Evoluten und Evolventen

(Anleitung zu einem Kleinprojekt)

Verwende die Ergebnisse des Kleinprojekts "Schläuche":

Rechnung

Remove["Global`*"];

from[uhu_] := 
      Module[{},  (*Rechnung*) 
v[t_] := {x[t], y[t], 
            0}; 
κ[t_] := 
          Sqrt[((v''[t] . v''[t]) (v'[t] . v'[
                          t]) - (v'[t] . v''[
                          t])^2)/((v'[t]) . (v'[
                        t]))^3]; 
ρ[t_] :=  
          1/κ[t]; 
tT[t_] := v'[t]/
            Sqrt[v'[t] . v'[t]]; 
nN[t_] := 
          1/κ[t] 1/
              Sqrt[v'[t] . v'[t]] D[v'[t]/
                Sqrt[v'[t] . v'[t]], t]; 
evolu[
            t_] := v[
              t] + ρ[t]  nN[t]/
                Sqrt[nN[t] . nN[t]]; 
evolv[t_, anf_] := 
          v[t] + (-1) tT[t]  Evaluate[
                N[Integrate[
                    Sqrt[v'[ti] . v'[ti]] , {ti, anf, 
                      t} ]]]; 
v2[
            t_] := {v[t][[1]], v[
                t][[2]]}; 
evolu2[
            t_] := {evolu[t][[1]], evolu[
                t][[2]]}; 
evolv2[t_, 
            anf_] := {evolv[t, anf][[1]], evolv[t, 
                anf][[2]]}; 
 (*Plot 3  
            D*) 
p1[anf_, end_] := 
          ParametricPlot[Evaluate[v2[t]], {t, anf, end}, 
            AspectRatio -> Automatic, 
            PlotStyle -> {Thickness[ .01]}, 
            DisplayFunction -> Identity]; 
p2[anf_, 
            end_] := ParametricPlot[Evaluate[evolu2[t]], {t, anf, end}, 
            AspectRatio -> Automatic, 
            PlotStyle -> {Thickness[ .005], Dashing[{0.1, 0.02}]}, 
            DisplayFunction -> Identity]; 
p3[anf_, 
            end_] := ParametricPlot[Evaluate[evolv2[t, anf]], {t, anf, end}, 
            AspectRatio -> Automatic, 
            PlotStyle -> {Thickness[ .005], Dashing[{0.05, 0.015}]}, 
            DisplayFunction -> 
              Identity]; 
Show[
p1[anf, 
            end], 
p2[anf, end], 
p3[anf, 
            end], DisplayFunction -> $DisplayFunction]
]

Demos Kurvendefinitionen

1

x[t_] := 
        4 Cos[t];   

    y[t_] := 2  Sin[t];

    z[t_] := 0;

    anf = 0;  

    end  = Pi/2;

    from[uhu];

2

x[t_] := 
        4 Cos[t];   

    y[t_] := 2  Sin[t];

    z[t_] := 0;

    anf = 0;  

    end  = 2  Pi;

    from[uhu];

3

(*Kurvendefinitionen*)
x[t_]:= Cos[t];   
y[t_]:= Sin[t];
z[t_]:= 0;
anf=0;  
end =2Pi;
from[uhu];

4

(*Kurvendefinitionen*)
x[t_]:= 1+Cos[t];   
y[t_]:= Sin[t^2];
z[t_]:= 0;
anf=0.1;  
end =Pi;
from[uhu];

Ergebnisse von Untersuchungen der Klasse M2p: Schlauchdefinitionen für interessante Formen

Rechnung mit Zwischenergebnissen

Remove["Global`*"];

\(\(\(from[uhu_] := 
        Module[{},  (*Rechnung*) 
v[t_] := {x[t], y[t], 
              0}; 
Print["v[t]  ", 
            v[t]]; 
κ[t_] := 
            Sqrt[\((\((\(v''\)[t] . \(v''\)[t])\) \((\(v'\)[t] . \(v'\)[
                            t])\) - \((\(v'\)[t] . \(v''\)[
                            t])\)^2)\)/\((\((\(v'\)[t])\) . \((\(v'\)[
                          t])\))\)^3]; 
ρ[t_] := \ 
            1/κ[t]; 
tT[t_] := \(v'\)[t]/
              Sqrt[\(v'\)[t] . \(v'\)[
                    t]]; 
Print["tT[t]  ", 
            tT[t]]; 
nN[t_] := 
            1/κ[t]\ 1/
                Sqrt[\(v'\)[t] . \(v'\)[t]]\ D[\(v'\)[t]/
                  Sqrt[\(v'\)[t] . \(v'\)[t]], 
                t]; 
Print["nNT[t]  ", 
            nN[t]]; 
evolu[t_] := 
            v[t] + ρ[t]\ \ nN[t]/
                  Sqrt[nN[t] . 
                      nN[t]]; 
Print["evolu[t]  ", 
            evolu[t]]; 
evolv[t_, anf_] := 
            v[t] + \((\(-1\))\)\ tT[t]\ \ Evaluate[
                  N[Integrate[
                      Sqrt[\(v'\)[ti] . \(v'\)[ti]]\ , {ti, anf, 
                        t}\ ]]]; 
Print["evolv[t,anf]  ", evolv[t, anf]]; 
v2[
              t_] := {\(v[t]\)[\([1]\)], \(v[
                  t]\)[\([2]\)]}; 
Print["v2[t]  ", 
            v2[t]]; 
evolu2[
              t_] := {\(evolu[t]\)[\([1]\)], \(evolu[
                  t]\)[\([2]\)]}; 
Print["evolu2[t]  ", 
            evolu2[t]]; 
evolv2[t_, 
              anf_] := {\(evolv[t, anf]\)[\([1]\)], \(evolv[t, 
                  anf]\)[\([2]\)]}; 
Print["evolv2[t,anf]  ", evolv2[t, anf]]; 
 (*Plot\ 3  
              D*) 
\(p1[anf_, end_] := 
                ParametricPlot[Evaluate[v2[t]], {t, anf, end}, 
                  AspectRatio -> Automatic, 
                  PlotStyle -> {Thickness[ .02]}];\)
\(p2[anf_, end_] := 
                ParametricPlot[Evaluate[evolu2[t]], {t, anf, end}, 
                  AspectRatio -> Automatic, 
                  PlotStyle -> {Thickness[ .01], 
                      Dashing[{0.1, 0.02}]}];\)
\(p3[anf_, 
                  end_] := ParametricPlot[
                  Evaluate[evolv2[t, anf]], {t, anf, end}, 
                  AspectRatio -> Automatic, 
                  PlotStyle -> {Thickness[ .01], 
                      Dashing[{0.05, 
                          0.015}]}];\)
Show[
p1[anf, end], 
p2[anf, 
                end], 
p3[anf, 
                end]]
];\)\(
\)
    \) (*Kurvendefinitionen*) \)
x[t_] := 4 Cos[t];   
y[t_] := 2  Sin[t];
z[t_] := 0;
anf = 0;  
end  = Pi;
from[uhu];