Getting Started with 3D-XplorMath-J

Think of 3D-XplorMath-J as a museum -- a Virtual Math Museum


Galleries and Exhibits

As with a bricks-and-mortar museum, the objects of this museum are arranged into Galleries of related Exhibits; to choose a Gallery you wish to visit, just select its name from the "Gallery" menu, and it will become the current Gallery. This means its name becomes the name of the Exhibit menu -- the one just to the right of the Gallery menu -- and the names of the Exhibits that it contains will be listed in the Exhibit menu or in one of its sub-menus. (When the program starts up you will find yourself in the Surface Gallery.)


Viewing Exhibits

When an Exhibit is selected by name from the Exhibit menu, the program immediately displays a default representation of that Exhibit. In many cases, an animation will play that shows the "creation" of the object. This illustrates a basic feature of the program: wherever possible, the program assigns carefully chosen default values to various parameters and options needed to display an Exhibit; it creates the view of the Exhibit corresponding to these default choices; and it then allows the user to alter one or more of these defaults by making various menu choices, by entering values in dialogs, or by clicking buttons in control panels. The program will then display the new representation of the same Exhibit appropriate for these user choices. We next explain in more detail some of the more important methods of modifying an Exhibit in this way.


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. (On the other hand if you just want to pause an animation, tap the space bar; then tap it again to continue the animation.)

In two-dimensional Galleries, 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.

In three-dimensional Galleries, 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

While most menus of the program change little if at all when the current Exhibit is changed to another in the same Gallery, or even when the Gallery changes, the "Action" menu changes substantially when the Gallery changes, and certain individual Exhibits within a Gallery often have associated to them special Action menu items that help explain some unusual detail of that Exhibit. As a result, after selecting an Exhibit for the first time, 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 generally show the same animation that is played when the Exhibit is first selected, 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 in the Plane Curves, Polyhedra, and Ordinary Differential Equation Galleries) 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.


The File Menu: Settings Files, Snapshots, and Multiple Windows

Using the "File" menu, it is possible to save a snapshot of the current visualization of an Exhibit, either as a JPEG or PNG file. It is also possible to save the identity of the current Exhibit together with all its parameters and options in a so-called settings file, by selecting "Save Exhibit to File..." in the "File" menu. If you later open this settings file, using the "File" menu command "Open Settings File...", the program will return to the state it had when the file was created. (And you can share your work by sending the settings file as an email attachment to a friend.) Saving an Exhibit to a file also works with user-defined Exhibits (see below), and is particularly useful in this case since it in effect allows you to create and distribute entirely new Exhibits that are not in the 3D-XPlorMath-J repertory.


User Defined Exhibits

When we began this introduction to 3D-XplorMath-J, we said that you should think of it as a museum. But now we want to explain in what sense you can also think of it as a mathematical laboratory, using so called user-defined Exhibits.

Most (but not all) Galleries contain at least one User Defined Exhibit. These at first glance seem similar to other Exhibits of the Gallery. However when you first select a User Defined Exhibit, you will see a dialog box, much like a "Set Parameters..." dialog, except that in addition to the usual boxes to reset the values of various parameters, there will also be text boxes where you can enter mathematical formulas that describe an Exhibit of the Gallery. As usual, these will have default values, and you can just click OK and see the Exhibit these defaults describe. But of course it is more interesting to replace these default formulas by formulas of your own. For example, if in the Plane Curves Gallery, you select User Plane Curve (parametric), the default equations are x(t) = a * cos(t) and y(t) = b * sin(2*t) (where a has the value 3 and b the value 1.5). If you accept these values by clicking OK you will get a "figure eight", but if you replace sin(2*t) by sin(t) you will get an ellipse.

By the way, once you have created a user-defined exhibit, it you want to edit your formulas and parameters, do not re-select the User Defined Exhibit from the Exhibit menu -- this will give back the original defaults. Instead, select "Change User Data..." from the Settings menu.