[right][size=50]Diese Aktivität ist eine Seite des [color=#980000]geogebra-books[/color] [url=https://www.geogebra.org/m/ggvbukux]"Loxodrome" ? Oder nicht ?[/url] [color=#ff7700](16/18.12.2019)[/color][/size][br][/right][size=85]In [url=https://de.wikipedia.org/wiki/Loxodrome]wikipedia[/url] werden neben den [color=#0000ff][i][b]Kugel-Loxodromen[/b][/i][/color] ("Winkelgleiche") auch andere Kurven dieses Typs erwähnt: [br] [math]\nearrow[/math] "Allgemeiner gibt es zu jedem [url=https://de.wikipedia.org/wiki/Rotationsk%C3%B6rper]Rotationskörper[/url] Loxodromen als Kurven konstanten Kurses."[br]Loxodromen findet man also auch auf Zylindern, Kegeln und eben auch auf einem [color=#20124D][i][b]Torus[/b][/i][/color] ![br][b]Walter Wunderlich[/b] hat 1952 in dem Artikel "[i]Über die Torusloxodromen[/i]" für diese Kurven Parameterdarstellungen angegeben, die wir oben im Applet verwenden. ([i]Monatsh. Math. 56 (1952) 313-334[/i])[br][b]W. Wunderlich[/b] bezieht sich u.a. auf einen Artikel von [b]Georg Holzmüller[/b] aus dem Jahr [b]1899[/b]: "[i]Elementares über die Dupin'schen Cykliden und die Grundlagen der Krümmungstheorie[/i]" (Zeitschrift d. Mathematik u. Physik 44.4 Heft 1899). Dort findet man übrigens ein schönes und genaues Bild über ein [color=#ff7700][i][b]6-Eck-Netz[/b][/i][/color] von [b]Villarceau[/b]-schen Kreisen auf einem [color=#20124D][i][b]Torus[/b][/i][/color] - erstellt ganz ohne Computer und moderner Software![br][table][tr][td] [i][b]Parameterdarstellung[/b][/i] eines [color=#20124D][i][b]Torus[/b][/i][/color]: [math]\mathbf{Torus}\left(u,v\right):=\left(\begin{matrix}\left(R+r\cdot\ \cos\left(u\right)\right)\cdot \cos\left(v\right)\\\left(R+r\cdot \cos\left(u\right)\right)\cdot \sin\left(v\right)\\r\cdot \sin\left(u\right)\end{matrix}\right)[/math];[/td] [td]R > r: [color=#20124D][i][b]Ringtorus[/b][/i][/color][br]R = r: [color=#20124D][i][b]Horntorus[/b][/i][/color][br]R < r: [color=#20124D][i][b]Spindeltorus[/b][/i][/color][/td][/tr][/table]In allen drei Fällen gibt es "[color=#0000ff][i][b]Loxodrome[/b][/i][/color]"![br][br][center]*********************************************************************************************************************************************[/center][size=100] [math]\rightharpoonup[/math] Was sind "[color=#0000ff][i][b]Loxodrome[/b][/i][/color]"?[/size][br][br] Für die [color=#0000ff][i][b]Standard-Loxodrome[/b][/i][/color] auf der [color=#980000][i][b]Kugel[/b][/i][/color] liegt ein [i][b]orthogonales Netz[/b][/i] von [color=#ff0000][i][b]Kreisen[/b][/i][/color] vor: [br] die [color=#1155Cc][i][b]Breitenkreise[/b][/i][/color] und die [color=#ff0000][i][b]Meridianen[/b][/i][/color] ([color=#ff0000][i][b]Längenkreise[/b][/i][/color]).[br][color=#20124D][i][b] Loxodrome[/b][/i][/color] sind diejenigen Kurven, welche die [color=#ff0000][i][b]Meridianen[/b][/i][/color], bzw. die [/size][size=85][i][b][color=#1e84cc][size=85][color=#1155Cc][i][b]Breitenkreise[/b][/i][/color][/size][/color][/b][/i] jeweils unter einem konstanten Winkel[br] schneiden.[br] Diese [u][b]Definition[/b][/u] läßt sich verallgemeinern: Liegt ein [i][b]orthogonales Netz[/b][/i] von Kurven vor, so kann man die Kurven, welche [br] die [i][b]Netz-Kurven [/b][/i]unter konstantem Winkel schneiden, [color=#0000ff][i][b]Loxodrome[/b][/i][/color] nennen. [br][center]**********************************************************************************************************************************************[/center]Das [i][b]orthogonale Netz [/b][/i]auf einem [color=#073763][i][b]Torus[/b][/i][/color] besteht aus den Kreisen, welche bei der Rotation der [color=#073763][i][b]Torus[/b][/i][/color]-Punkte um die Achse entstehen:[br]wir haben sie im Applet [color=#0B5394][i][b]Längskreise[/b][/i][/color] genannt, und die dazu orthogonalen [color=#ff0000][i][b]Meridiankreise[/b][/i][/color] ([color=#ff0000][i][b]Querkreise[/b][/i][/color]).[br][br][table][tr][td]Ein Kandidat für die gesuchen [color=#0000ff][i][b]Loxodromen[/b][/i][/color] könnten die [br]Kurven mit der nebenstehenden Parameterdarstellung sein:[/td][td][math]t\mapsto\mathbf{Loxo}\left(t\right):=\left(\begin{matrix}\left(R+r\cdot\ \cos\left(m\cdot t\right)\right)\cdot \cos\left(n\cdot t\right)\\\left(R+r\cdot \cos\left(m\cdot t\right)\right)\cdot \sin\left(n\cdot t\right)\\r\cdot \sin\left(m\cdot t\right)\end{matrix}\right)[/math] mit [math]t\in\mathbb{R}[/math] [/td][/tr][/table][br]Für ganzzahlige [math]m[/math] und [math]n[/math] entstehen geschlossene Kurven, welche sich [color=#999999][i][b]spiralig[/b][/i][/color] um den [color=#073763][i][b]Torus[/b][/i][/color] winden. Diese Kurven stimmen jedoch nicht mit den von [b]Walter Wunderlich[/b] angegebenen [color=#0000ff][i][b]Torusloxodromen[/b][/i][/color] überein. Insbesondere enthalten sie im Falle des [color=#073763][i][b]Ringtorus[/b][/i][/color] nicht die [b]Villarceau[/b]schen Kreise, welche bekannterweise [color=#0000ff][i][b]Loxodrome[/b][/i][/color] ("Winkelgleiche") des [color=#073763][i][b]Torus[/b][/i][/color] sind.[br][br]Im folgenden geben wir die [color=#B45F06][i]Parameterdarstellungen[/i][/color] der drei Kurventypen an. An einem Nachweis, dass es sich wirklich um die "[color=#0000ff][i][b]Winkelgleichen[/b][/i][/color]" handelt, versuchen wir uns noch![br][table][tr][td][color=#0000ff][i][b] [/b][/i][/color][size=150][color=#0000ff][i][b]Loxodromen[/b][/i][/color] [br] des [color=#073763][i][b][br] Ringtorus[br][br] [math]R>r[/math][/b][/i][/color][/size][/td][td] 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[/img][/td][td][/td][/tr][tr][td] mit [math]Rr=\sqrt{R^2-r^2}[/math] [/td][td][math]\mathbf{rLoxo}\left(\varphi\right):=\frac{Rr}{R-r\cdot \cos\left(b\cdot\varphi\right)}\cdot\left(Rr\cdot \cos\left(\varphi\right),Rr\cdot \sin\left(\varphi\right),r\cdot \sin\left(b\cdot\varphi\right)\right)[/math][/td][td]und [math]\varphi\in\left[-k\cdot\pi,k\cdot\pi\right][/math][/td][/tr][/table] Für [math]b=\frac{m}{n}[/math] mit ganzzahligen [math]m[/math] und [math]n[/math] ergeben sich geschlossene [color=#0000ff][i][b]Loxodrome[/b][/i][/color]; für [math]b=\pm1[/math] erhält man die [b]Villarceau[/b]schen Kreise, die es für die beiden anderen [color=#073763][i][b]Torus-[/b][/i][/color]Typen nicht gibt.[br][table][tr][td][size=85][size=150][color=#0000ff][i][b] Loxodromen[/b][/i][/color] [br] des [color=#073763][i][b][br] Spindeltorus[br] [br] [math]r>R[/math][/b][/i][/color][/size][/size][/td][td] [img]data:image/png;base64,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[/img][/td][td][/td][/tr][tr] [td] mit [math]rR=\sqrt{r^2-R^2}[/math][/td][td][math]\mathbf{sLoxo}\left(\varphi\right):=\frac{rR}{r\cdot \cosh\left(b\cdot\varphi\right)\pm R}\cdot\left(rR\cdot \cos\left(\varphi\right),rR\cdot \sin\left(\varphi\right),r\cdot \sinh\left(b\cdot\varphi\right)\right)[/math][/td][td]und [math]\varphi\in\left[-k\cdot\pi,k\cdot\pi\right][/math][/td][/tr][/table] Wieder erhält man für [math]b[/math] wie oben mit ganzzahligen [math]m[/math] und [math]n[/math] geschlossene [color=#0000ff][i][b]Loxodrome[/b][/i][/color], die allerdings nicht immer angezeigt werden. Das Vorzeichen im Nenner entscheidet, welcher Teil der Kurve im Inneren, bzw. im Äußeren angezeigt wird.[br][br][table][tr][td][size=85][size=85][size=150][color=#0000ff][i][b] Loxodromen[/b][/i][/color] [br] des [color=#073763][i][b][br] Horntorus[br] [math]r=R[/math][/b][/i][/color][/size][/size][/size][/td][td][math]\mathbf{hLoxo}\left(\varphi\right):=\frac{2\cdot R}{1+b^2\cdot \varphi^2}\cdot\left( \cos\left(\varphi\right), \sin\left(\varphi\right),b\cdot\varphi\right)[/math][/td][td] mit [math]\varphi\in\left[-k\cdot\pi,k\cdot\pi\right][/math][/td][/tr][/table][math]b,m,n[/math] wie oben; wieder werden in einigen Fällen die Kurven nicht angezeigt (?!?)[br]Für die beiden letzten [i][b]Torus[/b][/i]-typen haben wir die [color=#1e84cc][i][b]Längs[/b][/i][/color]- und [color=#ff0000][i][b]Querkreise[/b][/i][/color] für die Schnittpunkte nicht berechnet, der Rechenaufwand ist insgesamt für die Parameterkurven sehr hoch![br][br]Der Grenzfall [math]R\rightarrow0[/math] führt übrigens zu den [color=#0000ff][i][b]Standard-Loxodromen[/b][/i][/color] auf der [color=#980000][i][b]Kugel[/b][/i][/color], das kann man im Applet gut erkennen![/size][size=85][br][table][tr][td][color=#38761D][i][b]Bemerkungen:[/b][/i][/color] [br] - In dem Faktor [math]b[/math] steckt der Schnittwinkel der [color=#0000ff][i][b]Loxodromen[/b][/i][/color] mit den [color=#ff0000][i][b]Meridianen[/b][/i][/color].[br] - [b]G. Holzmüller[/b] erwähnt, dass es "bekanntlich" eine "einfache" [color=#ff7700][i][b]konforme Abbildung[/b][/i][/color] eines [color=#B45F06][i][b]Rechtecks[/b][/i][/color] auf den [color=#0000ff][i][b]Torus[/b][/i][/color] gäbe. [br]Damit ist wohl gemeint, dass eine [color=#38761D][i][b]winkeltreue[/b][/i][/color] Abbildung eines Rechtecks auf den [size=85][size=85][color=#0000ff][i][b]Torus[/b][/i][/color][/size] existiert, für welche die achsenparallelen Srecken die [/size][size=85][size=85][color=#1e84cc][i][b]Längs[/b][/i][/color]- und [color=#ff0000][i][b]Querkreise[/b][/i][/color][/size], und die schrägen Parallel-Strecken die [/size][size=85][size=85][color=#0000ff][i][b]Loxodrome[/b][/i][/color][/size] ergeben.[br]Diese Abbildung haben wir nicht gefunden, das im Falle der [color=#0B5394][i][b]Ring-Tori [/b][/i][/color]angezeigte [color=#38761D][i][b]Gitter[/b][/i][/color] könnte ein Urbild für diese Abbildung sein! Wie [b]Holzmüller[/b] angibt, muss nur der Winkel der [color=#f6b26b][i][b]Diagonalen[/b][/i][/color] stimmen!?! (Hierzu jedoch die Seite [math]\hookrightarrow[/math] [url=https://www.geogebra.org/m/ggvbukux#material/rjp5dfcn]Torus mit[/url] [math]\sqrt{2}[/math] [url=https://www.geogebra.org/m/ggvbukux#material/rjp5dfcn]wickeln[/url].)[br] - [b]W. Wunderlich[/b] verdeutlicht in einem Bild, dass die Projektion der [color=#0000ff][i][b]Loxodromen[/b][/i][/color] vom Ursprung aus auf einen ko-achsialen [color=#9900ff][i][b]Zylinder[/b][/i][/color] einfache [i][b]Sinus-Linien[/b][/i] ergibt.[br] [math]\hookrightarrow[/math] Eine[url=https://www.geogebra.org/m/ggvbukux#material/sy2kxrgf] hübsche Aufgabe[/url] zur Veranschaulichung mit [color=#980000][i][b]ge[/b][/i][/color][icon]/images/ggb/toolbar/mode_cylinder.png[/icon][color=#980000][i][b]gebra[/b][/i][/color]-Hilfsmitteln![/size][/td][td][img]data:image/png;base64,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[/td][/tr][/table][/size][size=85] - [color=#0000ff][i][b]Loxodrome[/b][/i][/color] dürfte es auf allen hinreichend glatten Rotationskörpern geben: zB. auf rotationssymmetrischen [color=#0B5394][i][b]Zylindern[/b][/i][/color], [color=#0B5394][i][b]Paraboloiden[/b][/i][/color], [color=#0B5394][i][b]Hyperboloiden[/b][/i][/color], [color=#0B5394][i][b]Ellipsoiden[/b][/i][/color] etc. [br]Allerdings kommt es hier wirklich auf die [i][b]Definition[/b][/i] dieses Kurventyps an: wenn [color=#0000ff][i][b]loxodromisch[/b][/i][/color] "[i][b]winkelgleich[/b][/i]" meint, so ist die [color=#ff0000][i][b]Metri[/b][/i][i][b]k[/b][/i][/color], vor allem die [color=#ff0000][i][b]Winkelmessung[/b][/i][/color] bestimmend. Legt man die Metrik des umliegenden [color=#BF9000][i][b]euklidischen Raumes[/b][/i][/color] zugrunde, [br]so ändern sich die Winkel durch Streckung in Richtung der Rotationsachse, die [color=#0000ff][i][b]Loxodrome[/b][/i][/color] können also nicht einfach durch diese Streckung als "[i][b]Winkelgleiche[/b][/i]" mitbewegt werden![br][br][i][b]Allgemein[/b][/i]: winkeltreue Abbildungen sind die [color=#0000ff][i][b]konformen[/b][/i][/color] Transformationen. Im Raum sind das zB. die räumlichen [color=#0000ff][i][b]Möbiustransformationen[/b][/i][/color]: diese sind kreis- und winkeltreu![/size]