6.1 Introduction

There are many different ways to display numerical results and these different ways depend on whether the data are discrete, are obtained from expressions, require logarithmic compression, are to be represented in 2D or 3D, or are specific to a given application area such as statistics (histograms, bar charts, and pie charts), computational geometry (Voronoi diagrams and convex hulls), wavelet analysis, or controls (Bode, Nichols, and root locus plots). To accommodate these wide-ranging needs, Mathematica provides a large set of high-level plotting commands that require little user involvement to generate a basic figure; that is, one with minimum annotation. However, for each plot command, one is given the means to control virtually all aspects of the graph, thereby providing a high degree of flexibility in enhancing a figure. We shall discuss a subset of these plot commands and introduce instructions that can be used to modify, enhance, and individualize the graph’s curves and the overall figure.

Mathematica also provides a set of straightforward commands for creating an interactive environment for the presentation of numbers, expressions, and graphics by using Manipulate. This command is introduced in Chapter 7.

The introduction and usage of the various 2D and 3D plotting commands and their enhancement are introduced, primarily, via tables. The usage of these enhancements is then illustrated by examples from engineering topics.

6.2 2D Graphics

6.2.1 Basic Plotting

The basic forms of the 2D plotting commands that will be introduced in this chapter are presented in Table 6.1. They are divided into four categories: those that display the numerical evaluation of expressions using linear axes and those using logarithmic axes; and those that display lists of data using linear axes and those using logarithmic axes. It is mentioned that each plotting command that numerically evaluates an expression uses an internal procedure to determine the number and spacing of the values of the independent variable. These values are determined in a recursive fashion and the number of recursions that are used is, by default, determined automatically. However, if one wants to specify the number of recursions to n recursions, where n is a positive integer, this is done with the option MaxRecursion->n which is one of the optional arguments in the plotting command. Examples of the use of the 2D plot commands in Table 6.1 are shown in Tables 6.2 to 6.4.

Plotting type	Mathematica function*
Linear axes
Basic	Plot[{f,g,…},{x,xs,xe},enh]
Parametric	ParametricPlot[{f,g},{x,xs,xe}},enh]
Polar	PolarPlot[r,{th,ths,the}},enh]
Contour	ContourPlot[h,{x,xs,xe},{y,ys,ye}},enh]
Region	RegionPlot[reg+,{x,xs,xe},{y,ys,ye}},enh]
Plotting expressions with logarithmic axes
y-axis logarithmic, x-axis linear	LogPlot[{f,g,…},{x,xs,xe}},enh]
y-axis linear, x-axis logarithmic	LogLinearPlot[{f,g,…},{x,xs,xe}},enh]
y-axis logarithmic, x-axis logarithmic	LogLogPlot[{f,g,…},{x,xs,xe}},enh]
Plotting of lists of coordinate pairs with linear axes
Basic: plot points only	ListPlot[{list1,list2,…} },enh]
Basic: connect data values	ListLinePlot[{list1,list2,…} },enh]
Polar	ListPolarPlot[{listr1,listr2,…} },enh]
Contour	ListContourPlot[lst,enh]
Plotting lists of coordinate pairs with logarithmic axes
y-axis logarithmic, x-axis linear	ListLogPlot[{list1,list2,…} },enh]
y-axis linear, x-axis logarithmic	ListLogLinearPlot[{list1,list2,…} },enh]
y-axis logarithmic, x-axis logarithmic	ListLogLogPlot[{list1,list2,…} },enh]

^af = f(x), g = g(x); r = r(θ); h = h(x,y);

list1, list2,  … , are lists of Cartesian coordinate pairs {{x1,y1},{x2,y2},…} ;

listr1 , listr2 , … , are lists of polar coordinate pairs {{th1,r1},{th2,r2},…} ;

xs or ys or ths = start value;

xe or ye or the = end value;

lst = {{x1,y1,f1},{x2,y2,f2},…}

^b reg = is a set of relational equations each of which is separated by && ; reg is plotted only for those values of x and y that satisfy the set of relational equations.

Plot command	Expressions plotted and the plotting program	Figure created
Plot	f[t_]:=Exp[-0.15 t] Sin[t] Plot[{f[t],f’[t]},{t,0,25},     PlotRange->All]
ParametricPlot	f[t_]:=Exp[-0.15 t] Sin[t] ParametricPlot[{f[t],f’[t]},     {t,0,25},PlotRange->All]
PolarPlot	r(θ) = |J0(2π(0.6 − cos θ))| r[θ_]:=Abs[BesselJ[0,     2 (0.6-Cos[ θ])]] PolarPlot[r[θ],{ θ,0,2 }]
ContourPlot	w(x, y) = sin (2πx)sin (3πy) w[x_,y_]:=Sin[2 x] *     Sin[3 y] ContourPlot[w[x,y],     {x,0,1},{y,0,1}]
RegionPlot	13cos 2(x + y) ⩽ 92(x − 3)2 + 0.3(y − 1)2 ⩽ 3 RegionPlot[     13 Cos[x+y]^2 <=9&&     2(x-3)^2+0.3 (y-1)^2 <=3,     {x,1.3,4.5},     {y,-2.5,4.5},     AspectRatio->Automatic]

Plot command	Expressions plotted and the plotting program	Figure created
LogPlot	H(Ω) = {(1 − Ω2)2 + (0.1Ω)2}− 1/2 h[_]:=1/Sqrt[(1-^2)^2+     (0.1 )^2] LogPlot[h[],{,0.01,10}]
LogLinearPlot	H(Ω) = {(1 − Ω2)2 + (0.1Ω)2}− 1/2 h[_]:=1/Sqrt[(1-^2)^2+     (0.1 )^2] LogLinearPlot[h[],     {,0.01,10},     PlotRange->All]
LogLogPlot	H(Ω) = {(1 − Ω2)2 + (0.1Ω)2}− 1/2 h[_]:=1/Sqrt[(1-^2)^2+     (0.1 )^2] LogLogPlot[h[],     {,0.01,10}]

Plot command	Expressions used to generate discrete data and the plotting program	Figure created
ListPlot	f[t_]:=Exp[-0.15 t] Sin[t] tau=Range[0,25,0.5]; lst1=Table[{t,f[t]},     {t,tau}]; lst2=Table[{t,f’[t]},     {t,tau}]; ListPlot[{lst1,lst2},     PlotRange->All]
ListLinePlot	f[t_]:=Exp[-0.15 t] Sin[t] tau=Range[0,25,0.5]; lst1=Table[{t,f[t]},     {t,tau}]; lst2=Table[{t,f’[t]},     {t,tau}]; ListLinePlot[{lst1,lst2},     PlotRange->All]

All figures can have their displayed size adjusted by employing the option

…[…, ImageSize->sz,…]

where typically sz is either Tiny, Small, Medium, or Large.

6.2.2 Basic Graph Enhancements

To enhance a figure, the arguments of the basic plotting commands given in Table 6.1 can be augmented with a very large set of optional instructions. We shall introduce several of them in this section and several others in the following sections.

Color

The named color attributes available for equations, text, curves, fill, regions, points, background, and graphics primitives are listed in Table 6.5. In addition to these named colors, each of the main colors can be made lighter by using Lighter or darker by using Darker as indicated in this table.

There is also available a very large number of color schemes for use by such commands as ContourPlot, ListContourPlot, ParamentricPlot, RegionPlot, Plot3D, SphericalPlot3D, and the Filling option in Plot, ListPlot, and others. The color schemes are accessed with the Color Schemes palette obtained from the Palettes menu as shown in Figure 6.1 in either of two ways. The first way is to use

ColorFunction->ColorData["Color scheme name"]

where “Color scheme name” (quotation marks required) can be entered manually from any of the choices appearing in the palettes in any of the four headings: Gradient, Physical, Named, and Indexed. Shown in Figure 6.1 are the first few choices appearing under Gradient. The other choices become visible by scrolling down with the slider. The second way is to use the palette directly by typing the plotting option as

ColorFunction->

Leaving the cursor after the “>”, one goes to the palette menu and clicks on the desired selection and then clicks on Insert. When this is done for the palette selection shown in Figure 6.1, the following will appear

ColorFunction->ColorData["DarkRainbow"]

Curve Characteristics

Mathematica automatically selects the properties of the curves that it displays such that each curve is a different color, but each line is solid and has the same thickness. This automation can be overridden and line colors, line thickness, and the line type – solid or dashed – can be selected for each curve. The instruction to perform these changes is PlotStyle and illustrations of its usage are given in Table 6.6.

PlotStyle->{{curve 1 attribute #1, curve 1 attribute #2,…},                                     {curve 2 attribute #1, curve 2 attribute #2,…},…}*
Option	curve k attribute #m
Dashed lines	Dashing[arg]       arg = Tiny, Small, Medium, or Large ; arg = {} specifies a solid line
Line thickness	Thickness[arg]       arg = Tiny, Small, Medium, or Large or decimal number 0 ≤ p ≤ 0.1†
Color	From Table 6.5
Usage
Default     h[t_,p_]:=Exp[-0.15 t] Sin[t-p]     Plot[{h[t,0],h[t,/3.]},         {t,0,15}]     (Note: Line colors are different)
Modified     h[t_,p_]:=Exp[-0.15 t] Sin[t-p]     Plot[{h[t,0],h[t,/3.]},{t,0,15},         PlotStyle->{{Dashing[{}],             Thickness[0.015],Black},             {Dashing[Small],             Thickness[Large],Red}}]

^aOrder of attributes is arbitrary.

^bTypical region for p.

Text Characteristics

The characteristics that text can have are listed in Table 6.7. These text enhancement attributes can be employed in the text creation portions of labeling the axes (AxisLabel), in creating a legend (PlotLegend), in giving a plot a title (PlotLabel), in framing the figure (FrameLabel), in identifying individual curves (Epilog, Inset, and Tooltip), and in placing text annotation anywhere within a figure (Epilog and Inset).

Option	Instruction
Font size	Large Small Tiny n (For text, equals number of points: default is 12) p (For points, decimal number where typically 0.01 ≤ p ≤ 0.1)
Font attribute	Bold (Can be used with Italic and Underlined) Italic (Can be used with Bold and Underlined) Underlined (Can be used with Bold and Italic) Plain (None of the above: default)
Font style	"Times" "Courier" (default) "Helvetica"

Text can include subscripts and superscripts and other mathematical notation as described in Section 1.5.2, numerical values can have the number of digits specified as described in Section 1.9.2, and the format of equations can be specified as described in Section 1.9.2.

In creating text for labels, Style is used to specify the attributes of the text or an equation. The general means of employing Style is shown in Figure 6.2. Several examples of how Style is used with text and equations are given in Table 6.8. Additional examples of the use of Style are given subsequent examples. Text can also be framed and its background color can be selected using Frame and Background as indicated in Figure 6.2 and shown in Table 6.8.

Mathematica statements	Output
modexpr=Style[Framed["Text"],Blue,12,     Bold,Italic,FontFamily->"Helvetica",         Background->Yellow]
expr=TraditionalForm[Sqrt[x+y^2]]; modexpr=Style[expr,Blue,14,Bold,         FontFamily->"Times"]
expr=NumberForm[Sqrt[5.],6]; modexpr=Style[Framed[expr],14,Bold,         FontFamily->"Times"]
n1=TraditionalForm[Sqrt[1+Cos[k θ]]]; n2=NumberForm[Sin[/5.]^2,4]; label=Row[{3,"=",         n1," and ",         Style["h",Italic],n2}]
n1=TraditionalForm[Sqrt[1+Cos[k θ]]]; n2=NumberForm[Sin[/5.]^2,4]; label=Column[{Row[{3,         " = ",n1}],Row[{         Style["h",Italic],         " = ",n2}]}]
m=1;    n=2;    Ω=3.512; str=StringJoin[ToString[n],ToString[m]] Row[{Ω," = ",NumberForm[^2,4]}]
n=3; ex=Row[{"-",n}]; label="α= 10"ex	α = 10−3

Axes Characteristics

There are two ways in which axes can appear. The first way is for the axes to appear with tick marks and with tick labels. This is the default way in which axes are displayed. The second way is for the axes to appear without tick marks and without labels and instead the entire figure is framed and the tick marks appear on the inside of the frame and the tick labels appear on the outside of the frame. This option is applied by using the option Frame->True. The characteristics of the axes can be changed as shown in Table 6.9 and those for frames as shown in Table 6.10.

		Axes-> Instruction
	Option	Instruction
Axes drawn	Draw axes (Default)	True or omit the option
	Omit axes	False
	Draw only x-axis	{True,False}
	Draw only y-axis	{False,True}
		PlotRange-> Instruction
	Option	Instruction
Axes limits	Use all values	All
	Use most points, omitting outliers	Automatic (default)
	Set specific limits	{{xmin,xmax},{ymin,ymax}}
	AxesStyle->{{x-axis attribute #1, x-axis attribute #2,…}, {y-axis attribute #1, y-axis attribute #2,…}}*
	Option	x-axis or y-axis attribute
Axes characteristics	Change color of axis, ticks, and tick labels	Color from Table 6.5
	Size of tick labels	Font size from Table 6.7
	Tick label style	Font attribute from Table 6.7
	Use default attributes	None or omit the option
	Axis thickness	Thickness[…] (see Table 6.6)
	Axis dashed	Dashing[…] (see Table 6.6)
	AxesLabel->{x-axis instruction, y-axis instruction}†
	Option	x-axis or y-axis instruction
Axes labels (appear at end of each axis)	No label (Default)	{None,None} or omit the option
	Labels	{labelx,labely}†
	Label only x-axis	{labelx,None}†
	Label only y-axis	{None,labely}†
	Ticks->{x-axis instruction, y-axis instruction}
	Option	x-axis or y-axis instruction
Tick mark characteristics and labels	System decides locations and their labels (default)	Automatic
	No tick marks	None or omit the option
	Specify locations and system determines labels	{loc} = list of x or y coordinates
	Specify locations and their respective labels	{{loc1,lab1},{loc2,lab2},…}‡

^aIf only one attribute is selected and it is to be applied to both axes, then all braces may be omitted

^b labelx and labely are, in general, given by label shown in Figure 6.2.

^clocN is either the value of the x-coordinate along the x-axis or the value of the y-coordinate along the y-axis

Option	Instruction
Frame->Instruction
No frame (default)	False or None or omit the option
Draw frame on all four sides	True
Draw any combination of portions of frame	{{True or False,True or False},     {True or False,True or False}} where the locations are {{left, right},{bottom, top}}
FrameLabel-> Instruction
No labels (default)	None or omit the option
Label at bottom edge	label*
Label at bottom and left edges	{bottom,left} †
Label all edges	{{left,right},{bottom,top}}†
FrameTicks-> Instruction
No tick marks	None or omit the option
Place tick marks automatically	Automatic
Place tick marks automatically on bottom and left edges	True
Place tick marks automatically on all edges	All
Specify locations and labels for each tick mark on each edge	{{left,right}{bottom,top}}†     left={{loc1,label1},{loc2,label2},…}‡                     or None or Automatic Similarly for right, bottom, and top

^alabel or labelN is given by the general form shown in Figure 6.2

^bbottom, top, left, and right specify edge location based on its position in the list; when implemented, an appropriate label of the general form given in Figure 6.2 replaces each of these identifiers

^clocN equals x-coordinate value for bottom and top and equals the y-coordinate value for left and right

6.2.3 Common 2D Shapes: Graphics[]

There are several commands that create common 2D shapes. These commands can be used to create separate figures or to enhance a figure. They can also be used to create plot markers as shown in Table 6.15. A list of these shapes is given in Table 6.11. They are created with the following command

Shape	Mathematica function
Lines	Line[{{x1,y1},{x2,y2},…}]
Connected lines with the last point containing an arrowhead (unless Arrowheads indicates otherwise)	Arrow[{{x1,y1},{x2,y2},…}]
Arrowhead size or size of two arrowheads, one at each end of line	Arrowheads[p] for arrowhead at last point (typically, 0 < p ≤ 0.1) Arrowheads[{-p1,p2}] for arrowheads at first and last points: p1 size of arrowhead of first point (minus sign required), p2 size of arrowhead of last point. Note: must be used to draw two arrowheads and this specification must precede Arrow.
Points	Point[{{x1,y1},{x2,y2},…}]
Circle or arc of circle	Circle[{x0,y0},r,{θ1,θ2}]     {x0,y0} = coordinates of circle center     r = radius of circle (if omitted, r = 1)     {θ1,θ2} = angles of an arc of the circle (if omitted, complete             circle)
Disk or disk sector	Disk[{x,y},r,{ θ1,θ2}]     {x,y} = coordinates of disk center     r = radius of disk (if omitted, r = 1)     {θ1,θ2} = angles of a sector of the disk (if omitted, complete             disk)
Ellipse	Disk[{x,y},{rx,ry}]     {x,y} = coordinates of ellipse center     rx = semi-axis length parallel to x-axis     ry = semi-axis length parallel to y-axis
Rectangle	Rectangle[{xl,yl},{xu,yu}]     {xl,yl} = coordinates of lower left hand corner     {xu,yu} = coordinates of upper right hand corner
Polygon	Polygon[list]     list = list of coordinates of the vertices

Enhancement	Instruction
Filling area under a curve, place a line under a point, or place a “volume” under a surface	Filling->loc     loc = None ; no filling (default)     loc = Axis ; fill to x-axis     loc = Top ; fill to top of plot     loc = Bottom ; fill to bottom of plot     loc = {1->{2}} ; fill between the first and second curves drawn
	FillingStyle->typ     typ = Automatic (default)     typ = {col,opa,thk,dash}         opa = Opacity[n] (0 ≤ n ≤ 1)         col = color from Table 6.5         thk = Thickness[…] from Table 6.6         dash = Dashing[…] from Table 6.6
Grid lines	GridLines->spec     spec = None (default)     spec = Automatic     spec = {listx,listy}, where listx and listy specify                      locations of grid lines in each direction: listx or                      listy can equal None or can equal Automatic
Plot title	PlotLabel[label]*
Place additional objects on a figure	Epilog->{{obj1},{obj2},…}     objN is typically a graphics primitive or an Inset object
Rotate a graphics object including an entire figure created by any plot command	Rotate[obj,th,{xo,yo}]     obj = object to be rotated     th = counterclockwise angle of rotation (radians)     {xo,yo} = optional point about which rotation occurs

^alabel is, in general, of the form given in Figure 6.2.

h[t_,p_]:=Exp[-0.15 t] Sin[t-p]; gx=Table[g,{g,Range[1,15,1]}]; (* Coordinates of grid lines for         x-axis *) lab=Style["Two curves",Blue]; eq=Table[{n,h[n,0]},{n,2./3,14./3,}]; (* See footnote *) p1=Plot[{h[t,0],h[t,/3.]},{t,0,15},PlotStyle->{{Thickness[Large],     Red},{Thickness[0.01],Black}},options]
options: PlotLabel->lab	options: PlotLabel->lab,     Filling->{1->{2}},     FillingStyle->Orange,     Frame->True
options: PlotLabel->lab,     Filling->{1->{2}},     FillingStyle->Orange,     Frame->True,     GridLines->{gx,Automatic}	options: PlotLabel->lab,     Filling->{1->{2}},     FillingStyle->Orange,     Frame->True,     GridLines->{gx,Automatic},     Epilog->{PointSize[0.02],         Yellow,Point[eq]}
Rotate[p1,/2]     using in p1 the options:     PlotLabel->lab,     Filling->{1->{2}},     FillingStyle->Orange,     Frwme->True,     GridLines->{gx,Automatic},     Epilog->{PointSize[0.02],         Yellow,Point[eq]}

^a Coordinates of curves’ intersections

plt=Sequence[{BesselJ[0,x],Exp[-(x-4)^2/2]},{x,0,10}]; lst1=Table[{x,BesselJ[0,x]},{x,Range[0,10,0.15]}]; lst2=Table[{x,Exp[-(x-4)^2/2]},{x,Range[0,10,0.15]}]; pstyl=Sequence[PlotStyle->{{Dashing[{}],Black},{Dashing[Medium],Blue}}]; lstl=Sequence[{TraditionalForm[BesselJ[0,x]],TraditionalForm[Exp[-(x-4)^2/2]]}]; frd=Sequence[{RoundingRadius->10,Background->Yellow,FrameStyle->Blue}];
Plot[Evaluate[plt],Evaluate[pstyl],     PlotLegends->Placed["Expressions",Right]]	Plot[Evaluate[plt],Evaluate[pstyl],     PlotLegends->Placed[LineLegend[     {"Bessel","Exponential"}],Right]]
Plot[Evaluate[plt],Evaluate[pstyl],PlotLegends->     Placed[LineLegend["Expressions",     LegendLabel->"Functions"],Right]]	Plot[Evaluate[plt],Evaluate[pstyl],     PlotLegends->Placed[LineLegend["Expressions",     LegendLabel->"Functions",     LegendFunction->Framed],Right]]
Plot[Evaluate[plt],Evaluate[pstyl],     PlotLegends->Placed["Expressions",Top]]	Plot[Evaluate[plt],Evaluate[pstyl],     PlotLegends->Placed[LineLegend["Expressions",         LegendLabel->"Functions",LegendFunction->         (Framed[#,Evaluate[frd]] &)],Right]]
ListLinePlot[{lst1,lst2},PlotLegends->     Placed[LineLegend[Sequence[lstl],     LegendLabel->"Functions"],Right]]	ListPlot[{lst1,lst2},PlotLegends->     Placed[PointLegend[Sequence[lstl],     LegendLabel->"Functions"],Right]]
ListLinePlot[{lst1,Legended[lst2,     TraditionalForm[Exp[-(x-4)^2/2]]]},PlotLegends->     Placed[LineLegend[None,LegendLabel->"Function",     LegendFunction->Framed],Right]]	ContourPlot[Sin[2 x] Sin[y],{x,0,1},{y,0,1},     ColorFunction->ColorData["Rainbow"],     PlotLegends->Placed[BarLegend[Automatic],             Right]]
Plot[Evaluate[plt],Evaluate[pstyl],     PlotLegends->Placed["Expressions",{0.8,0.8}]]

Objective	Program*	Output
Use default values	lst1=Table[{n /4.,Sin[n /4.]},{n,0,16,0.5}]; lst2=Table[{n /4.,Exp[-n /16.]},{n,0,16}]; ListPlot[{lst1,lst2},PlotMarkers->Automatic]
Use default values	lst1=Table[{n /4.,Sin[n /4.]},{n,0,16,0.5}]; lst2=Table[{n /4.,Exp[-n /16.]},{n,0,16}]; ListLinePlot[{lst1,lst2},PlotMarkers->     Automatic]
Select fill and marker characteristics	lst1=Table[{n /4.,Sin[n /4.]},{n,0,16}]; ListPlot[lst1,Filling->Axis,     FillingStyle->{Dashing[Medium],         Blue,Thickness[Large]},     PlotMarkers->{Graphics[{Red,Rectangle[]}],         0.03}]
Place a series of points whose locations differ from those used to plot curve	lst1=Table[{n /5.,Sin[n /5.]},{n,0,20}]; Plot[Sin[x],{x,0,4 },},Filling->Axis,     Epilog->{Red,PointSize[0.025],Point[lst1]}]
Place different markers at same x location	lst1=Table[{n /5.,Sin[n /5.]},{n,0,20}]; lst2=Table[{n /5.,0.8 Sin[n /5.]},{n,0,20}]; Show[ListPlot[lst1,Filling->Axis,     FillingStyle->{Blue,Thickness[Medium]},     PlotMarkers->         {Graphics[{Red,Rectangle[]}],0.03}],     ListPlot[lst2,PlotMarkers->         {Graphics[{Black,Disk[]}],0.03}]]

^aAdditional plot marker shapes, which include the shapes shown above, can be obtained by using Polygon and symbols from the palette shown in Figure 1.2b.

Graphics[{s1,s2,…c1,c2,…},opt]

where opt are options such as Axes and Frame that can be used to further modify the figure and

sK={col1K,EdgeForm[{col2K,thkK,dashK}],shape,Opacity[n]}
cK={colK,thk,dash,cla}

In these expressions,

•     colK, col1K, and col2K are colors from Table 6.5;
•     thkK, thk = Thickness[…] specifies the line thickness as shown in Table 6.6;
•     dashK, dash = Dashing[…] specifies a dashed line as shown in Table 6.6;
•     Opacity[n] is the degree of opaqueness, where n is a number from 0 to 1 with 1         being opaque and 0 being invisible: default is opaque.
•     shape = Disk (Ellipse), Rectangle, or Polygon
•     cla = Circle, Line, Point, or Arrow
•     EdgeForm controls the properties of the perimeter of the shape

With regard to Point, the options dash and thk are replaced with

PointSize[arg]

where arg is either Tiny, Small, Medium, Large, or a decimal number that typically is in the range 0 < p ≤ 0.1.

The usage of these shapes is illustrated with the following statement

Graphics[{{Red,Rectangle[{0,0},{1,1}]},
  {Blue,Opacity[0.4],Rectangle[{0.5,0.5},{1.5,1}]},
  {Thick,Yellow,Dashing[Large],Circle[{0.5,0.5},0.5]},
  {Purple,Polygon[{{1,0},{1.5,0},{1.5,0.5}}]},
  {Dashing[Small],Arrow[{{1,0.5},{1.25,0.25}}]},
  {Black,PointSize[0.05],Point[{1,0.5}]}}]

the execution of which yields Figure 6.3.

For another illustration, two of these built-in shapes are used to create n concentric circles and n concentric unfilled rectangles as follows. It is assumed that the rectangle and the circle are centered at (x_c,y_c), the width of the rectangle is a and its height is b, and the radius of the circle is r. The coordinates of the end points of the rectangle in terms of x_c and y_c are given by (x_c − na/2, y_c − nb/2) for the lower left corner and by (x_c + na/2, y_c + nb/2) for the upper right corner, where n = 1, 2, … , is the number of rectangles to be drawn. If it is assumed that x_c = 1, y_c = 1, b = 0.5, a = r and r varies from 0.5 to 2 in increments of 0.5, then the program to draw these concentric shapes is

xc=1;    yc=1;    a=1;    b=0.5;
Graphics[{Table[{EdgeForm[Black],Opacity[0.],
  Rectangle[{xc-n a/2,yc-n b/2},{xc+a n/2,yc+b n/2}]},
   {n,1,4}],
  Table[Circle[{xc,yc},r],{r,0.5,2,0.5}]}]

where we have used EdgeForm to define the perimeter of the rectangle independently of the rectangle’s opaqueness, which has been set to 0 (invisible). The execution of this program produces Figure 6.4.

6.2.4 Additional Graph Enhancements

In addition to the enhancements described above, there are several other enhancements that are often used to increase the visual impact of a figure. These are: filling regions under curves, drawing grid lines, placing additional objects within the figure, placing a figure title, and rotating the figure. Typical objects are text, graphics shapes, or another figure. These figure enhancements are listed in Table 6.12 and illustrated in Table 6.13.

Legends

The basic legend commands that are introduced are: PlotLegend s, which creates legends; LegendLabel, which places a title over the legends; LineLegend, which selects the characteristics of the legend identifiers such as line colors and their labels; and LegendFunction, which specifies the characteristics of the box that contains the legends. To have additional flexibility when using legends, two additional commands are often employed. The first is

Placed[expr,{xp,yp}]

which places expr at the location {xp,yp}. The magnitude of {xp,yp} is a function of the command in which it is used. For graphs, it is typically a pair of positive numbers between 0 and 1. The second command is

Legended[expr,label]

which, when used with the appropriate legend commands, will associate label with expr. It is used when only selected curves are to be identified in a legend.

Examples of the use of these commands are given in Table 6.14 for Plot, ListPlot, ListLinePlot, and ContourPlot. In addition, to improve readability of the programs in Table 6.14, the command

Sequence[arg]

is used. The quantity arg represents a sequence of comma-separated arguments that can be placed into any function. In Table 6.14, it is used to specify the characteristics of line and point styles of the graphs and is implemented with Evaluate.

Plot Markers

Plot markers are a way to specify the shape and size of the symbol used to plot the data values that appear in the ListPlot family of plotting functions. The basic option is

PlotMarkers->Automatic

To specify a specific shape, color, and size of the plot marker, the following is used

PlotMarkers->{Graphics[{color,shape}],n}

wherecolor is a the color selected from Table 6.5, shape is a graphic shape selected from Table 6.11, and n is a number, a typical value of which is 0.03. Examples of the usage of PlotMarkers are given in Table 6.15.

Placement of an Object within a Figure

One or more objects can be placed anywhere in a figure by using Epilog, Insert, and the definitions in Figure 6.5. The Inset command is

Inset[obj,loc]

where obj is a graphics object or text, loc = {xo,yo} is the location where the center of the object’s coordinate system will be placed, sft={xs,ys} is the location with respect to the inserted figure’s coordinate system, and sz={ Δx,Δy}. When used with LogPlot, LogLinearPlot, and LogLogPlot, the following transformation has to be performed on the location coordinates:

LogPlot → loc = {x1,Log[y1]}
LogLinearPlot → loc = {Log[x1],y1}
LogLogPlot → loc = {Log[x1],Log[y1]}

As an example, consider following statements

p2=Plot[2-x^2-x^3,{x,-1,1}];
Plot[2+x^2+x^3,{x,-2,2},PlotRange->All,
  Epilog->Inset[p2,{-1,6},{0,0},{8,8}]]

which produces Figure 6.6.

These graph enhancement techniques are now illustrated with the following examples.

Plot[h[x],{x,-1,2},PlotRange->{{-1,2},{-10,100}},options] opt1=Sequence[PlotLabel->plab,Filling->Axis,FillingStyle->Orange]; opt2=Sequence[AxesLabel->{"x","h(x)"},LabelStyle->{14,Blue}]; opt3=Sequence[{PointSize[Large],Point[{xmin,hmin}]}]; opt4=Sequence[{Dashing[Medium],Line[{{0,hmax},{1.5,hmax}}]},     {Arrowheads[0.03],Arrow[{{0.7,80},{xmax,hmax}}]},     Inset[hminlab,{0.8,7.}],Inset[hmaxlab,{0.4,80}]];
options:none	options: Evaluate[opt1]
options: Evaluate[opt1],     Evaluate[opt2],     Epilog->{Evaluate[opt3]}	options: Evaluate[opt2],     Evaluate[opt2],     Epilog->{Evaluate[opt3],     Evaluate[opt4]}

6.2.5 Combining Figures: Show[] and GraphicsGrid[]

The command Show allows one to combine several graphical entities into one figure, including the combination of those created with different plotting commands such as Plot and ListPlot. However, the major labeling and overall appearance of the final graphic will be determined by the first plotting or graphics command appearing in Show. Therefore, one should place the desired overall graphic options in this command.

Show combines several distinct graphs onto one graph. If one wants to keep several distinct graphs separate, but arrange them in a specific manner based in some sort of grid, then there are three other commands to perform this function. The first command is GraphicsGrid, which arranges the graphs on an n × m grid. The second is GraphicsColumn, which place the graphs in a column one below the other. The third command is GraphicsRow, which places the graphs adjacent to each other in a row. These commands and their respective output are illustrated in Table 6.17.

The use of Show is illustrated with the following example.

6.2.6 Tooltip[]

Tooltip is a graph enhancement that allows one to use a mouse pointer to display information about curves or data points when the mouse pointer is over a curve or a data point. Its form is

Tooltip[expr,label]

where label will be displayed in a framed box. The use of Tooltip in plotting is shown in Table 6.18.

Tooltip objective	Instructions	Figure*
Identify each curve	c1=Tooltip[Sin[x],     TraditionalForm[Sin[x]]]; c2=Tooltip[Cos[2 x],     TraditionalForm[Cos[2 x]]]; Plot[{c1,c2},{x,0,5 }]
Display values of individual data points [unformatted]	lst1=Table[Tooltip[{x,Sin[x]}],     {x,Range[0,5. ,5 /51.]}]; ListPlot[lst1]
Display values of individual data points [formatted]	lst1=Table[Tooltip[     {x,Sin[x]},Column[{Row[{"x = ",     NumberForm[x,3]}],     Row[{"sin(x) = ",     NumberForm[Sin[x],3]}]}]],     {x,Range[0,5. ,5 /51.]}]; ListPlot[lst1]
Display values of individual data points and connect data points [unformatted]	ListLinePlot[Table[Tooltip[     {x,Sin[x]},{x,Sin[x]}],     {x,Range[0,5. ,5 /102.]}],     PlotMarkers->{Automatic,5}]†
Display values of individual data points, connect data points, and identify data set [unformatted]	lst1=Table[Tooltip[{x,Sin[x]},     Column[{"Data set #1",     {x,Sin[x]}}]],     {x,Range[0,5. ,5 /102.]}]; lst2=Table[Tooltip[{x,Cos[x]},     Column[{"Data set #2",     {x,Cos[x]}}]],     {x,Range[0,5. ,5 /102.]}]; ListLinePlot[{lst1,lst2},     PlotMarkers->{Automatic,5}]

^aArrow indicates cursor position and text in box is produced by Tooltip. For graphs with two curves, both tooltips are shown; however, in actuality, only one can appear at a time.

^bPlotmarkers is required otherwise Tooltip will not appear.

Export["Path/FileName.ext",fig,"Graphics"]

6.3 3D Graphics

There are numerous three-dimensional plotting functions that are available in Mathematica. The ones that we shall discuss are listed in Tables 6.19. The plotting can be enhanced in the same manner that the 2D plotting functions were; that is, one may use the options appearing in Tables 6.5, 6.6, 6.7, 6.9, 6.12, and 6.18. However, there are three additional commands that are specific to 3D graphs: Boxed, Mesh, and ViewPoint. These are listed in Table 6.20. Also, for 3D graphics, one can rotate the display interactively by placing the cursor over the figure. A pair of curved lines, each with an arrowhead appearing head to tail, will become visible. Depressing the mouse button and moving this cursor will cause the figure to change orientation.

As was the case for 2D graphics, Mathematica provides commands that can create common 3D objects. These commands can be used to create separate figures or to enhance a figure. A list of these shapes is given in Table 6.21. They are created with the following command

Graphics3D[{s1,s2,…c1,c2,…},opt]

where opt are options such as Axes, Mesh, Boxed, ViewPoint, and MaxRecursion that can be used to modify the figure and

•     sK={col1K,EdgeForm[{col2K,thkK,dashK}],Opacity[n],shape}
•     cK={colK,thk,dash,pla}
•     colK, col1K, col2K = color from Table 6.5
•     thkK = Thickness[…] : line thickness from Table 6.6
•     dashK = Dashing[…] : dashed lines from Table 6.6
•     Opacity[n] = degree of opaqueness: n is a number from 0 to 1, where 1 is opaque and
                                  0 is invisible. If omitted, the shape is opaque.
•     shape = Cone, Cuboid, Cylinder, Sphere, or Tube
•     pla = Point, Line, Polygon, or Arrow

In addition to the 3D shapes listed in Table 6.21, one can create a three-dimensional object of arbitrary complexity by using

GraphicsComplex[vec,prim]

where vec is a list of K triplets of numbers and prim denotes the connectivity of the vertices of the object. Thus,

vec={{x1,y1,z1},{x2,y2,z2},…{xK,yK,zK}};

The argument of prim is a list of lists of integers, with each integer corresponding to a point in vec. The order of these integers is used by ComplexGraphics to create a surface with these vertices for each element of prim. For example,

prim[{{1,2,5,6},…}

indicates that the vertices of the first implementation of prim is composed of the points connected in the order given; that is,

{{x1,y1,z1},{x2,y2,z2},{x5,y5,z5},{x6,y6,z6}};

To illustrate the use of this command, we shall create a frustum of a right pyramid whose base is 1 by 1 units and whose top is 0.25 by 0.25 units and 2 units from the base. This object consists of six 4-sided polygons and its creation, which is shown in Figure 6.12, is obtained from

vec={{0,-1/4,-1/4},{0,1/4,-1/4},{0,1/4,1/4},{0,-1/4,1/4},
  {2,-1,-1},{2,1,-1},{2,1,1},{2,-1,1}};
n={{1,2,3,4},{5,6,7,8},{1,2,6,5},{3,4,8,7},{1,4,8,5},
  {2,3,7,6}};
Graphics3D[{Opacity[1],GraphicsComplex[vec,Polygon[n]]},
  Boxed->False,ViewPoint->{Bottom,Left}]

The detailed characteristics of a three-dimensional shape can be significantly altered by the options selected. To give an indication of how these options can alter a three-dimensional figure, we have chosen to show the effects of several of them on the parametric surface

for 0 ≤ u ≤ 2π, −15 ≤ u ≤ 6, a = 1.13, and b = 1.14. The program that displays this shape is given in Table 6.22 as a function of several options.

We shall now illustrate the use of the three-dimensional plot commands appearing in Tables 6.19 to 6.21. ParametricPlot3D has been illustrated in Table 6.22. The following examples illustrate the use of Graphics3D, Plot3D, ListPlot3D, and RevolutionPlot3D.

6.4 Summary of Functions Introduced in Chapter 6

The 2D plot commands are listed in Tables 6.1 to 6.4 and the built-in 2D geometric shapes are listed in Table 6.11. The options that are available to these graphics commands are listed in Tables 6.5, 6.6, 6.7, 6.9, 6.10, 6.12, 6.14, 6.15, and 6.18. The 3D plotting functions are given in Table 6.19 and the options specific to 3D graphics are listed in Table 6.20. The built-in 3D geometric shapes are listed in Table 6.21.

References

1. P. K. Kundu and I. M. Cohen, Fluid Mechanics, 4th edn, Academic Press, Burlington, Massachusetts, 2008, p. 753.
2. E. W. Weisstein, CRC Concise Encyclopedia of Mathematics, 2nd edn, Chapman Hall/CRC, Boca Raton, 2003, p. 2129–30.
3. A. Leissa, Vibration of Shells, NASA Sp-288, 1973, p. 44.
4. F. P. Incropera and D. P. Dewitt, Introduction to Heat Transfer, 4th edn, John Wiley & Sons, New York, 2002, p. 675.

Exercises

Section 6.2

• 6.1 The relationship between an oblique shock wave at an angle σ and the deflection angle δ as a function of the incident Mach number M₁ is [1]
  (6.1)

  where γ is the ratio of the specific heats of the gas. For a given M₁, there is a value of σ for which δ is a maximum. We denote these values as σ_max and δ_max. Then for a given M₁ those values of σ > σ_max for which δ < δ_max the downstream Mach number M₂ is less than 1 and the shock waves are called strong shock waves. For those values σ < σ_max for which δ < δ_max, the downstream Mach number M₂ is greater than 1 and the shock waves are called weak shock waves. With this information, use Eq. (6.1) to obtain Figure 6.18. This figure was obtained with γ = 1.4.

• 6.2 For a given value of y, the value of p is determined from
  

  where h = 10^−p, k₁ = 2.5 × 10⁻⁴, k₂ = 5.6 × 10⁻¹¹, and k₃ = 1.7 × 10⁻³. Plot the value of p when 0.00001 ≤ y ≤ 2.5.

• 6.3 The Cartesian locations of bi-cylinder coordinates are given by
  

  If a = 1.5, −1 ≤ η ≤ 1, and 0 ≤ ϕ ≤ 2π, replicate Figure 6.19.

  

  

• 6.4 One can draw n circles, n ≥ 3, that are tangent to a central circle of radius r_b and to each adjacent circle as shown in Figure 6.20 for n = 10. The radius of the outer circles is
  

  Write a program that replicates this figure and similar figures for arbitrary n. In this case, the outer circles (disks) are blue.

  

• 6.5 A Pappus chain is a series of n tangent circles inscribed in the area between three semicircles as shown in Figure 6.21. The radius of the outer circle is R_o and the radii of the other two circles are R_L and R_R, R_L ≥ R_R. The location of the center of the nth circle is [2]
  

  and its radius is

  

  where r = 2R_L/(R_L + R_R). The ellipse that connects the centers of the circles is given by

  

  Replicate Figure 6.21, which has been obtained for n = 4, R_o = 1/2, R_L = 1/4, and R_R = 1/4. Color the four complete circles yellow and their background green. The right semicircle is colored magenta and the left one red.

• 6.6 A piece-wise linear map is defined by the points with the coordinates
  

  Obtain a plot for the coordinate points when x₁ = 1, y₁ = 3.65, and N = 15,000. The resulting figure is sometimes referred to as the gingerbread man.

• 6.7 The following relation is used to determine the value of a pipe’s coefficient of friction λ for a given Reynolds’ number Re, pipe diameter d, and surface roughness k
  

  Using this relation, replicate Figure 6.22.

  

  

  

• 6.8 The following equation will plot a Joukowsky airfoil over the range −0.5 ≤ x ≤ 0.5
  

  where the plus sign describes the top of the airfoil and the negative sign its bottom. With h = 0.08 and t = 0.13, replicate the Figure 6.23.

• 6.9 A normalized structural damping factor describing the losses in a vibrating thin plate with a damping layer is given by
  

  where α is the ratio of the layers’ moduli and ξ is the ratio of the layers’ thickness. Replicate the logarithmic plot shown in Figure 6.24. The x-locations of the curves’ labels and their individual orientations have been done iteratively.

• 6.10 Given the following polynomial [3]
  (6.2)

  where

  

  For ν = 0.3, h = 10⁻⁵, and n = 1, 2, 3, 4, 5, plot the smallest positive value of Ω that satisfies Eq. (6.2) for 0.5 ≤ π/λ ≤ 100. The results should look like those shown in Figure 6.25. The placement of the labels on the curves should appear as shown even if the value of h changes.

  

  

  

• 6.11 The spectral emissive power is given by the Planck distribution [4]
  

  where c₁ =3.742 × 10⁸ W μm⁴·m⁻² and c₂ = 1.4388 × 10⁴ μm K. For T = {2000., 1000., 400., 100.} K, obtain a plot of E versus λ. The result should look like that shown in Figure 6.26. Show in tabular form that for the four temperatures given, λ_maxT_n = 2898 μm K.

Section 6.3

• 6.12 Plot the surface described by
  

  over the region indicated. Label the axes.

• 6.13 Plot the surface described by
  

  over the region indicated when a = 0.25. Label the axes.

• 6.14 The curve that results from the intersection of a sphere of radius 2a centered at the origin and a cylinder of radius a centered at (a,0,0) is given by the parametric equations
  

  where 0 ≤ ϕ ≤ 4π. If a = 1 and the length of the cylinder is 2, then replicate Figure 6.27. Use a tube radius of 0.04. Color the sphere blue, the cylinder magenta, and the tube yellow.

• 6.15 Plot the following parametric equations over the regions indicated.
  1. Figure eight torus
     
  2. Seashell
     
  3. Astroidal ellipsoid