How to create a 3D plot of a path from equations of motion in wxMaxima?

Question

I have the following equations of motion of a point:

x(t):=r*cos(t^2)$
y(t):=r*sin(t^2)$
z(t):=b*t$

I already calculated the velocities and accelerations but now I would like to plot a path in 3D for this data:

r:5;
b:2;

It should look like a kind of spiral.

I tried various commands to create this 3D plot but none of them worked. All the examples of 3D plots in wxMaxima that I've found are surfaces while in this case, I want to create a curve. Is it possible in this software?

Here are two of the failed attempts:

wxplot3d([x(t), y(t), z(t)], [t, 0, 45]);
wxplot3d([parametric, x(t), y(t), z(t), [t, 0, 45]]);

UPDATE: The command below works but the plot is incorrect for some reason (I attached it as a picture). Is that because of the characteristics of these functions? Do I need some additional input?

wxdraw3d(parametric (x(t), y(t), z(t), t, 0, 45));

UPDATE 2: I tried the following command:

wxdraw3d(nticks = 10, parametric (x(t), y(t), z(t), t, 0, 45));

and the plot looks better (only with nticks=10):

But it's still not what I would expect. Here's a reference plot from a Polish book describing MathCAD use in mechanics (so I can't utilize the code presented there directly):

Maybe the problem lies in the fact that the authors of this book use some tricks ("auxiliary variables scaling the argument of the function") to obtain the plot. But I assume that it's only necessary in MathCAD. I can be wrong though...

If you know how to define a range variable in Maxima, I can try to replicate the approach from the book in Maxima.

UPDATE 3: Here's what was done in the book to obtain that plot using MathCAD:

define auxiliary variables scaling the argument of the function:

M:=1000
K:=0,1.. 45

for which the time domain is given by:

t_k:=k*sqrt(π/M)

define the functions for plotting as:

X_k:=r*cos(((k^2)/M)*π)
Y_k:=r*sin(((k^2)/M)*π)
Z_k:=b*k

And here's my attempt to translate this to Maxima:

M:1000$
assume(k >= 0, k <= 45);
t_k:k*sqrt(%pi/M);
X_k(t_k):=r*cos(((k^2)/M)*%pi);
Y_k(t_k):=r*sin(((k^2)/M)*%pi);
Z_k(t_k):=b*k;
wxdraw3d(nticks = 10, parametric (X_k(t_k), Y_k(t_k), Z_k(t_k), t_k, 0, 45));

Unfortunately, I get the following error:

draw3d (parametric): non defined variable

That's likely because of the way k was defined. Can such a range variable be defined differently in Maxima?

Answer 1

I've adapted the code you showed and it seems to work okay with some modifications.

r:5;
b:2;
M:1000$
X(k):=r*cos(((k^2)/M)*%pi);
Y(k):=r*sin(((k^2)/M)*%pi);
Z(k):=b*k;
draw3d(nticks = 1000, parametric (X(k), Y(k), Z(k), k, 0, 45));

The major changes are that foo_k := ... in the MathCAD code is translated as foo(k) := ... in Maxima, and that the plotting variable is k (which is constructed to vary much more slowly than t) instead of t or t(k).

Also I increased nticks by a lot, and cut out the mention of t_k since it doesn't come into the picture now.

By the way, I think if you say draw3d in wxMaxima, it will launch an external viewer instead of embedding an image in the notebook (with wxdraw2d, I think). You can drag with the mouse to rotate the plot in the external viewer, I think. That's helpful with 3-d plots. I could be mistaken about how to launch the viewer, since I don't use wxMaxima very much.