7.1 Two-Dimensional Graphics

The FriCAS two-dimensional graphics package provides the ability to graphics:two-dimensional display

  • curves defined by functions of a single real variable
  • curves defined by parametric equations
  • implicit non-singular curves defined by polynomial equations
  • planar graphs generated from lists of point components.

These graphs can be modified by specifying various options, such as calculating points in the polar coordinate system or changing the size of the graph viewport window.

7.1.1 Plotting Two-Dimensional Functions of One Variable

curve:one variable function The first kind of two-dimensional graph is that of a curve defined by a function y=f(x) over a finite interval of the x axis.

The general format for drawing a function defined by a formula f(x) is:

draw(f(x), x = a..b, options)

where a..b defines the range of x, and where options prescribes zero

or more options as described in ugGraphTwoDOptions . An example of an option is curveColor==brightred(). An alternative format involving functions f and g is also available.

A simple way to plot a function is to use a formula. The first argument is the formula. For the second argument, write the name of the independent variable (here, x), followed by an =, and the range of values.

Display this formula over the range 0≤x≤6. FriCAS converts your formula to a compiled function so that the results can be computed quickly and efficiently.

draw(sin(tan(x)) - tan(sin(x)),x = 0..6)

Once again the formula is converted to a compiled function before any points were computed. If you want to graph the same function on several intervals, it is a good idea to define the function first so that the function has to be compiled only once.

This time we first define the function.

f(x) == (x-1)*(x-2)*(x-3)

Type: Void

To draw the function, the first argument is its name and the second is just the range with no independent variable.

draw(f, 0..4)

7.1.2 Plotting Two-Dimensional Parametric Plane Curves

The second kind of two-dimensional graph is that of parametric plane curve curves produced by parametric equations. curve:parametric plane Let x=f(t) and y=g(t) be formulas or two functions f and g as the parameter t ranges over an interval [a,b]. The function curve takes the two functions f and g as its parameters.

The general format for drawing a two-dimensional plane curve defined by parametric formulas x=f(t) and y=g(t) is:

draw(curve(f(t), g(t)), t = a..b, options)

where a..b defines the range of the independent variable t, and where options prescribes zero or more options as described in ugGraphThreeDOptions . An example of an option is curveColor==brightred().

Here’s an example:

Define a parametric curve using a range involving %pi, FriCAS’s way of saying π. For parametric curves, FriCAS compiles two functions, one for each of the functions f and g.

draw(curve(sin(t)*sin(2*t)*sin(3*t), sin(4*t)*sin(5*t)*sin(6*t)), t = 0..2*%pi)

The title may be an arbitrary string and is an optional argument to the draw command.

draw(curve(cos(t), sin(t)), t = 0..2*%pi)

If you plan on plotting x=f(t), y=g(t) as t ranges over several intervals, you may want to define functions f and g first, so that they need not be recompiled every time you create a new graph. Here’s an example:

As before, you can first define the functions you wish to draw.

f(t:DFLOAT):DFLOAT == sin(3*t/4)
Function declaration f : DoubleFloat -> DoubleFloat has been
   added to workspace.

Type: Void

FriCAS compiles them to map DoubleFloat values to DoubleFloat values.

g(t:DFLOAT):DFLOAT == sin(t)
Function declaration f : DoubleFloat -> DoubleFloat has been added
   to workspace.

Type: Void

Give to curve the names of the functions, then write the range without the name of the independent variable.


Here is another look at the same curve but over a different range. Notice that f and g are not recompiled. Also note that FriCAS provides a default title based on the first function specified in curve.


7.1.3 Plotting Plane Algebraic Curves

A third kind of two-dimensional graph is a non-singular solution curve curve:plane algebraic in a rectangular region of the plane. A solution curve is a curve defined by a polynomial equation p(x,y)=0. plane algebraic curve Non-singular means that the curve is smooth in that it does not cross itself or come to a point (cusp). Algebraically, this means that for any point (x,y) on the curve, that is, a point such that p(x,y)=0, the partial derivatives ∂p∂x(x,y) and ∂p∂y(x,y) are not both zero. curve:smooth curve:non-singular smooth curve non-singular curve

The general format for drawing a non-singular solution curve given by a polynomial of the form p(x,y)=0 is:

draw(p(x,y) = 0, x, y, range == [a..b, c..d], options)

where the second and third arguments name the first and second

independent variables of p. A range option is always given to designate a bounding rectangular region of the plane a≤x≤b,c≤y≤d. Zero or more additional options as described in ugGraphTwoDOptions may be given.

We require that the polynomial has rational or integral coefficients. Here is an algebraic curve example (Cartesian ovals): Cartesian:ovals

p := ((x^2 + y^2 + 1) - 8*x)^2 - (8*(x^2 + y^2 + 1)-4*x-1)

Type: Polynomial Integer

The first argument is always expressed as an equation of the form p=0 where p is a polynomial.

draw(p = 0, x, y, range == [-1..11, -7..7])

7.1.4 Two-Dimensional Options

The draw commands take an optional list of options, such as title shown above. Each option is given by the syntax: name == value. Here is a list of the available options in the order that they are described below.

adaptive clip unit
clip curveColor range
toScale pointColor coordinates

The adaptive option turns adaptive plotting on or off. adaptive plotting Adaptive plotting uses an algorithm that traverses a graph and computes more points for those parts of the graph with high curvature. The higher the curvature of a region is, the more points the algorithm computes. graphics:2D options:adaptive

The adaptive option is normally on. Here we turn it off.

draw(sin(1/x),x=-2*%pi..2*%pi, adaptive == false)

The clip option turns clipping on or off. graphics:2D options:clipping If on, large values are cut off according to clipPointsDefaultclipPointsDefaultGraphicsDefaults.

draw(tan(x),x=-2*%pi..2*%pi, clip == true)

Option toScale does plotting to scale if true or uses the entire viewport if false. The default can be determined using drawToScaledrawToScaleGraphicsDefaults. graphics:2D options:to scale

draw(sin(x),x=-%pi..%pi, toScale == true, unit == [1.0,1.0])

Option clip with a range sets point clipping of a graph within the graphics:2D options:clip in a range ranges specified in the list [xrange,yrange]. clipping If only one range is specified, clipping applies to the y-axis.

draw(sec(x),x=-2*%pi..2*%pi, clip == [-2*%pi..2*%pi,-%pi..%pi], unit == [1.0,1.0])

Option curveColor sets the color of the graph curves or lines to be the graphics:2D options:curve color indicated palette color curve:color (see ugGraphColor and ugGraphColorPalette ). color:curve

draw(sin(x),x=-%pi..%pi, curveColor == bright red())

Option pointColor sets the color of the graph points to the indicated graphics:2D options:point color palette color (see ugGraphColor and ugGraphColorPalette ). color:point

draw(sin(x),x=-%pi..%pi, pointColor == pastel yellow())

Option unit sets the intervals at which the axis units are plotted graphics:2D options:set units according to the indicated steps [ x interval, y interval].

draw(curve(9*sin(3*t/4),8*sin(t)), t = -4*%pi..4*%pi, unit == [2.0,1.0])

Option range sets the range of variables in a graph to be within the ranges graphics:2D options:range for solving plane algebraic curve plots.

draw(y^2 + y - (x^3 - x) = 0, x, y, range == [-2..2,-2..1], unit==[1.0,1.0])

A second example of a solution plot.

draw(x^2 + y^2 = 1, x, y, range == [-3/2..3/2,-3/2..3/2], unit==[0.5,0.5])

Option coordinates indicates the coordinate system in which the graph graphics:2D options:coordinates is plotted. The default is to use the Cartesian coordinate system. Cartesian:coordinate system For more details, see ugGraphCoord or CoordinateSystems. coordinate system:Cartesian

draw(curve(sin(5*t),t),t=0..2*%pi, coordinates == polar)

7.1.5 Color

The domain Color Color provides operations for manipulating graphics:color colors in two-dimensional graphs. color Colors are objects of Color. Each color has a hue and a weight. hue Hues are represented by integers that range from 1 to the numberOfHues()numberOfHues()Color, normally graphics:color:number of hues 27. weight Weights are floats and have the value 1.0 by default.

color (integer)
creates a color of hue integer and weight 1.0.
hue (color)
returns the hue of color as an integer. graphics:color:hue function
red ()
blue(), green(), and yellow() graphics:color:primary color functions create colors of that hue with weight 1.0.
color1 + color2
returns the color that results from additively combining the indicated color1 and color2. Color addition is not commutative: changing the order of the arguments produces different results.
integer * color
changes the weight of color by integer without affecting its hue. graphics:color:multiply function For example, red()+3*yellow() produces a color closer to yellow than to red. Color multiplication is not associative: changing the order of grouping color:multiplication produces different results.

These functions can be used to change the point and curve colors for two- and three-dimensional graphs. Use the pointColor option for points.

draw(x^2,x=-1..1,pointColor == green())

Use the curveColor option for curves.

draw(x^2,x=-1..1,curveColor == color(13) + 2*blue())

7.1.6 Palette


Domain Palette is the domain of shades of colors: dark, dim, bright, pastel, and light, designated by the integers 1 through 5, respectively. Palette

Colors are normally bright.

shade red()

Type: PositiveInteger

To change the shade of a color, apply the name of a shade to it. color:shade shade

myFavoriteColor := dark blue()

Type: Palette

The expression shade(color) returns the value of a shade of color.

shade myFavoriteColor

Type: PositiveInteger

The expression hue(color) returns its hue.

hue myFavoriteColor

Type: Color

Palettes can be used in specifying colors in two-dimensional graphs.

draw(x^2,x=-1..1,curveColor == dark blue())

7.1.7 Two-Dimensional Control-Panel

graphics:2D control-panel Once you have created a viewport, move your mouse to the viewport and click with your left mouse button to display a control-panel. The panel is displayed on the side of the viewport closest to where you clicked. Each of the buttons which toggle on and off show the current state of the graph. Transformations

graphics:2D control-panel:transformations

Object transformations are executed from the control-panel by mouse-activated potentiometer windows.

To scale a graph, click on a mouse button graphics:2D control-panel:scale within the Scale window in the upper left corner of the control-panel. The axes along which the scaling is to occur are indicated by setting the toggles above the arrow. With X On and Y On appearing, both axes are selected and scaling is uniform. If either is not selected, for example, if X Off appears, scaling is non-uniform.
To translate a graph, click the mouse in the graphics:2D control-panel:translate Translate window in the direction you wish the graph to move. This window is located in the upper right corner of the control-panel. Along the top of the Translate window are two buttons for selecting the direction of translation. Translation along both coordinate axes results when X On and Y On appear or along one axis when one is on, for example, X On and Y Off appear. Messages

graphics:2D control-panel:messages

The window directly below the transformation potentiometer windows is

used to display system messages relating to the viewport and the control-panel. The following format is displayed:

[scaleX, scaleY] >graph < [translateX, translateY]

The two values to the left show the scale factor along the X and Y coordinate axes. The two values to the right show the distance of translation from the center in the X and Y directions. The number in the center shows which graph in the viewport this data pertains to. When multiple graphs exist in the same viewport, the graph must be selected (see Multiple Graphs, below) in order for its transformation data to be shown, otherwise the number is 1. Multiple Graphs

graphics:2D control-panel:multiple graphs The Graphs window contains buttons that allow the placement of two-dimensional graphs into one of nine available slots in any other two-dimensional viewport. In the center of the window are numeral buttons from one to nine that show whether a graph is displayed in the viewport. Below each number button is a button showing whether a graph that is present is selected for application of some transformation. When the caret symbol is displayed, then the graph in that slot will be manipulated. Initially, the graph for which the viewport is created occupies the first slot, is displayed, and is selected.

The Clear button deselects every viewport graph slot. graphics:2D control-panel:clear A graph slot is reselected by selecting the button below its number.
The Query button is used to display the scale and graphics:2D control-panel:query translate data for the indicated graph. When this button is selected the message Click on the graph to query appears. Select a slot number button from the Graphs window. The scaling factor and translation offset of the graph are then displayed in the message window.
The Pick button is used to select a graph graphics:2D control-panel:pick to be placed or dropped into the indicated viewport. When this button is selected, the message Click on the graph to pick appears. Click on the slot with the graph number of the desired graph. The graph information is held waiting for you to execute a Drop in some other graph.
Once a graph has been picked up using the Pick button, graphics:2D control-panel:drop the Drop button places it into a new viewport slot. The message Click on the graph to drop appears in the message window when the Drop button is selected. By selecting one of the slot number buttons in the Graphs window, the graph currently being held is dropped into this slot and displayed. Buttons

graphics:2D control-panel:buttons

turns the coordinate axes on or off. graphics:2D control-panel:axes
turns the units along the x and y axis on or off. graphics:2D control-panel:units
encloses the area of the viewport graph in a bounding box, or removes the box if already enclosed. graphics:2D control-panel:box
turns on or off the display of points. graphics:2D control-panel:points
turns on or off the display of lines connecting points. graphics:2D control-panel:lines
writes the current viewport contents to graphics:2D control-panel:ps a file axiom2D.ps or to a name specified in the user’s graphics:.Xdefaults:PostScript file name .Xdefaults file. file:.Xdefaults @ .Xdefaults The file is placed in the directory from which FriCAS or the viewAlone program was invoked. PostScript
resets the object transformation characteristics and attributes back to their initial states. graphics:2D control-panel:reset
makes the control-panel disappear. graphics:2D control-panel:hide
queries whether the current viewport graphics:2D control-panel:quit session should be terminated.

7.1.8 Operations for Two-Dimensional Graphics

Here is a summary of useful FriCAS operations for two-dimensional graphics. Each operation name is followed by a list of arguments. Each argument is written as a variable informally named according to the type of the argument (for example, integer). If appropriate, a default value for an argument is given in parentheses immediately following the name.

adaptive ([boolean(true)])
adaptive plotting sets or indicates whether graphs are plotted graphics:set 2D defaults:adaptive according to the adaptive refinement algorithm.
axesColorDefault ([color(dark blue())])
sets or indicates the default color of the graphics:set 2D defaults:axes color axes in a two-dimensional graph viewport.
clipPointsDefault ([boolean(false)])
sets or indicates whether point clipping is graphics:set 2D defaults:clip points to be applied as the default for graph plots.
drawToScale ([boolean(false)])
sets or indicates whether the plot of a graph graphics:set 2D defaults:to scale is to scale or uses the entire viewport space as the default.
lineColorDefault ([color(pastel yellow())])
sets or indicates the default color of the graphics:set 2D defaults:line color lines or curves in a two-dimensional graph viewport.
maxPoints ([integer(500)])
sets or indicates the default maximum number of graphics:set 2D defaults:max points possible points to be used when constructing a two-dimensional graph.
minPoints ([integer(21)])
sets or indicates the default minimum number of graphics:set 2D defaults:min points possible points to be used when constructing a two-dimensional graph.
pointColorDefault ([color(bright red())])
sets or indicates the default color of the graphics:set 2D defaults:point color points in a two-dimensional graph viewport.
pointSizeDefault ([integer(5)])
sets or indicates the default size of the graphics:set 2D defaults:point size dot used to plot points in a two-dimensional graph.
screenResolution ([integer(600)])
sets or indicates the default screen graphics:set 2D defaults:screen resolution resolution constant used in setting the computation limit of adaptively adaptive plotting generated curve plots.
unitsColorDefault ([color(dim green())])
sets or indicates the default color of the graphics:set 2D defaults:units color unit labels in a two-dimensional graph viewport.
viewDefaults ()
resets the default settings for the following graphics:set 2D defaults:reset viewport attributes: point color, line color, axes color, units color, point size, viewport upper left-hand corner position, and the viewport size.
viewPosDefault ([list([100,100])])
sets or indicates the default position of the graphics:set 2D defaults:viewport position upper left-hand corner of a two-dimensional viewport, relative to the display root window. The upper left-hand corner of the display is considered to be at the (0, 0) position.
viewSizeDefault ([list([200,200])])
sets or indicates the default size in which two graphics:set 2D defaults:viewport size dimensional viewport windows are shown. It is defined by a width and then a height.
viewWriteAvailable ([list([“pixmap”,”bitmap”, “postscript”, “image”])])
indicates the possible file types graphics:2D defaults:available viewport writes that can be created with the writewriteTwoDimensionalViewport function.
viewWriteDefault ([list([])])
sets or indicates the default types of files, in graphics:set 2D defaults:write viewport addition to the data file, that are created when a write function is executed on a viewport.
units (viewport, integer(1), string(“off”))
turns the units on or off for the graph with index integer.
axes (viewport, integer(1), string(“on”))
turns the axes on graphics:2D commands:axes or off for the graph with index integer.
close (viewport)
closes viewport. graphics:2D commands:close
connect (viewport, integer(1), string(“on”))
declares whether lines graphics:2D commands:connect connecting the points are displayed or not.
controlPanel (viewport, string(“off”))
declares whether the two-dimensional control-panel is automatically displayed or not.
graphs (viewport)
returns a list graphics:2D commands:graphs describing the state of each graph. If the graph state is not being used this is shown by “undefined”, otherwise a description of the graph’s contents is shown.
graphStates (viewport)
displays graphics:2D commands:state of graphs a list of all the graph states available for viewport, giving the values for every property.
key (viewport)
returns the process graphics:2D commands:key ID number for viewport.
move (viewport, integerx(viewPosDefault), integery(viewPosDefault))
moves viewport on the screen so that the graphics:2D commands:move upper left-hand corner of viewport is at the position (x,y).
options (viewport)
returns a list graphics:2D commands:options of all the DrawOptions used by viewport.
points (viewport, integer(1), string(“on”))
specifies whether the graph points for graph integer are graphics:2D commands:points to be displayed or not.
region (viewport, integer(1), string(“off”))
declares whether graph integer is or is not to be displayed with a bounding rectangle.
reset (viewport)
resets all the properties of viewport.
resize (viewport, integerwidth, integerheight)
graphics:2D commands:resize resizes viewport with a new width and height.
scale (viewport, integern(1), integerx(0.9), integery(0.9))
scales values for the graphics:2D commands:scale x and y coordinates of graph n.
show (viewport, integern(1), string(“on”))
indicates if graph n is shown or not.
title (viewport, string(“FriCAS 2D”))
designates the title for viewport.
translate (viewport, integern(1), floatx(0.0), floaty(0.0))
graphics:2D commands:translate causes graph n to be moved x and y units in the respective directions.
write (viewport, stringdirectory, [strings])
if no third argument is given, writes the data file onto the directory with extension data. The third argument can be a single string or a list of strings with some or all the entries “pixmap”, “bitmap”, “postscript”, and “image”.

7.1.9 Addendum: Building Two-Dimensional Graphs

In this section we demonstrate how to create two-dimensional graphs from lists of points and give an example showing how to read the lists of points from a file. Creating a Two-Dimensional Viewport from a List of Points

FriCAS creates lists of points in a two-dimensional viewport by utilizing the GraphImage and TwoDimensionalViewport domains. In this example, the makeGraphImagemakeGraphImageGraphImage function takes a list of lists of points parameter, a list of colors for each point in the graph, a list of colors for each line in the graph, and a list of sizes for each point in the graph.

The following expressions create a list of lists of points which will be read by FriCAS and made into a two-dimensional viewport.

p1 := point [1,1]$(Point DFLOAT)

Type: Point DoubleFloat

p2 := point [0,1]$(Point DFLOAT)

Type: Point DoubleFloat

p3 := point [0,0]$(Point DFLOAT)

Type: Point DoubleFloat

p4 := point [1,0]$(Point DFLOAT)

Type: Point DoubleFloat

p5 := point [1,.5]$(Point DFLOAT)

Type: Point DoubleFloat

p6 := point [.5,0]$(Point DFLOAT)

Type: Point DoubleFloat

p7 := point [0,0.5]$(Point DFLOAT)

Type: Point DoubleFloat

p8 := point [.5,1]$(Point DFLOAT)

Type: Point DoubleFloat

p9 := point [.25,.25]$(Point DFLOAT)

Type: Point DoubleFloat

p10 := point [.25,.75]$(Point DFLOAT)

Type: Point DoubleFloat

p11 := point [.75,.75]$(Point DFLOAT)

Type: Point DoubleFloat

p12 := point [.75,.25]$(Point DFLOAT)

Type: Point DoubleFloat

Finally, here is the list.

llp := [ [p1,p2], [p2,p3], [p3,p4], [p4,p1], [p5,p6], [p6,p7], [p7,p8],

[p8,p5], [p9,p10], [p10,p11], [p11,p12], [p12,p9] ]


Type: List List Point DoubleFloat

Now we set the point sizes for all components of the graph.

size1 := 6::PositiveInteger

Type: PositiveInteger

size2 := 8::PositiveInteger

Type: PositiveInteger

size3 := 10::PositiveInteger
lsize := [size1, size1, size1, size1, size2, size2, size2, size2, size3,

size3, size3, size3]


Type: List Polynomial Integer

Here are the colors for the points.

pc1 := pastel red()

Type: Palette

pc2 := dim green()

Type: Palette

pc3 := pastel yellow()

Type: Palette

lpc := [pc1, pc1, pc1, pc1, pc2, pc2, pc2, pc2, pc3, pc3, pc3, pc3]

Type: List Palette

Here are the colors for the lines.

lc := [pastel blue(), light yellow(), dim green(), bright red(), light

green(), dim yellow(), bright blue(), dark red(), pastel red(), light blue(), dim green(), light yellow()]


Type: List Palette

Now the GraphImage is created according to the component specifications indicated above.

g := makeGraphImage(llp,lpc,lc,lsize)$GRIMAGE

The makeViewport2DmakeViewport2DTwoDimensionalViewport function now creates a TwoDimensionalViewport for this graph according to the list of options specified within the brackets.


This example demonstrates the use of the GraphImage functions componentcomponentGraphImage and appendPointappendPointGraphImage in adding points to an empty GraphImage.

)clear all
g := graphImage()$GRIMAGE

Type: GraphImage

p1 := point [0,0]$(Point DFLOAT)

Type: Point DoubleFloat

p2 := point [.25,.25]$(Point DFLOAT)

Type: Point DoubleFloat

p3 := point [.5,.5]$(Point DFLOAT)

Type: Point DoubleFloat

p4 := point [.75,.75]$(Point DFLOAT)

Type: Point DoubleFloat

p5 := point [1,1]$(Point DFLOAT)

Type: Point DoubleFloat


Type: Void


Type: Void


Type: Void


Type: Void


Type: Void

g1 := makeGraphImage(g)$GRIMAGE

Here is the graph.

makeViewport2D(g1,[title(“Graph Points”)])$VIEW2D

A list of points can also be made into a GraphImage by using the operation coercecoerceGraphImage. It is equivalent to adding each point to g2 using componentcomponentGraphImage.

g2 := coerce([ [p1],[p2],[p3],[p4],[p5] ])$GRIMAGE

Now, create an empty TwoDimensionalViewport.

v := viewport2D()$VIEW2D
options(v,[title("Just Points")])$VIEW2D

Place the graph into the viewport.


Take a look.

makeViewport2D(v)$VIEW2D Creating a Two-Dimensional Viewport of a List of Points from a File

The following three functions read a list of points from a file and then draw the points and the connecting lines. The points are stored in the file in readable form as floating point numbers (specifically, DoubleFloat values) as an alternating stream of x- and y-values. For example,

0.0 0.0     1.0 1.0     2.0 4.0
3.0 9.0     4.0 16.0    5.0 25.0
drawPoints(lp:List Point DoubleFloat):VIEW2D ==
  g := graphImage()$GRIMAGE
  for p in lp repeat
  gi := makeGraphImage(g)$GRIMAGE
drawLines(lp:List Point DoubleFloat):VIEW2D ==
  g := graphImage()$GRIMAGE
  component(g, lp, pointColorDefault(), lineColorDefault(),
  gi := makeGraphImage(g)$GRIMAGE
plotData2D(name, title) ==
  f:File(DFLOAT) := open(name,"input")
  lp:LIST(Point DFLOAT) := empty()
  while ((x := readIfCan!(f)) case DFLOAT) repeat
    y : DFLOAT := read!(f)
    lp := cons(point [x,y]$(Point DFLOAT), lp)

This command will actually create the viewport and the graph if the point data is in the file “file.data”.

plotData2D("file.data", "2D Data Plot")

7.1.10 Addendum: Appending a Graph to a Viewport Window Containing a Graph

This section demonstrates how to append a two-dimensional graph to a viewport already containing other graphs. The default draw command places a graph into the first GraphImage slot position of the TwoDimensionalViewport.

This graph is in the first slot in its viewport.

v1 := draw(sin(x),x=0..2*%pi)

So is this graph.

v2 := draw(cos(x),x=0..2*%pi, curveColor==light red())

The operation getGraphgetGraphTwoDimensionalViewport retrieves the GraphImage g1 from the first slot position in the viewport v1.

g1 := getGraph(v1,1)

Now putGraphputGraphTwoDimensionalViewport places g1 into the the second slot position of v2.


Display the new TwoDimensionalViewport containing both graphs.