new curve last night
- To: mathgroup at smc.vnet.net
- Subject: [mg102879] new curve last night
- From: "rlbagulatftn" <rlb at tftn.net>
- Date: Tue, 1 Sep 2009 03:52:16 -0400 (EDT)
e: I got something from a "Differential Geometry" book last night! http://www.amazon.com/Differential-Geometry-Heinrich-W-Guggenheimer/dp/0486= 634337/ref=sr_1_1?ie=UTF8&qid=1250868201&sr=8-1 New kind of variable Cartan matrix ( not the kind in Lie algebras) from differential geometry: Rotation matrix as O(2): m[t_] = Cos[t]*{{1, 0}, {0, 1}} + Sin[t]*{{0, 1}, {-1, 0}} Cartan matrix defined in my new book; FullSimplify[D[m[t], {t, 1}].Inverse[m[t]], Trig -> True]={{0, 1}, {-1, 0= }} Two inventions last night: 1) diaxial rotation: (Using Cos[t+2*Pi/3] instead of Sin[t]) m[t_] = Cos[t]*{{1, 0}, {0, 1}} + Cos[t + 2*Pi/3]*{{0, 1}, {-1, 0}} The new Cartan matrix of it: c[t_] = D[m[t], {t, 1}].Inverse[m[t]] 2) the transform of the diaxial ellipse by this Cartan matrix: ParametricPlot[c[t].{Cos[t], Cos[t + 2* Pi/3]}, {t, -Pi, Pi}, AspectRatio -> Automatic] The result is a curve I call a double Limacon (two loops inside): http://www.geocities.com/rlbagulatftn/doublelimacon.jpg Mathematica: m[t_] = Cos[t]*{{1, 0}, {0, 1}} + Cos[t + 2*Pi/3]*{{0, 1}, {-1, 0}} ParametricPlot[{Cos[t], Cos[t + 2*Pi/3]}, {t, -Pi, Pi}, AspectRatio -> Automatic] ParametricPlot[m[t].{Cos[t], Cos[t + 2*Pi/3]}, {t, -Pi, Pi}, AspectRatio -> Automatic] c[t_] = D[m[t], {t, 1}].Inverse[m[t]] FullSimplify[c[t].{Cos[t], Cos[t + 2*Pi/3]}] ParametricPlot[c[t].{Cos[t], Cos[t + 2*Pi/3]}, {t, -Pi, Pi}, AspectRatio -> Automatic] Respectfully, Roger L. Bagula 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :http://www.g= eocities.com/rlbagulatftn/Index.html alternative email: rlbagula at ...
