In math mode, type    and bring up the Plot Builder from the Context Menu. In the section "Select Plot Type and Functions" choose "Interactive Plot with 2 parameters." (See Figure 1.) In the "select Plot" section, the Plot Builder defaults to "2-D plot" but in the "Select Variable Purposes, Ranges, and Plot Options" section, adjust as per Figure 1.

Click on the Options button and in the ensuing dialog, Adjust the "Range from - to" options as shown in Figure 2. Also, check the box for Constrained Scaling, and in the section "Coordinate System", select "polar". Then click either the Preview or Plot buttons to obtain the 2-slider graph shown in Figure 3.

The 2-slider graph produced by the Plot Builder is shown in Figure 3. In exchange for the simplicity with which this tool was built, there are two shortcomings. First, it is a Maplet, so it cannot be saved or stored, and as long as it's "up," there is no access to the underlying copy of Maple. Second, as the sliders are moved, it will become apparent that the "Range from - to" setting applies only to the vertical axis. As the horizontal width of the curve changes, so too does the horizontal axis. It is just not possible to fix both the horizontal and vertical axes of this graph using just the Plot Builder.

Note the plot command in the "Plot Command" window. This command draws a single curve according to the values of  and  set by the sliders. In fact, if the "Done" button is clicked, a single curve such as shown in Figure 4 results. As has already been pointed out, there is no way to "save" the 2-slider tool constructed via the Interactive Parameter Maplet.

If, instead of returning the Interactive Parameter Maplet, the command

is returned. (Either select "On 'Plot" return plot command" - see Figure 1, or press the Command button in the Options panel - see Figure 2.) This suggests examining the interactiveparams command in the plots package.

With a slight modification, the interactiveparams command in the plots package will generate an improved 2-slider graph that bounds the horizontal range as well as the vertical.

Unfortunately, no matter how the Interactive Parameter Maplet is launched, it is still a Maplet, and has the shortcoming of disappearing once it is closed.

The Explore command can be applied to a complete graphing command either as a command, or as an option selected from the Context Menu. The result is a slider-controlled graph resident in a separate worksheet. For example, the following command will first generate the dialog shown in Figure 5, then, in a new worksheet, create the application shown in Figure 6.

Notice that in the dialog Maple has been instructed to skip creating a slider for "constrained." (Obviously, some of the options to the plot command are interpreted as variables.) The default ranges of 0 to 10 have been modified. Pressing the "Explore" button generates (in a new worksheet) the application pictured in Figure 6.

The great advantage of the application produced by the Explore command is that it consists of Embedded Components, so is not subject to the major defect of the Interactive Parameter Maplet: it can be made to "persist." However, there is a trick to this.

It is not enough to copy and paste the application created by the Explore command. When the application is inserted into a new Maple worksheet, code written by the command is executed. This code is what "programs" the sliders. Copying and pasting the application back into the launching worksheet copies this hidden code, but does not re-execute it. Hence, the pasted application will not function.

To re-execute this hidden code for the on-going session, select Execute_Repeat Autoexecute from the Edit menu. The Autoexecute Warning dialog in Figure 7 will appear, a dialog magically suppressed when the application is first created. Clicking "Yes" will execute the hidden code and allow the pasted application to function. Autoexecute for this worksheet has been suppressed by selecting Autoexecute_Clear All from the Format menu.)

But where is this code hidden? It's in autoexecute regions, regions that can be found if the "Markers" option has been selected in the View menu. This opens a column on the left of the workspace where opposing triangles delineate Document Blocks. The Document Block corresponding to the blank line beneath the header (where it says "Maplesoft" and "Exploration Assistant") contains the bulk of the hidden code. Place the cursor on the Marker column at the location of this Document Block, and right-click. In the pop-up, select the option "Expand Document Block" to see the code. Of course, going back to the Marker column and selecting "Collapse Document Block" will again hide this code. There is also autoexecute code in the Document Blocks containing the two sliders. This code simply prints the slider labels  and .

Right-clicking on the Marker for a Document Block that contains autoexecute code provides the Autoexecute option. Selecting the "Set" option and saving the worksheet means that when the worksheet is relaunched, the Autoexecute dialog shown in Figure 7 will appear. The Autoexecute option for each of the three Autoexecute regions inherited from the pasted Explore worksheet have been set to "Clear" so that the Autoexecute dialog does not appear upon the initial launch of this worksheet.

The astute reader will realize that the Explore functionality is an efficient way to generate a variety of applications controlled by sliders. Simply create the application with the Explore command (or select it from the Context Menu), copy/paste the application back into the launching worksheet, and activate its code by selecting Execute_Repeat Autoexecute from the Edit menu.

Of course, the 2-slider graph generated with the Explore command could be built interactively by inserting and coding the appropriate components from the Components palette. One advantage of such a construction would be a bit more flexibility over what is to be graphed. Simply add a MathContainer into which a function can be entered, and code the sliders to read this component for its content. However, building a robust graphing application means providing access to a multitude of plotting options. Just look at the Options panel in the Plot Builder to see just how many options have to be addressed. Each option requires at least one component, and each such component takes up space on the screen. The Plot Builder, which uses Maplet technology, handles the space constraint with secondary Maplets. This option is not yet available for embedded components, so screen space becomes the limiting factor in the design and construction of applications based on this technology.

For these reasons, we pass over the option of building a 2-slider graph with the Component palette, and proceed to an investigation of the conchoid of a curve.