# Plot 3D vector that is a function of one variable

I have a function, r[t], which is a piecewise function that generates a 3D vector. It looks like this:

r[t_] = Piecewise[{{{5 t, 0, 3 (1 + Cos[t])}, 0 <= t <= π},
{{5 Cos[t - 3  π/2] + 5*π, 5 Sin[t - 3 π/2] + 5, 0},  π < t <= 2 π},
{{5*π + 3*Cos[t - 3 π/2], 2*5 - 3 + 3*Sin[t - 3 π/2],  1/(2 π) (t - 2 π)^2},
2 π < t <= 4 π},
{{-3 (t - 17 π/3), 10, -1/π t^2 + 10 t - 22 π},   4 π < t <= 5 π},
{{-3 t + 17 π, 10, -972 π + 540 t - (99 t^2)/π + (6 t^3)/π^2},
5 π < t <= 6 π},
{{-π - 3 Sin[t], 9/40 (1/3 (20 + 18 π) - t)^2, 3 Cos[t] - 3}, 6 π < t <= 8 π},
{{-25 π + 25 t - (19 t^2)/(4 π) + t^3/(4 π^2),  -(25/2) (140 - 132 π + 27 π^2) +
(15 (80 - 74 π + 15 π^2) t)/(2 π) -  (3 (180 - 164 π + 33 π^2) t^2)/(8 π^2) -
((-50 + 45 π - 9 π^2) t^3)/(20 π^3), 1056 - (360 t)/π + (81 t^2)/(2 π^2) - (3 t^3)/(2 π^3)},
8 π < t <= 10 π}}];


I am very new to Mathematica (this is my first day using it), and I am unsure of how to plot this function in a 3D space. I have tried Plot3D and VectorPlot3D, but neither seems to offer a good way to do this (that I can think of). How might I go about plotting the function?

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Have you looked at ParametricPlot3D? That's the function you need... i.stack.imgur.com/WC16T.png –  rm -rf Sep 13 '12 at 1:43
@R.M That, sir, is the answer I was looking for. I feel really stupid now, I guess knew it was parametric really, but... I don't know, there's no excuse for a brain fart of that proportion. –  Ben7005 Sep 13 '12 at 1:48
No worries; to brain fart is human, after all :) You can write down the answer yourself, explain it and include a picture... Would be a good way to start out :) –  rm -rf Sep 13 '12 at 1:50
How would I add points to my plot? Is that an option in ParametricPlot3D or do you feed the produced graph to another function? –  Ben7005 Sep 13 '12 at 1:54
See this question and its answers... I would suggest using a sphere for 3D (see JM's answer) –  rm -rf Sep 13 '12 at 2:02

r[t_] := Piecewise[
{{{5 t, 0, 3 (1 + Cos[t])}, 0 <= t <= Pi},
{{5 Cos[t - 3 Pi/2] + 5*Pi, 5 Sin[t - 3 Pi/2] + 5, 0}, Pi < t <= 2 Pi},
{{5*Pi + 3*Cos[t - 3 Pi/2], 2*5 - 3 + 3*Sin[t - 3 Pi/2], 1/(2 Pi) (t - 2 Pi)^2}, 2 Pi < t <= 4 Pi},
{{-3 (t - 17 Pi/3), 10, -1/Pi t^2 + 10 t - 22 Pi}, 4 Pi < t <= 5 Pi},
{{-3 t + 17 Pi, 10, -972 Pi + 540 t - (99 t^2)/Pi + (6 t^3)/Pi^2}, 5 Pi < t <= 6 Pi},
{{-Pi - 3 Sin[t], 9/40 (1/3 (20 + 18 Pi) - t)^2, 3 Cos[t] - 3}, 6 Pi < t <= 8 Pi},
{{-25 Pi + 25 t - (19 t^2)/(4 Pi) + t^3/(4 Pi^2), -(25/2) (140 - 132 Pi + 27 Pi^2) + (15 (80 - 74 Pi + 15 Pi^2) t)/(2 Pi) - (3 (180 - 164 Pi + 33 Pi^2) t^2)/(8 Pi^2) - ((-50 + 45 Pi - 9 Pi^2) t^3)/(20 Pi^3), 1056 - (360 t)/Pi + (81 t^2)/(2 Pi^2) - (3 t^3)/(2 Pi^3)}, 8 Pi < t <= 10 Pi}}];

Show[ParametricPlot3D[r[t], {t, 0, 10 Pi}, PlotStyle -> Directive[Purple, Thick], PlotRange -> Full,
Boxed -> False, Axes -> False],
ListPointPlot3D[Table[r[x], {x, 0, 10 Pi, Pi/5}], PlotStyle -> {PointSize[Large], Red}]]


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Thanks, I already had the solution from R.M, but this made it easier to see how to plot points on it. –  Ben7005 Sep 13 '12 at 2:20
@Ben7005 another way to add points to the plot is to use the Mesh option in ParametricPlot3D. There are some examples in the Examples > Options > Mesh section of the documentation of ParametricPlot3D. –  Heike Sep 13 '12 at 6:43