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Customizing the wireframe / strides of a 3D plot

asked 2016-06-26 11:57:57 -0500

Lazza gravatar image

updated 2016-09-27 07:42:24 -0500

I am experimenting with 3D plotting to see how I can customize the output. In particular, I am trying to obtain something more similar to how Mathematica or Wolfram Alpha do things.

For example, I plotted Taubin's heart surface:

$$ \left(x^2+\frac{9 y^2}{4} + z^2 - 1\right)^3-x^2 z^3 - \frac{9 y^2 z^3}{80} = 0 $$

Using the following code:

var("x y z")

f = (x^2 + 2.25*y^2 + z^2 - 1)^3 - (x^2)*(z^3) - 0.1125*(y^2)*(z^3)
xint = (x, -1.5, 1.5)
yint = (y, -1.5, 1.5)
zint = (z, -1.5, 1.5)

my_plot = implicit_plot3d(f, xint, yint, zint, opacity=.7, mesh=2)
my_plot.show()

The mesh corresponds to the triangulation of points that Sage performs on the data calculated with implicit_plot3d. However, Wolfram Alpha gives a different result (https://www.wolframalpha.com/input/?i...), because it does not use all the edges of the triangulation:

What I think they are doing is they are using a high number of points to draw the surface, however they are "slicing" the object with planes parallel to the three axes (at fixed distances) to draw the contours.

How can I replicate these lines?

Matplotlib calls them "strides" as documented for Axes3d.plot_surface(http://matplotlib.org/mpl_toolkits/mp...). My final goal is exporting 3D vector drawings which I managed to do with Matplotlib thanks to the numpy-stlpackage (https://github.com/WoLpH/numpy-stl):

my_plot.triangulate()
my_plot.save("/tmp/test.stl")
from mpl_toolkits import mplot3d
from matplotlib import pyplot

# Create a new plot
figure = pyplot.figure()
axes = mplot3d.Axes3D(figure)

# Load the STL files and add the vectors to the plot
your_mesh = mesh.Mesh.from_file('/tmp/test.stl')
axes.add_collection3d(mplot3d.art3d.Poly3DCollection(your_mesh.vectors))

# Auto scale to the mesh size
scale = your_mesh.points.flatten(-1)
axes.auto_scale_xyz(scale, scale, scale)

# Show the plot to the screen
pyplot.show()
pyplot.savefig("/home/andrea/Scrivania/wow.pdf")

In this case the presence of too much grid lines looks even worse:

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Sorry for the ugly formatting but the website wouldn't allow me to post the Q until I dropped all links. Someone with enough rep please fix the links and images. Thank you.

Lazza gravatar imageLazza ( 2016-06-26 11:58:39 -0500 )edit

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answered 2016-06-26 19:02:04 -0500

One thing you can do for starters is turn off the mesh with mesh=false since that keyword argument is Boolean. You can also improve the smoothness of the surface by setting plot_points higher than it's default of 40 in each direction.

If the surface could always be solved for each independent variable, one way to draw contour-like lines would be setting one variable to a constant, solving for the second in terms of the third and using parametric_plot. Since you can't do that for this surface, here's a workaround using a series of implicit_plot3d for lines with constant y-value:

for i in [ .1*j - .6 for j in range(0,13)]:
    g = (x^2 + 2.25*i^2 + z^2 - 1)^3 - (x^2)*(z^3) - 0.1125*(i^2)*(z^3)
    my_plot += implicit_plot3d(g, xint, (y,i-.01,i+.01), zint, color='black')

Here's a link to a live example. The function threejs at the end is a SageMathCell alternative to show.

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Thanks for your answer! Unfortunately I cannot upvote it yet. I have tweaked it a bit, obtaining this: sagecell.sagemath.org/?q=jrogwx. There is still the problem that those are actually "slices" of a 3D object and therefore have one side which is a line and one side which is thick (depending on the width). I will try to figure out how to draw just lines for them.

Lazza gravatar imageLazza ( 2016-06-28 08:44:17 -0500 )edit

@Lazza, new users can't upvote but they can accept answers (click the tick mark next to the answer).

slelievre gravatar imageslelievre ( 2016-09-27 10:54:04 -0500 )edit

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Asked: 2016-06-26 11:57:57 -0500

Seen: 376 times

Last updated: Sep 27 '16