{"id":842,"date":"2026-06-23T16:58:35","date_gmt":"2026-06-23T16:58:35","guid":{"rendered":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/ohniskove-konchoidy-kuzelosecek-focal-conchoids-of-conic-sections\/"},"modified":"2026-06-23T17:14:12","modified_gmt":"2026-06-23T17:14:12","slug":"ohniskove-konchoidy-kuzelosecek-focal-conchoids-of-conic-sections","status":"publish","type":"post","link":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/en\/ohniskove-konchoidy-kuzelosecek-focal-conchoids-of-conic-sections\/","title":{"rendered":"Ohniskov\u00e9 konchoidy ku\u017eelose\u010dek \u2013 Focal conchoids of conic sections"},"content":{"rendered":"

Roman Ha\u0161ek<\/em>
\nJiho\u010desk\u00e1 univerzita v \u010cesk\u00fdch Bud\u011bjovic\u00edch, Pedagogick\u00e1 fakulta, \u010cesk\u00e9 Bud\u011bjovice<\/small><\/p>\n

(English version follows.)<\/em><\/p>\n

Je tomu 10 let, co byla na 17. mezin\u00e1rodn\u00ed konferenci o geometrii a grafice v Pekingu a n\u00e1sledn\u011b na 2. \u010cesko-slovensk\u00e9 konferenci o geometrii a grafice v Ro\u017enov\u011b pod Radho\u0161t\u011bm p\u0159edstavena precl\u00edkov\u00e1 k\u0159ivka, angl. pretzel curve. Vyjevila se tehdy p\u0159i \u0159e\u0161en\u00ed probl\u00e9mu \u010d. 35 ze sb\u00edrky Exercitationes Geometricae (1773) od pod\u011bbradsk\u00e9ho rod\u00e1ka Jana Holfelda. O t\u00e9to pozoruhodn\u00e9 k\u0159ivce bylo mimo jin\u00e9 prok\u00e1z\u00e1no, \u017ee je konchoidou paraboly s p\u00f3lem um\u00edst\u011bn\u00fdm v jej\u00edm ohnisku. Pat\u0159\u00ed proto mezi tak zvan\u00e9 ohniskov\u00e9 konchoidy ku\u017eelose\u010dek, kter\u00e9 zm\u00ednil nap\u0159\u00edklad E. H. Lockwood v jednom kr\u00e1tk\u00e9m odstavci sv\u00e9 knihy A book of curves (1961). Ani\u017e by tak s\u00e1m u\u010dinil, vyzval \u010dten\u00e1\u0159e k zobrazen\u00ed t\u011bchto k\u0159ivek pro konkr\u00e9tn\u00ed nastaven\u00ed hodnoty jejich parametru fixn\u00ed vzd\u00e1lenosti vzhledem k hodnot\u00e1m ur\u010duj\u00edc\u00edch prvk\u016f ku\u017eelose\u010dky. Jako ohniskov\u00e1 konchoida paraboly s parametrem rovn\u00fdm jej\u00edmu latus rectum se pak zobraz\u00ed v\u00fd\u0161e zm\u00edn\u011bn\u00e1 precl\u00edkov\u00e1 k\u0159ivka. Zasazen\u00edm do rodiny p\u0159\u00edbuzn\u00fdch k\u0159ivek se tak uzav\u00edr\u00e1 cesta jej\u00edho zkoum\u00e1n\u00ed.
\nP\u0159\u00edsp\u011bvek je symbolickou te\u010dkou za t\u00edmto procesem. Vych\u00e1z\u00ed z Lockwoodova odstavce, aby formou dynamick\u00fdch GeoGebra applet\u016f uspo\u0159\u00e1dan\u00fdch do GeoGebra knihy Focal conchoids of conics, umo\u017enil \u010dten\u00e1\u0159i naplnit Lockwoodovu v\u00fdzvu, a v\u0161echny pozoruhodn\u00e9 p\u0159\u00edpady ohniskov\u00fdch ku\u017eelose\u010dek, jak pro parabolu a elipsu, tak i pro hyperbolu, n\u00e1le\u017eit\u011b zobrazit. Krom\u011b samotn\u00e9 GeoGebra knihy budou stru\u010dn\u011b p\u0159edstaveny geometrick\u00e9 z\u00e1klady jejich tvorby, svou n\u00e1ro\u010dnost\u00ed nep\u0159ekra\u010duj\u00edc\u00ed hranice kurikula st\u0159edo\u0161kolsk\u00e9 matematiky.<\/p>\n

Focal conchoids of conic sections<\/strong><\/p>\n

It has been ten years since the pretzel curve was first introduced at the 17th International Conference on Geometry and Graphics in Beijing and subsequently at the 2nd Czech\u2013Slovak Conference on Geometry and Graphics in Ro\u017enov pod Radho\u0161t\u011bm. The curve emerged during the solution of Problem No. 35 from Exercitationes Geometricae (1773) by the Pod\u011bbrady-born mathematician Jan Holfeld. Among its remarkable properties, it was shown to be the conchoid of a parabola with the pole located at the parabola's focus. It therefore belongs to the family of so-called focal conchoids of conics, briefly mentioned by E. H. Lockwood in A Book of Curves (1961). Although he did not himself draw these curves, Lockwood invited readers to do so for particular values of the constant-distance parameter relative to the defining parameters of the conic. When this parameter is chosen to be equal to the parabola's latus rectum, the resulting focal conchoid is precisely the above-mentioned pretzel curve. Identifying the pretzel curve as a member of the broader family of focal conchoids thus completes the investigation of its geometric nature.
\nThe present contribution serves as a symbolic conclusion to this line of research. Taking Lockwood's brief discussion as its point of departure, it presents a collection of dynamic GeoGebra applets organized into the GeoGebra Book Focal Conchoids of Conics. These interactive constructions enable readers to take up Lockwood's invitation and draw all the remarkable focal conchoids of the parabola, ellipse, and hyperbola. In addition to the GeoGebra Book itself, the paper briefly outlines the geometric principles underlying the construction of these applets, using only methods that remain within the scope of the secondary-school mathematics curriculum.<\/p>\n","protected":false},"excerpt":{"rendered":"

Roman Ha\u0161ek Jiho\u010desk\u00e1 univerzita v \u010cesk\u00fdch Bud\u011bjovic\u00edch, Pedagogick\u00e1 fakulta, \u010cesk\u00e9 Bud\u011bjovice (English version follows.) Je tomu 10 let, co byla na 17. mezin\u00e1rodn\u00ed konferenci o geometrii a grafice v Pekingu a n\u00e1sledn\u011b na 2. \u010cesko-slovensk\u00e9 konferenci o geometrii a grafice v Ro\u017enov\u011b pod Radho\u0161t\u011bm p\u0159edstavena precl\u00edkov\u00e1 k\u0159ivka, angl. pretzel curve. Vyjevila se tehdy p\u0159i \u0159e\u0161en\u00ed […]<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-842","post","type-post","status-publish","format-standard","hentry","category-presentations"],"_links":{"self":[{"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/posts\/842","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\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/comments?post=842"}],"version-history":[{"count":3,"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/posts\/842\/revisions"}],"predecessor-version":[{"id":846,"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/posts\/842\/revisions\/846"}],"wp:attachment":[{"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/media?parent=842"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/categories?post=842"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/home.pf.jcu.cz\/~csgg2026\/index.php\/wp-json\/wp\/v2\/tags?post=842"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}