\u00edk\u016f.<\/p>\n\n\n\n

<\/div>\n\n\n\n

Petr\u2019s theorem and harmonic analysis of polygons<\/strong><\/h2>\n\n\n\n

Over 120 years have elapsed since K. Petr, the professor of Charles\u2019s university in Prague, established a theorem which is called after him Petr\u2019s theorem. So, it is a good occasion to remind some results which are connected with this theorem.<\/p>\n\n\n\n

First, plane closed 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\u2019s theorem is studied. <\/p>\n\n\n\n

Then we will investigate plane linear polygon transformations. We will study two kinds of transformations \u2013 S<\/em>-transformation which is described by means of complex numbers and preserves similarity and A<\/em>-transformation which is defined by real numbers and is invariant under any affinity.<\/p>\n\n\n\n

In the next part, we use A<\/em>-transformations to generalization of linear polygon transformations into a space which leads to a space analog of harmonic analysis of polygons.<\/p>\n","protected":false},"excerpt":{"rendered":"

prof. RNDr. Pavel Pech, CSc. Jiho\u010desk\u00e1 univerzita v \u010cesk\u00fdch Bud\u011bjovic\u00edch Uplynulo v\u00edce ne\u017e 120 let od doby, kdy Karel Petr, profesor Univerzity Karlovy v Praze, formuloval v\u011btu, kter\u00e1 se dnes naz\u00fdv\u00e1 Petrova v\u011bta. Je proto vhodn\u00e1 p\u0159\u00edle\u017eitost p\u0159ipomenout n\u011bkter\u00e9 v\u00fdsledky s touto v\u011btou souvisej\u00edc\u00ed. Nejprve se zam\u011b\u0159\u00edme na rovinn\u00e9 uzav\u0159en\u00e9 mnoho\u00faheln\u00edky a podrob\u00edme je harmonick\u00e9 […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[],"class_list":["post-800","post","type-post","status-publish","format-standard","hentry","category-uncategorized-cs"],"_links":{"self":[{"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/posts\/800","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/comments?post=800"}],"version-history":[{"count":2,"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/posts\/800\/revisions"}],"predecessor-version":[{"id":804,"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/posts\/800\/revisions\/804"}],"wp:attachment":[{"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/media?parent=800"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/categories?post=800"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/tags?post=800"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}