Algebra Fenster:[br][br][b]Achsenvektor v[br]Drehwinkel w[/b][br][br]Drehe [math]\large e_x\;\mapsto\; e_x',e_y\; \mapsto \; e'_y,e_z\; \mapsto \; e'_z[/math] um Achse v mit Winkel w [br]Drehkreismittelpunkt e_k: [math]M_{e_k}:=v\cdot\frac{\left(v\cdot e_k\right)}{v^2}[/math] , k=x,y,z [br]Drehkreismittelpunkt P: [size=85]M_p:=v ((v P) / v²) [/size][br]Die Bildvektoren e' bilden die Drehmatrix[br][br]Wickipedia:[br] v=[math]\large \frac{v}{\left|v\right|}[/math] = (n1,n2,n3).[br][math]R_n(a, n1, n2, n3) \, := \left(\begin{array}{rrr}n1^{2} \; \left(1 - \operatorname{cos} \left( a \right) \right) + \operatorname{cos} \left( a \right)&n1 \; n2 \; \left(1 - \operatorname{cos} \left( a \right) \right) - n3 \; \operatorname{sin} \left( a \right)&n1 \; n3 \; \left(1 - \operatorname{cos} \left( a \right) \right) + n2 \; \operatorname{sin} \left( a \right)\\n2 \; n1 \; \left(1 - \operatorname{cos} \left( a \right) \right) + n3 \; \operatorname{sin} \left( a \right)&n2^{2} \; \left(1 - \operatorname{cos} \left( a \right) \right) + \operatorname{cos} \left( a \right)&n2 \; n3 \; \left(1 - \operatorname{cos} \left( a \right) \right) - n1 \; \operatorname{sin} \left( a \right)\\n3 \; n1 \; \left(1 - \operatorname{cos} \left( a \right) \right) - n2 \; \operatorname{sin} \left( a \right)&n3 \; n2 \; \left(1 - \operatorname{cos} \left( a \right) \right) + n1 \; \operatorname{sin} \left( a \right)&n3^{2} \; \left(1 - \operatorname{cos} \left( a \right) \right) + \operatorname{cos} \left( a \right)\\\end{array}\right)[/math][br][br]mit [math]spur\left(R \right) = 2 \; \operatorname{cos} \left( \alpha \right) + 1 \quad \to \quad \alpha =\operatorname{acos} \left( \frac{1}{2}\;(spur\left(R \right) - 1)\right) [/math][br][br]R(w) stellt die ermittelten e' dar[br]und [br]R_w stellt das Ergebnis mittels der (wikipedia) Formel dar:[br]

o=x,y,z, e_o Vektor Standardbasis, Achsenvektor v=(n1,n2,n3), Drehwinkel t (w)[br][br]E_o Ebene mit Richtungsvektor der Achsengeraden v als Normalenvektor durch e_o[br]E_o: v((x,y,z)-e_o)=0[br][br]Drehkreismittelpunkt D_o: Schnittpunkt Achsengerade t v [math]\subset[/math] E_o für[br]v (t v - e_o)=0 ===> t= v e_o/ v^2 ===> D_o=(v e_o/ v^2) v[br]D_o=(n1,n2,n3) e_o/(n1,n2,n3)^2 * (n1,n2,n3)[br][br][math]D_x:= \left\{ \frac{n1^{2}}{n1^{2} + n2^{2} + n3^{2}}, \frac{n1 \cdot n2}{n1^{2} + n2^{2} + n3^{2}}, \frac{n1 \cdot n3}{n1^{2} + n2^{2} + n3^{2}} \right\} [/math][br][math]D_y:= \, \left( \frac{n1 \cdot n2}{n1^{2} + n2^{2} + n3^{2}}, \frac{n2^{2}}{n1^{2} + n2^{2} + n3^{2}}, \frac{n2 \cdot n3}{n1^{2} + n2^{2} + n3^{2}} \right)[/math][br][math]D_z:= \, \left( \frac{n1 \cdot n3}{n1^{2} + n2^{2} + n3^{2}}, \frac{n2 \cdot n3}{n1^{2} + n2^{2} + n3^{2}}, \frac{n3^{2}}{n1^{2} + n2^{2} + n3^{2}} \right)[/math][br][br]Konstruiere einen Vektor (x,y,z)=u [math]\perp[/math] (e_o-D_o)[br] (e_o - D_o) (x,y,z)=0 [br] für einen Kreis in Ebene E_o (u [math]\in[/math] E_o)[br] (n1,n2,n3) ((x, y, z) - D_o), [br]mit dem Radius (e_o-D_o)[br]((x,y,z)-D_o)^2=(e_o - D_o)^2 }[br]Löse GLS:[br][br][math]u_x:=\left(\frac{n1^{2}}{n1^{2} + n2^{2} + n3^{2}}, \frac{n1 \; n2 - n3 \; \sqrt{n1^{2} + n2^{2} + n3^{2}}}{n1^{2} + n2^{2} + n3^{2}}, \frac{n1 \; n3 + n2 \; \sqrt{n1^{2} + n2^{2} + n3^{2}}}{n1^{2} + n2^{2} + n3^{2}} \right)[/math][br][math]u_y:=\left(\frac{-n3 \; \sqrt{n1^{2} + n2^{2} + n3^{2}} + n1 \; n2}{n1^{2} + n2^{2} + n3^{2}}, \frac{n2^{2}}{n1^{2} + n2^{2} + n3^{2}}, \frac{n1 \; \sqrt{n1^{2} + n2^{2} + n3^{2}} + n2 \; n3}{n1^{2} + n2^{2} + n3^{2}} \right)[/math][br][math]u_z:=\left(\frac{n2 \; \sqrt{n1^{2} + n2^{2} + n3^{2}} + n1 \; n3}{n1^{2} + n2^{2} + n3^{2}}, \frac{-n1 \; \sqrt{n1^{2} + n2^{2} + n3^{2}} + n2 \; n3}{n1^{2} + n2^{2} + n3^{2}}, \frac{n3^{2}}{n1^{2} + n2^{2} + n3^{2}} \right)[/math][br][br]Drehkreis: [br][math]M= D_o, r=\sqrt{e_o-D_o}, t=w[/math][br][math]K_o(t):= D_o +( ( u_o-D_o )\cdot sin(t) + ( e_o-D_o)\cdot cos(t) )[/math][br][br]D_x(n1,n2,n3) + ((u_x(n1,n2,n3)-D_x(n1,n2,n3)) sin(t)+ (e_x-D_x(n1,n2,n3))cos(t))[br][math]e'_x=\left(\begin{array}{r}\frac{n1^{2} + n2^{2} \; \operatorname{cos} \left( t \right) + n3^{2} \; \operatorname{cos} \left( t \right)}{n1^{2} + n2^{2} + n3^{2}}\\\frac{n1 \; n2 - n1 \; n2 \; \operatorname{cos} \left( t \right) + n3 \; \sqrt{n1^{2} + n2^{2} + n3^{2}} \; \operatorname{sin} \left( t \right)}{n1^{2} + n2^{2} + n3^{2}}\\\frac{n1 \; n3 - n1 \; n3 \; \operatorname{cos} \left( t \right) - n2 \; \sqrt{n1^{2} + n2^{2} + n3^{2}} \; \operatorname{sin} \left( t \right)}{n1^{2} + n2^{2} + n3^{2}}\\\end{array}\right)[/math][br]D_y(n1,n2,n3) + ((u_y(n1,n2,n3)-D_y(n1,n2,n3)) sin(t)+ (e_y-D_y(n1,n2,n3))cos(t))[br][math]e'_y=\left(\begin{array}{r}\frac{n1 \; n2 - n1 \; n2 \; \operatorname{cos} \left( t \right) - n3 \; \sqrt{n1^{2} + n2^{2} + n3^{2}} \; \operatorname{sin} \left( t \right)}{n1^{2} + n2^{2} + n3^{2}}\\\frac{n2^{2} + n1^{2} \; \operatorname{cos} \left( t \right) + n3^{2} \; \operatorname{cos} \left( t \right)}{n1^{2} + n2^{2} + n3^{2}}\\\frac{n2 \; n3 + n1 \; \sqrt{n1^{2} + n2^{2} + n3^{2}} \; \operatorname{sin} \left( t \right) - n2 \; n3 \; \operatorname{cos} \left( t \right)}{n1^{2} + n2^{2} + n3^{2}}\\\end{array}\right)[/math][br]D_z(n1,n2,n3) + ((u_z(n1,n2,n3)-D_z(n1,n2,n3)) sin(t)+ (e_z-D_z(n1,n2,n3))cos(t))[br][math]e'_z=\left(\begin{array}{r}\frac{n1 \; n3 - n1 \; n3 \; \operatorname{cos} \left( t \right) + n2 \; \sqrt{n1^{2} + n2^{2} + n3^{2}} \; \operatorname{sin} \left( t \right)}{n1^{2} + n2^{2} + n3^{2}}\\\frac{n2 \; n3 - n1 \; \sqrt{n1^{2} + n2^{2} + n3^{2}} \; \operatorname{sin} \left( t \right) - n2 \; n3 \; \operatorname{cos} \left( t \right)}{n1^{2} + n2^{2} + n3^{2}}\\\frac{n3^{2} + n1^{2} \; \operatorname{cos} \left( t \right) + n2^{2} \; \operatorname{cos} \left( t \right)}{n1^{2} + n2^{2} + n3^{2}}\\\end{array}\right)[/math][br][br]für n1[sup]2[/sup]+n2[sup]2[/sup]+n3[sup]2[/sup] = 1 ergibt {e'_x, e'_y, e'_z} = R([math]\large \alpha[/math],n1,n2,n3)[br][br]

Wickipedia:[br]R(a,n1,n2,n3)[size=85] [br]{{n1^2*(1 - cos(a)) + cos(a), n1*n2*(1 - cos(a)) - n3*sin(a), n1*n3*(1 - cos(a)) + n2*sin(a)}, [br]{n2*n1*(1 - cos(a)) + n3*sin(a), n2^2*(1 - cos(a)) + cos(a), n2*n3*(1 - cos(a)) - n1*sin(a)}, [br]{n3*n1*(1 - cos(a)) - n2*sin(a), n3*n2*(1 - cos(a)) + n1*sin(a), n3^2*(1 - cos(a)) + cos(a)}}[br][/size][br]$7: Formelkonstruktor[br]D_x(n1, n2, n3) + (u_x(n1, n2, n3) - D_x(n1, n2, n3)) sin(t) + (e_x - D_x(n1, n2, n3)) cos(t)[br][br]Drehmatrix R für Achsenvektor v=(n1,n2,n3) und Winkel t[br]{[br][size=85]{(n1^2*(1-cos(t))+(n1^2+n2^2+n3^2)*cos(t)), (n1*n2*(1-cos(t))-n3*sin(t)*sqrt(n1^2+n2^2+n3^2)), (n2*n3*(1-cos(t))-n1*sin(t)*sqrt(n1^2+n2^2+n3^2))}, [br]{(n1*n2*(1-cos(t))+n3*sin(t)*sqrt(n1^2+n2^2+n3^2)), (n2^2*(1-cos(t))+(n1^2+n2^2+n3^2)*cos(t)), (n2*n3*(1-cos(t))-n1*sin(t)*sqrt(n1^2+n2^2+n3^2))}, [br]{(n1*n3*(1-cos(t))-n2*sin(t)*sqrt(n1^2+n2^2+n3^2)), (n2*n3*(1-cos(t))+n1*sin(t)*sqrt(n1^2+n2^2+n3^2)), (n3^2*(1-cos(t)))+(n1^2+n2^2+n3^2)*cos(t)}[/size][br]}*1/(n1^2+n2^2+n3^2)[br][br][size=85]n1^2+n2^2+n3^2[/size]=1[br][table][tr][td](n1^2*(1-cos(t))+cos(t))[br][/td][td]((n1*n2*(1-cos(t))-n3*sin(t))[br][/td][td](n1*n3*(1-cos(t))+n2*sin(t)) [br][/td][/tr][tr][td]( n1*n2*(1- cos(t))+n3*sin(t)) [br][/td][td](n2^2*(1-cos(t))+cos(t))[br][/td][td](n2*n3*(1-cos(t))-n1*sin(t))[br][/td][/tr][tr][td](n1*n3*(1- cos(t))-n2*sin(t))[br][/td][td](n2*n3*(1-cos(t))+n1*sin(t))[br][/td][td](n3^2*(1-cos(t)))+cos(t)[br][/td][/tr][/table][br][br][/size]

Exkurs (vollständigkeitshalber)[br][br][size=150]Drehungen um Koordinatenachsen[/size][br][br][code]Do(χ,a):=Take({{ (1, cos(a),cos(a))χ, (0,0,-sin(a))χ ,(0, sin(a), 0)χ}, {(0,0, sin(a))χ, ( cos(a) ,1, cos(a) )χ, (-sin(a),0,0)χ}, {(0,-sin(a),0)χ, (sin(a),0,0)χ, (cos(a) , cos(a) ,1)χ }} ,1,3)[/code][br][br]D[sub]z[/sub](a)=Do((0,0,1),a)[br][math]\left(\begin{array}{rrr}\operatorname{cos} \left( a \right)&-\operatorname{sin} \left( a \right)&0\\\operatorname{sin} \left( a \right)&\operatorname{cos} \left( a \right)&0\\0&0&1\\\end{array}\right)[/math][br][br][size=150]Spiegelung[/size][br][br]n=(n1,n2,n3) normierter Achsenvektor (Ursprungsgerade g(t):=t v, v=(1,1,1)/sqrt(3), ) [br][code]S_n(n1,n2,n3):=Take({{2n1² - 1, 2n1 n2 , 2n1 n3 }, {2n1 n2 , 2n2² - 1, 2n2 n3 }, {2n1 n3 , 2n2 n3 , 2n3² - 1}},1,3)[/code][br][br]S_v:=S_n(x(v),y(v),z(v))[br][math]S_v \, := \, \left(\begin{array}{rrr}\frac{-1}{3}&\frac{2}{3}&\frac{2}{3}\\\frac{2}{3}&\frac{-1}{3}&\frac{2}{3}\\\frac{2}{3}&\frac{2}{3}&\frac{-1}{3}\\\end{array}\right)[/math][br][br]ov Ortsvektor, rv Richtungsvektor der Spiegel-Geraden g[br][code]GS(po,oo,ro):=2 (oo+(Dot(po-oo,ro))/(Dot(ro,ro))*ro)-po[/code][br][br][br]Ursprungs-Ebene Normalenvektor no=(n1,n2,n3)[br][br][code]SP(no,vo):=vo-2Dot(vo,no)/Dot(no,no)*no[/code][br][code][/code][br][code]S_N(n1,n2,n3):=Take({{((-n1^(2)) + n2^(2) + n3^(2)) / (n1^(2) + n2^(2) + n3^(2)), (((-2) * n1) * n2 / (n1^(2) + n2^(2) + n3^(2))), (((-2) * n1) * n3 / (n1^(2) + n3^(2) + n2^(2)))}, {(((-2) * n2) * n1 / (n2^(2) + n1^(2) + n3^(2))), ((-n2^(2)) + n1^(2) + n3^(2)) / (n2^(2) + n1^(2) + n3^(2)), (((-2) * n2) * n3 / (n2^(2) + n3^(2) + n1^(2)))}, {(((-2) * n3) * n1 / (n3^(2) + n1^(2) + n2^(2))), (((-2) * n3) * n2 / (n3^(2) + n2^(2) + n1^(2))), ((-n3^(2)) + n1^(2) + n2^(2)) / (n3^(2) + n1^(2) + n2^(2))}},1,3)[br][/code][br][br]mit [math]n1^{2} + n2^{2} + n3^{2} = 1[/math][br][math]S_N(n1,n2,n3)=\left(\begin{array}{rrr}-n1^{2} + n2^{2} + n3^{2}&-2 \; n1 \; n2&-2 \; n1 \; n3\\-2 \; n1 \; n2&n1^{2} - n2^{2} + n3^{2}&-2 \; n2 \; n3\\-2 \; n1 \; n3&-2 \; n2 \; n3&n1^{2} + n2^{2} - n3^{2}\\\end{array}\right)[/math][br][br][br]Sn:=S_N(x(v),y(v),z(v))[br][math]Sn \, := \, \left(\begin{array}{rrr}\frac{1}{3}&\frac{-2}{3}&\frac{-2}{3}\\\frac{-2}{3}&\frac{1}{3}&\frac{-2}{3}\\\frac{-2}{3}&\frac{-2}{3}&\frac{1}{3}\\\end{array}\right)[/math][br][br][br]Beliebige Ebene E:=no ((x,y,z)-p)=0, R Urbild, R' Bild[br]d:=no p (Abstand Ebene vom Ursprung)[br]R'=Sn (R-no p) + no p[br]