Rotation R³-Achse als elementare x,y,z-Achsendrehungen

Rotation ∢φ um beliebige Achse
App[br]Die Winkelauswahl der Achsrotationen (9),(15) muß ggf. für andere Rotations-Achsen angepasst werden, um die richtige Richtung zur zAchse zu treffen! [br][list][*](12) y-Koordinate = 0[/*][*](18) x,y-Koordinaten = 0[/*][/list]Ich verwende homogene Koordinaten um die Translation in den Ursprung als Matrix schreiben zu können.[br][br]Voraussetzung: [br]Die Rotationsachse stimme nicht mit einer Koordinatenachse überein.[br][br]Idee: [br]Transformiere Rotationsachse und Objekt damit die Rotationsachse mit der z-Achse ̈uberein-[br]stimmt, rotiere um Rotations-Winkel φ, transformiere zurück.[br][br]1. Translation der Rotationsachse in den Ursprung.[br]2. Rotation der Rotationsachse um die x-Achse in die xz-Ebene.[br]3. Rotation der Rotationsachse um die y-Achse in die z-Achse.[br]4. Rotation des Objekts um die z-Achse mit Winkel φ.[br]5. Rucktransformation durch Anwendung der inversen Transformationen der Schritte 3,2 und 1.[br][br]Sei [math]\large \vec{r}[/math] der normierte Richtungsvektor der Geraden g als Rotationsachse.[br][br]Schritt 1 Translation To Ortsvektor o=g(0) der Geraden in Ursprung verschieben. [br][table][tr][td]Schritt 2 [br]Rotationswinkels α berechnen [br]für Rotation um x-Achse, [br]α liegt zwischen der Projektion[br]r[sub]y[/sub] von r auf der yz-Fläche [br]und der z-Achse .[br][br]Nach Schritt 2 befindet sich der [br]Drehachsen-Vektor r als r′ [br]in der xz-Ebene:[br][br]Schritt 3 ------------------[br]Rotation um y-Achse[br]mit dem Rotationswinkel β. [br]Positive Winkel ergeben eine [br]Rotation gegen den Uhrzeigersinn, [br]wenn man aus Richtung der [br]Positiven y-Achse auf die [br]xz-Ebene schaut:[br]⇒ cos(β) = −cos(360°−β) = -√(r[sub]2[/sub][sup]2[/sup]+r[sub]3[/sub][sup]2[/sup])[br]⇒ sin(β) = −sin(360°−β) = −r[sub]1[/sub] [br][/td][td] 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[/img] [/td][/tr][/table][br]Beispiel:[br]g(t):=(1,1,1) + t (1,1,1) | φ=120°[br]R_g: T(v) D[sub]x[/sub]⁻¹ D[sub]y[/sub]⁻¹ D[sub]z[/sub](φ) D[sub]y[/sub] D[sub]x[/sub] T(-v)[br][math]T(-v) \, := \, \left(\begin{array}{rrrr}1&0&0&-1\\0&1&0&-1\\0&0&1&-1\\0&0&0&1\\\end{array}\right) [/math][br] [math]D_z \, := \, \left(\begin{array}{rrrr}\frac{-1}{2}&\frac{-1}{2} \; \sqrt{3}&0&0\\\frac{1}{2} \; \sqrt{3}&\frac{-1}{2}&0&0\\0&0&1&0\\0&0&0&1\\\end{array}\right)[/math] [math]D_y \, := \, \left(\begin{array}{rrrr}\frac{1}{3} \; \sqrt{6}&0&\frac{-1}{3} \; \sqrt{3}&0\\0&1&0&0\\\frac{1}{3} \; \sqrt{3}&0&\frac{1}{3} \; \sqrt{6}&0\\0&0&0&1\\\end{array}\right)[/math] [math]D_x \, := \, \left(\begin{array}{rrrr}1&0&0&0\\0&\frac{1}{2} \; \sqrt{2}&\frac{-1}{2} \; \sqrt{2}&0\\0&\frac{1}{2} \; \sqrt{2}&\frac{1}{2} \; \sqrt{2}&0\\0&0&0&1\\\end{array}\right)[/math] [br] [math]T(v) \, := \, \left(\begin{array}{rrrr}1&0&0&1\\0&1&0&1\\0&0&1&1\\0&0&0&1\\\end{array}\right) [/math], [br][br][size=85]http://www-lehre.inf.uos.de/~cg/2008/PDF/kap-13.pdf[/size]
simple example

Information: Rotation R³-Achse als elementare x,y,z-Achsendrehungen