Classroom Tips and Techniques: Shading between Curves, and Integrating Vectors Componentwise 

Robert J. Lopez
Emeritus Professor of Mathematics and Maple Fellow
Maplesoft 

Introduction 

This month, we will provide a method for shading between two arbitrary curves, and will show how to integrate a vector componentwise.  The first issue is a long-standing one in Maple.  Maple can shade between a curve and the horizontal axis, but cannot easily provide shading between two curves.  Over the years, a variety of methods have been proposed by different users, methods typically requiring special coding to implement.  In Maple 11, the implicitplot command now supports the option filledregions = true/false, and we will show how this new option can be used to shade between curves. 

Our second topic was the subject of a recent inquiry by a colleague I have know for many years.  He's new to the use of the VectorCalculus package, and wanted to know how to integrate componentwise a vector in this package. 

Initializations 

 

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Shading between Curves 

Shading with the plot Command 

Maple's plot command has long had the functionality of shading between a curve and the horizontal axis.  However, it has never had the ability to shade between two arbitrary curves.  Consider the curves defined by the functions 

>

 

The -coordinates of the intersections of these curves are given by 

>

 
Our ultimate objective will be to explore ways to shade the region bounded by these two curves and the vertical lines and .  Maple easily shades from the graph of to the -axis, as we see in Figure 1. 

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Figure 1   Shading between the graph of and the -axis

Unfortunately, provision was never made for displaying the curve in a color different from the color of the shading.  To show the curve in black, for example, takes a superposition of the shaded and unshaded versions of the graph, as we see in Figure 2. 

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Figure 2   Graph of shaded between the curve and the -axis

In Figure 3, we show what happens if we include the graph of , along with shading between this curve and the -axis.  Note the use of the transparency option, without which the pink shading would cover the cyan, making the cyan invisible. 

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Figure 3   Shaded graphs of and

Although the region between the bounding curves is shaded in pink, the image in Figure 3 suffers from the cyan shading between the graph of and the -axis.  In Figure 4, we deal with this by shading the region under the graph of in white. 

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Figure 4   Shading between the graphs of and

Of course, the graphs of the curves themselves can be made visible by adding them to Figure 4.  This is done in Figure 5. 

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Figure 5   Graphs of and with shading of the region between the curves

Shading with the implicitplot Command 

The following remarks should clarify some aspects of the behavior of the implicitplot command. 

The implicitplot command can be applied to an expression, an equation, or a single inequality. 

As per Table 1, setting the filledregions option to true causes implicitplot to shade regions in the graph.   

Rule 1	Expression	Region satisfying shaded one color; satisfying another.
Rule 2	Equation	All terms are moved to the left.  Calling this left side the shading is as in Rule 1.
Rule 3	Inequality	Just the region satisfying the inequality is shaded.
Table 1   Rules governing the behavior of the filledregions option

These basic functionalities are illustrated in Figure 6, where the options 

>

are used. 

>	>	>
Figure 6   Rules in Table 1 are illustrated respectively in Figures 6(a), 6(b), and 6(c)

To extract from these basic functionalities the ability to shade between two arbitrary curves, write the function 

>

where and are as defined in the previous section.  Since the filledregions option will shade the region where in red and the region where in yellow, Figure 7 shows in red the region satisfying and .  The first factor in is therefore negative; the second, positive.  Hence, the region between the two curves is graph in red because on this region In the upper yellow region, because the inequalities and are both satisfied.  In the lower yellow region, again because now the inequalities and  

 Figure 8 sets the colors to red and white so that just the region between the curves is shaded. 

>	>
Figure 7   Shading between and via the device of plotting the product	Figure 8   The plot in Figure 7 with modified color scheme

Other Solutions 

 

Here are two other ways to shade between two curves.  The first is the command FillBetweenCurves generated by the following Maple code.  It was written by one of the Maple programmers in response to my constant badgering for a complete solution to the the problem of shading between two curves.  It's not built into Maple because it is as yet only a partial solution to the problem. 

 

>	FillBetweenCurves := proc(f1, f2, {[color, colour] := red, transparency := 0.0})

 

>	local p, c, n, pts1, pts2, polys, i, x, ytop, ybottom, yvals;

 

>

 

>	p := plot([f1, f2], _rest, _options['color'], adaptive=false);

 

>	c := select(type, p, 'specfunc'('anything', 'CURVES'));

 

>	if nops(c) <> 2 then

 

>	error "cannot fill between curves";

 

>

end if;

 

>

 

>	pts1 := op([1,1], c);

 

>	pts2 := op([2,1], c);

 

>	n := nops(pts1);

 

>	if n <> nops(pts2) then

 

>	error "cannot fill between curves";

 

>

end if;

 

>

 

>	polys := NULL;

 

>	x := Vector(n, datatype=float);

 

>	ytop := Vector(n, datatype=float);

 

>	ybottom := Vector(n, datatype=float);

 

>	for i to n do

 

>	x[i] := op(1, pts1[i]);

 

>	yvals := op(2, pts1[i]), op(2, pts2[i]);

 

>	ytop[i] := max(yvals);

 

>	ybottom[i] := min(yvals);

 

>

end do;

 

>

 

>	polys := [seq([x[i], ytop[i]], i=1..n), seq([x[i], ybottom[i]], i=n..1, -1),

 

>	[x[1], ytop[1]]];

 

>	plots[display](plottools[polygon](polys, _options['color'],

 

>	_options['transparency']), p);

 

>

end proc:

 

An example of its use is provided by Figure 9. 

 

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Figure 9   Shading between and via FillBetweenCurves command

On obvious shortcoming of this FillBetweenCurves command is that the curves are colored with the same color as the shading. 

A second solution is provided by the inequalityplot command in the InequalityGraphics package written by Robert Ipanaqu? Chero, available from the Maple Applications Center (InequalityGraphics), and described in the Maple Reporter article A Package for Graphing Solution Sets of Nonlinear Inequalities, also available from the Maple Applications Center (Reporter Article).  The interested reader can follow these links to download the articles and the additional Maple code.