RE: scalar field visualization

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

Chris, 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 earthlink.net http://home.earthlink.net/~djmp/ From: Chris Chiasson [mailto:chris at chiasson.name] To: mathgroup at smc.vnet.net In your experience, what are some effective ways to visualize three dimensional scalar fields of real numbers in Mathematica 5.2? -- http://chris.chiasson.name/