Finding aspects of a plot of route in the tour de france for a project?

Questions
1

Hi, I have never used mathematica before and have an elementary knowledge in programming so please answer in layman’s terms. I have plotted a parametric equation for part of a route in the tour de france. The position function is r(t)=< 2.9Cos[3.2Pit] , Sin[4Pit] +5t> and the plot is:

ParametricPlot[{2.9 Cos[3.2 Pi t], Sin[4 Pi t] + 5 t}, {t, 0, 1}]

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Does anyone know how to:

  1. Plot the a point at the beginning, end, and the point where the equation intersects itself
  2. Find the time (t) and the point at which it has the sharpest curve
  3. Plot the speed as a function of time

Any help is appreciated. You don’t have to do my work for me but at least point me in the right direction.

2

  1. Can be done as…

     fn[t_] := {2.9 Cos[3.2 Pi t], Sin[4 Pi t] + 5 t};
 
 ans = NMinimize[{EuclideanDistance[fn[t1], fn[t2]],  0 < t1 < 1 && 0 < t2 < 1 && t1 < t2 - 0.1}, {t1, t2}]
 
 ParametricPlot[fn[t], {t, 0, 1},  Epilog -> 
     {PointSize[0.05], Point[fn[0]],    Point[fn[1]], {Point[fn[t1]], Point[fn[t2]]} /. ans[[2]]}]

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And 3. can be done as

Plot[Evaluate[Norm[D[fn[t], t]]], {t, 0, 1}]

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3

Thank you for your help Jon. So i realized that the plot can be represented better in ParametricPlot3D when the z-direction is set to 0.
f[t_] = ParametricPlot3D[{2.9 Cos[3.2 [Pi] (t)],Sin[4 [Pi] (t)] + 5 t, 0}, {t, 0, 1}]

Once this has been applied, it seems to not jive with your suggested material. Is there a way to apply what you did into the third dimension?

4

The code that I wrote should (in pricinple) just work in 3D since EuclideanDistance and Norm work with any dimensions. Just re-write fn