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RE: scalar field visualization

  • To: mathgroup at
  • Subject: [mg66671] RE: [mg66647] scalar field visualization
  • From: "David Park" <djmp at>
  • Date: Fri, 26 May 2006 04:17:48 -0400 (EDT)
  • Sender: owner-wri-mathgroup at


It all depends on the particular scalar field! And this is true of all
graphics work in general. There is no such thing as one size (or type) plot
that fits all functions and all communication or investigation needs. So one
has to ask what are the features of the function that you wish to represent
and how best to represent them.

The problem with 3D scalar function is that anything that represents the
value of the function at one point hides everything behind it.

The most common method is to take 2D slices. You could make contour plots
for fixed values of z, say, and then display multiple images. (Think medical
CAT scans.) Or, better yet, you could assemble them into an animation and
then move back and forth through the slices with the arrow keys.

Obviously, there are different ways that one could take slices. For
functions whose radial dependence is separate from the angular dependence,
one might use spherical slices.

The idea of plotting a 3D grid of points that are colored according to the
value of the function does not work very well. One could SpinShow it but it
will still look too confusing. On the other hand, if you wanted to show only
a few points, such as the local minima and maxima then that kind of plot
might work.

Another possibility might be to take a relatively small grid of points,
maybe even only one point, and sweep it through the 3D region in an
animation and vary the color. This might be even better with a 3D vector
field, where one could plot the vectors attached to a small grid of points
as they are swept through the 3D region. A small set of points, especially
if they are strategically choosen, is much less confusing to the viewer than
a complete 3D grid. A small set can be especially useful in showing the
behavior of the field at critical points. Such an animation also shows how
things are changing and this is valuable information about any function. It
is much more difficult to get this information from a static plot.

To go back to the scalar function, instead of color you could use point size
to represent the value of the function, or use spheres and radius to
represent the function. Or you could make a side bar chart plot for the
points that would show the value of the function at each point - as the
points are swept through the region.

So in general one should think multiple images and/or animation.

Repeating again, one has to think about what one wishes to show with the
graphics and what are the particular characteristics of the function and
tailor the graphics to that. There is plenty of room for invention and
innovation. (Many textbooks use the same graphical conventions and
representations that have been around for hundreds of years!)

David Park
djmp at

From: Chris Chiasson [mailto:chris at]
To: mathgroup at

In your experience, what are some effective ways to visualize three
dimensional scalar fields of real numbers in Mathematica 5.2?


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