3D-XplorMath-J Applets

3D-XplorMath-J is a program for "Exhibits" which are visualizations of mathematical objects. It is a rewrite of a similar program 3D-XplorMath, which runs on Macintosh computers. See http://3d-xplormath.org. 3D-XplorMath-J is written in the Java programming langauge, and it requires Java Version 5.0 or higher.

This Web site shows the same exhbitis as the program, but in in the form of "applets" that can appear on a Web page. The individual applets can be accessed by clicking on links in the navigation list to the left. Note that it takes a bit of time, after you click one of the links, for the applet to be downloaded and to start running.

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Using the Mouse

3D-XplorMath-J is designed to be a highly interactive program, and the mouse is an important tool for telling the program what you want. The first thing you should learn is that clicking the mouse during a drawn out action will usually abort it, possibly after a short delay.

For two-dimensional Exhibits, you can generally move the image around on the screen by clicking-and-dragging with the mouse. If you click-and-drag with the right-mouse button (or, on Mac OS, while holding down the Command key), you can select a rectangular portion of the image to be magnified. If you hold down the Shift key while dragging, or if you drag with the middle mouse button, you can zoom in and out on the image by dragging the mouse up and down.

For three-dimensional Exhibits, clicking-and-dragging with the mouse will generally rotate the object in three dimensions, and if you release the mouse button while you are in mid-drag, the object will continue to rotate. To move the object around, drag it with the right mouse button (or while holding down the Command key on Mac OS). And, just as in two dimensions, holding down the Shift key and dragging up and down will expand or contract the object on the screen.

There are some exceptions to these general rules about using the mouse. In some cases, the exception will be noted in the status line at the bottom of the window.

The Action Menu

The "Action" menu contains actions that are specific to a partcular Exhibit or typ of Exhibit. It is a good idea to check and see if the "Action" menu has any such special items. The top item of the "Action" menu does not vary; it is always "Create." Selecting "Create" will often show an animation that creates the Exhibit, but incorporating any changes that you have made to the properties of the Exhibit.

What sort of things are in the "Action" menu? Here are some typical examples. For Plane Curves there is an item "Show Osculating Circles". For Conformal Maps you can choose to draw a circle or a line and see how it gets mapped. And for Polyhedra you can choose among seeing a normal, truncated or stellated view.

The View Menu

One should keep in mind that mathematical objects are abstract concepts, so that the possibilities for representing them visually admit considerable flexibility and choice. There are simple options, such as whether to show an object on a white, black, or colored background, and in three dimensions there are important additional options, such as whether to use orthographic or perspective projection and whether to use mono- or stereo vision. These choices and more are controlled from the "View" menu.

The stereo views require some explanation. These views enable you to view true three-dimensional representations of an object. "Anaglyph Stereo Vision" is meant to be viewed using red/blue or red/green stereo glasses (with the red lens over the left eye); it gives a clear and easy-to-see three-dimensional effect, but the object appears in shades of yellow rather than in color. "Cross-eyed Stereo Vision" shows two copies of the object side-by-side, showing the object from the perspective of the left and right eyes. To get the three-dimensional effect, you have to cross your eyes to merge the two images. "Parallel Stereo Vision" is similar, but you must merge the two images by gazing as if at a point that lies a foot or so behind the screen. It takes some practice to learn how to merge the images, but the result should be a clear, full-color three-dimensional image.

The Animate Menu: Parameters and Morphing

Most Exhibits are not unique, single mathematical objects. Rather, Exhibits come in parameterized families. For example, the Ellipsoid Exhibit has three parameters, corresponding to the lengths of its three axes, and similarly a torus is described by two radii. When an object is selected from the Exhibit menu, the program displays it with special (but typical) default choices of these parameters that, as will be explained below, can then be altered with the "Set Parameters..." command in the "Settings" menu. But something more interesting can also be done to get a clear idea of how the appearance of an object changes as one or more of its parameters vary, and this is called morphing. Many Exhibit parameters are "morphable," meaning that one can choose "initial" and "final" values for these parameters. Then, when the "Morph" command is chosen from the "Animate" menu, the program creates an animation by rapidly drawing a sequence of images of the object, starting from the initial values of the parameters and ending with their final values, giving the illusion that the object is smoothly morphing from its initial to its final shape. In some cases (particularly for Plane Curves, Polyhedra, and Ordinary Differential Equations) the program is able to render individual images fast enough that the morph can be created on the fly, but in most cases, to get a sufficiently smooth morph, the individual images must be saved in memory as they are created and then played back in rapid succession as a a flip-book animation. (In general, it is best to vary just one parameter at a time, making the initial and final values of the others equal.)

The Settings Menu

The "Settings" menu is the center for changing the various parameters and options that define the current appearance of an Exhibit. Selecting an item from this menu, will in most cases bring up a dialog box where the new values can be entered. For example, as mentioned above, selecting "Set Parameters..." will bring up a dialog allowing you to change any of the parameters that are associated with the current Exhibit, while "Set Morphing..." will let you to set the initial and final values for these parameters during a morphing animation. "Set Number of Frames..." allows changing the number of frames in a morphing animation -- increasing this will make the animation take longer but it will be smoother. "Set Visible Ranges..." lets you adjust the size and the placement of the Exhibit by setting the horizontal and vertical edges of the rectangular region that is displayed in the window. In three dimensions, "Set Viewpoint and Up Direction..." allows you to set the location and orientation in space of the virtual camera that is used to image a three-dimensional Exhibit on its focal plane. And for surfaces, "Light Settings..." will bring up a dialog in which you can set the colors of the lights shining on the surface and the directions in which the lights point.