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Helping asked for Trochoïd and Frenet Coordinates

Cédric Mannu

Posted 2 years ago

I am trying to get a left trochoid (R=1, d=2) of a left trochoid of a left trochoid. To do so, I try to establish a Frenet system and get a curve derived from the first one, that makes two turn when the first make one and then by the same way, i try to get a second curve that makes four turns when the second makes two and the first makes one. But I can't get Mathematica do it. I must make a lot of mistakes, as I am not so familiar with Frenet and Mathematica. I need to obtain equations for C1, C2 and C3 (see graphic attached). I think C1 is a left trochoïd, where R=1 and d=2 for example (elongated trochoÏd), C2 and C3 are the same, only perioding faster , C2 along C1, and C3 along C2 along C1. Here is the code I made so far (see notebook) Attachments: Famille de Troch...jpg

POSTED BY: Cédric Mannu

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Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 2 years ago

Try with f1[t_] = {x2[t_], y2[t_], z2[t_]} = f0[t] + N1[t] Cos[2 t] + B1[t] Sin[2 t];

POSTED BY: Gianluca Gorni

Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 2 years ago

I am not sure what you want to do. Keep in mind that matrix multiplication is done with `.` and not with `*` This is my interpretation of your wish: f0[t_] = {x[t, 1, 2], y[t, 1, 2], z[t, 1, 2]}; f1[t_] = {x2[t_], y2[t_], z2[t_]} = f0[t] + T1[t] x[2 t, 1, 2] + N1[t] y[2 t, 1, 2] + B1[t] z[2 t, 1, 2]; curves = ParametricPlot3D[{f0[t], f1[t]}, {t, -4 Pi, 4 Pi}, PlotRange -> All] Manipulate[ Show[curves, Graphics3D[{{PointSize[Large], Blue, Point[f0[t]], Brown, Point[f1[t]]}, HalfLine[f0[t], T1[t]], Arrow[{f0[t], f0[t] + 15 T1[t]}], HalfLine[f0[t], N1[t]], Arrow[{f0[t], f0[t] + 15 N1[t]}], HalfLine[f0[t], B1[t]], Arrow[{f0[t], f0[t] + 15 B1[t]}]}]], {t, -4 Pi, 4 Pi}]

I am not sure what you want to do. Keep in mind that matrix multiplication is done with . and not with *

This is my interpretation of your wish:

f0[t_] = {x[t, 1, 2], y[t, 1, 2], z[t, 1, 2]};
f1[t_] = {x2[t_], y2[t_], z2[t_]} =
   f0[t] +
    T1[t] x[2 t, 1, 2] +
    N1[t] y[2 t, 1, 2] +
    B1[t] z[2 t, 1, 2];
curves = 
  ParametricPlot3D[{f0[t], f1[t]}, {t, -4 Pi, 4 Pi}, PlotRange -> All]
Manipulate[
 Show[curves, Graphics3D[{{PointSize[Large], Blue, Point[f0[t]],
     Brown, Point[f1[t]]}, HalfLine[f0[t], T1[t]],
    Arrow[{f0[t], f0[t] + 15 T1[t]}], HalfLine[f0[t], N1[t]],
    Arrow[{f0[t], f0[t] + 15 N1[t]}], HalfLine[f0[t], B1[t]],
    Arrow[{f0[t], f0[t] + 15 B1[t]}]}]],
 {t, -4 Pi, 4 Pi}]

POSTED BY: Gianluca Gorni

Cédric Mannu

Posted 2 years ago

POSTED BY: Cédric Mannu

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