Basic Plotting

Basic plotting functions.

This plots a graph of as a function of from 0 to:

You can plot functions that have singularities. The Wolfram Language will try to choose appropriate scales:

The singularities can be omitted from the plot by using theExclusions option:

You can give a list of functions to plot. A different color will automatically be used for each function:

This finds the numerical solution to a differential equation, as discussed in"Numerical Differential Equations":

Here is a plot of the solution:

Options for Graphics

When the Wolfram Language plots a graph for you, it has to make many choices. It has to work out what the scales should be, where the function should be sampled, how the axes should be drawn, and so on. Most of the time, the Wolfram Language will probably make pretty good choices. However, if you want to get the very best possible pictures for your particular purposes, you may have to help the Wolfram Language in making some of its choices.

There is a general mechanism for specifying "options" in Wolfram Language functions. Each option has a definite name. As the last arguments to a function likePlot, you can include a sequence of rules of the formname->value, to specify the values for various options. Any option for which you do not give an explicit rule is taken to have its "default" value.

Choosing an option for a plot.

A function likePlot has many options that you can set. Usually you will need to use at most a few of them at a time. If you want to optimize a particular plot, you will probably do best to experiment, trying a sequence of different settings for various options.

Each time you produce a plot, you can specify options for it."Redrawing and Combining Plots" will also discuss how you can change some of the options, even after you have produced the plot.

Some of the options forPlot. These can also be used inShow.

Here is a plot with all options having their default values:

This draws axes on a frame around the plot:

This specifies labels for the and axes. The expressions you give as labels are printed just as they would be if they appeared asTraditionalForm Wolfram Language output. You can give any piece of text by putting it inside a pair of double quotes:

You can give several options at the same time, in any order:

Setting theAspectRatio option changes the whole shape of your plot.AspectRatio gives the ratio of height to width. Its default value is the inverse of the Golden Ratio—supposedly the most pleasing shape for a rectangle:

Some common settings for various options.

When the Wolfram Language makes a plot, it tries to set the and scales to include only the "interesting" parts of the plot. If your function increases very rapidly, or has singularities, the parts where it gets too large will be cut off. By specifying the optionPlotRange, you can control exactly what ranges of and coordinates are included in your plot.

Settings for the optionPlotRange.

The setting for the optionPlotRange gives explicit limits for the graph. With the limits specified here, the bottom of the curve is cut off:

The Wolfram Language always tries to plot functions as smooth curves. As a result, in places where your function wiggles a lot, the Wolfram Language will use more points. In general, the Wolfram Language tries toadapt its sampling of your function to the form of the function. There is a limit, however, which you can set, to how finely the Wolfram Language will ever sample a function.

The function wiggles infinitely often when. the Wolfram Language tries to sample more points in the region where the function wiggles a lot, but it can never sample the infinite number that you would need to reproduce the function exactly. As a result, there are slight glitches in the plot:

It is important to realize that since the Wolfram Language can only sample your function at a limited number of points, it can always miss features of the function. The Wolfram Language adaptively samples the functions, increasing the number of samples near interesting features, but it is still possible to miss something. By increasingPlotPoints, you can make the Wolfram Language sample your function at a larger number of points. Of course, the larger you setPlotPoints to be, the longer it will take the Wolfram Language to plotany function, even a smooth one.

More options forPlot. These cannot be used inShow.

This usesPlotStyle to specify a dashed curve:

When plotting multiple functions,PlotStyle settings in a list are used sequentially for each function:

When aPlotStyle contains a sublist, the settings are combined:

By default nothing is indicated when thePlotRange is set, so that it cuts off curves:

SettingClippingStyle toAutomatic draws a dashed line where a curve is cut off:

SettingClippingStyle to a list defines the style for the parts cut off at the bottom and top:

This specifies filling between the curve and the axis:

The filling can be specified to extend to an arbitrary height, such as the bottom of the graphic. Filling colors are automatically blended where they overlap:

This specifies a specific filling to be used only for the first curve:

This shows a filling from the first curve to the second using a nondefault filling style:

Redrawing and Combining Plots

The Wolfram Language saves information about every plot you produce, so that you can later redraw it. When you redraw plots, you can change some of the options you use.

Functions for manipulating plots.

Here is a simple plot:

When you redraw the plot, you can change some of the options. This changes the choice of scale:

This takes the plot from the previous line, and changes another option in it:

By usingShow with a sequence of different options, you can look at the same plot in many different ways. You may want to do this, for example, if you are trying to find the best possible setting of options.

You can also useShow to combine plots. All of the options for the resulting graphic will be based upon the options of the first graphic in theShow expression.

This setsgj0 to be a plot of from to:

Here is a plot of from to :

Plot specifies an explicitPlotRange for each graphic:

This usesPlotRange to override the explicit value set forgj0:

All Wolfram Language graphics are expressions and can be manipulated in the same way as any other expression. Doing these kinds of manipulations does not require the use ofShow.

This replaces all instances of the symbolLine with the symbolPoint in the graphics expression represented bygj0:

UsingShow[plot₁,plot₂,…] you can combine several plots into one.GraphicsGrid allows you to draw several plots in an array.

Drawing arrays of plots.

This shows the previous plots in an array:

If you redisplay an array of plots usingShow, any options you specify will be used for the whole array, rather than for individual plots:

GraphicsGrid by default puts a narrow border around each of the plots in the array it gives. You can change the size of this border by setting the optionSpacings->{h,v}. The parametersh andv give the horizontal and vertical spacings to be used. TheSpacings option uses the width and height of characters in the default font to scale theh andv parameters by default, but it is generally more useful in graphics to useScaled coordinates.Scaled scales widths and heights so that a value of1 represents the width and height of one element of the grid.

This increases the horizontal spacing, but decreases the vertical spacing between the plots in the array:

When you make a plot, the Wolfram Language saves the list of points it used, together with some other information. Using what is saved, you can redraw plots in many different ways withShow. However, you should realize that no matter what options you specify,Show still has the same basic set of points to work with. So, for example, if you set the options so that the Wolfram Language displays a small portion of your original plot magnified, you will probably be able to see the individual sample points thatPlot used. Options likePlotPoints can only be set in the originalPlot command itself. (the Wolfram Language always plots the actual points it has; it avoids using smoothed or splined curves, which can give misleading results in mathematical graphics.)

Here is a simple plot:

This shows a small region of the plot in a magnified form. At this resolution, you can see the individual line segments that were produced by the originalPlot command:

Three‐Dimensional Surface Plots

Basic 3D plotting function.

This makes a three‐dimensional plot of the function:

Three-dimensional graphics can be rotated in place by dragging the mouse inside the graphic. Dragging inside the graphic causes the graphic to tumble in a direction that follows the mouse, and dragging around the borders of the graphic causes the graphic to spin in the plane of the screen. Dragging the graphic while holding down theShift key causes the graphic to pan. Use theCtrl key to zoom.

There are many options for three‐dimensional plots in the Wolfram System. Some are discussed here; others are described in"The Structure of Graphics and Sound".

The first set of options for three‐dimensional plots is largely analogous to those provided in the two‐dimensional case.

Some options forPlot3D. The first set can also be used inShow.

This redraws the previous plot with options changed. With this setting forPlotRange, only the part of the surface in the range is shown:

TheClippingStyle option ofPlot3D can be used to fill in the clipped regions:

When you make the original plot, you can choose to sample more points. The Wolfram System adaptively samples the plot, adding points for large variations, but occasionally you may still need to specify a greater number of points:

Here is the same plot, with labels for the axes, and grids added to each face:

Probably the single most important issue in plotting a three‐dimensional surface is specifying where you want to look at the surface from. TheViewPoint option forPlot3D andShow allows you to specify the point in space from which you view a surface. The details of how the coordinates for this point are defined are discussed in"Coordinate Systems for Three-Dimensional Graphics". When rotating a graphic using the mouse, you are adjusting theViewPoint value.

Here is a surface, viewed from the default view point{1.3,-2.4,2}. This view point is chosen to be "generic", so that visually confusing coincidental alignments between different parts of your object are unlikely:

This redraws the picture, with the view point directly in front. Notice the perspective effect that makes the back of the box look much smaller than the front:

TheViewPoint option also accepts various symbolic values that represent common view points:

Typical choices for theViewPoint option.

The human visual system is not particularly good at understanding complicated mathematical surfaces. As a result, you need to generate pictures that contain as many clues as possible about the form of the surface.

View points slightly above the surface usually work best. It is generally a good idea to keep the view point close enough to the surface that there is some perspective effect. Having a box explicitly drawn around the surface is helpful in recognizing the orientation of the surface.

Here is a plot with the default settings for surface rendering options:

This shows the surface without the mesh drawn. It is usually much harder to see the form of the surface if the mesh is not there:

To add an extra element of realism to three‐dimensional graphics, the Wolfram System by default colors three‐dimensional surfaces using a simulated lighting model. In the default case, the Wolfram System assumes that there are four point light sources plus ambient lighting shining on the object."Lighting and Surface Properties" describes how you can set up other light sources, and how you can specify the reflection properties of an object.

Lighting can also be specified using a string that represents a collection of lighting properties. For example, the option settingLighting->"Neutral" uses a set of white lights, and so can be faithfully reproduced on a black and white output device such as a printer.

TheColorFunction option by default usesLighting->"Neutral" so that the surface colors are not distorted by colored lights:

Plotting Lists of Data

The Wolfram Language can be used to make plots offunctions. You give the Wolfram Language a function, and it builds up a curve or surface by evaluating the function at many different points.

Here we describe how you can make plots from lists of data, instead of functions. ("Importing and Exporting Data" discusses how to read data from external files and programs.) The Wolfram Language commands for plotting lists of data are direct analogs of the ones for plotting functions.

Functions for plotting lists of data.

Here is a list of values:

This plots the values:

This joins the points with lines:

When plotting multiple datasets, the Wolfram Language chooses a different color for each dataset automatically:

This gives a list of pairs:

This plots the points:

This gives a rectangular array of values. The array is quite large, so we end the input with a semicolon to stop the result from being printed out:

This makes a three‐dimensional plot of the array of values:

This gives a density plot of the array of values:

Parametric Plots

"Basic Plotting" describes how to plot curves in the Wolfram Language in which you give the coordinate of each point as a function of the coordinate. You can also use the Wolfram Language to makeparametric plots. In a parametric plot, you give both the and coordinates of each point as a function of a third parameter, say.

Functions for generating parametric plots.

Here is the curve made by taking the coordinate of each point to beSin[t] and the coordinate to beSin[2t]:

Three‐dimensional parametric plots.

ParametricPlot3D[{f_x,f_y,f_z},{t,t_min,t_max}] is the direct analog in three dimensions ofParametricPlot[{f_x,f_y},{t,t_min,t_max}] in two dimensions. In both cases, the Wolfram Language effectively generates a sequence of points by varying the parametert, then forms a curve by joining these points. WithParametricPlot, the curve is in two dimensions; withParametricPlot3D, it is in three dimensions.

This makes a parametric plot of a helical curve. Varyingt produces circular motion in the- plane, and linear motion in the direction:

ParametricPlot3D[{f_x,f_y,f_z},{t,t_min,t_max},{u,u_min,u_max}] creates a surface, rather than a curve. The surface is formed from a collection of quadrilaterals. The corners of the quadrilaterals have coordinates corresponding to the values of thef_i whent andu take on values in a regular grid.

Here the and coordinates for the quadrilaterals are given simply byt andu. The result is a surface plot of the kind that can be produced byPlot3D:

This shows the same surface as before, but with the coordinates distorted by a quadratic transformation:

This produces a helicoid surface by taking the helical curve shown above, and at each section of the curve drawing a quadrilateral:

In general, it is possible to construct many complicated surfaces usingParametricPlot3D. In each case, you can think of the surfaces as being formed by "distorting" or "rolling up" the- coordinate grid in a certain way.

This produces a cylinder. Varying thet parameter yields a circle in the- plane, while varyingu moves the circles in the direction:

This produces a torus. Varyingu yields a circle, while varyingt rotates the circle around the axis to form the torus:

This produces a sphere:

You should realize that when you draw surfaces withParametricPlot3D, the exact choice of parametrization is often crucial. You should be careful, for example, to avoid parametrizations in which all or part of your surface is covered more than once. Such multiple coverings often lead to discontinuities in the mesh drawn on the surface, and may makeParametricPlot3D take much longer to render the surface.

Some Special Plots

As discussed in"The Structure of Graphics and Sound", the Wolfram Language includes a full graphics programming language. In this language, you can set up many different kinds of plots.

Some special plotting functions.

Here is an exponential function:

This generates linear-log and log-linear plots of the function:

This gathers some financial time series data:

This plots the financial data on linear and logarithmic scales:

This usesRevolutionPlot3D to revolve a curve around the and axes:

Here is a list of the first 10 primes:

Here is a bar chart of the primes:

This gives a pie chart:

Some of the plotting functions available are defined in theStandard Extra Packages included with the Wolfram System.

Some of the plotting functions in the Wolfram System standard extra packages.

ErrorListPlot is available in theError Bar Plotting Package:

This plots a dataset with its associated errors:

BoxWhiskerPlot,ParetoPlot, andPairwiseScatterPlot are available in theStatistical Plots Package:

This generates a box-and-whisker plot for samples from a normal distribution:

Here is a random dataset consisting of the letters a through f:

This creates a Pareto plot of the dataset:

PairwiseScatterPlot can be used to plot pairs of columns from trivariate data:

Sound

On most computer systems, the Wolfram System can produce not only graphics but also sound. The Wolfram System treats graphics and sound in a closely analogous way.

For example, just as you can usePlot[f,{x,x_min,x_max}] to plot a function, so also you can usePlay[f,{t,0,t_max}] to“play” a function.Play takes the function to define the waveform for a sound: the values of the function give the amplitude of the sound as a function of time.

Playing a function.

On a suitable computer system, this plays a pure tone with a frequency of 440 hertz for one second:

Sounds produced byPlay can have any waveform. They do not, for example, have to consist of a collection of harmonic pieces. In general, the amplitude function you give toPlay specifies the instantaneous signal associated with the sound. This signal is typically converted to a voltage, and ultimately to a displacement. Note thatamplitude is sometimes defined to be thepeak signal associated with a sound; in the Wolfram System, it is always theinstantaneous signal as a function of time.

This plays a more complex sound:

Play is set up so that the time variable that appears in it is always measured in absolute seconds. When a sound is actually played, its amplitude is sampled a certain number of times every second. You can specify the sample rate by setting the optionSampleRate.

Specifying the sample rate for a sound.

In general, the higher the sample rate, the better high‐frequency components in the sound will be rendered. A sample rate ofr typically allows frequencies up to hertz. The human auditory system can typically perceive sounds in the frequency range 20 to 22000 hertz (depending somewhat on age and sex). The fundamental frequencies for the 88 notes on a piano range from 27.5 to 4186 hertz.

The standard sample rate used for compact disc players is 44100. The effective sample rate in a typical telephone system is around 8000. On most computer systems, the default sample rate used by the Wolfram System is around 8000.

You can usePlay[{f₁,f₂,…}] to produce stereo sound. In general, the Wolfram System supports any number of sound channels.

Playing sampled sounds.

The functionListPlay allows you simply to give a list of values which are taken to be sound amplitudes sampled at a certain rate.

Play noise generated from random values:

When sounds are actually rendered by the Wolfram System, only a certain range of amplitudes is allowed. The optionPlayRange inPlay andListPlay specifies how the amplitudes you give should be scaled to fit in the allowed range. The settings for this option are analogous to those for thePlotRange graphics option discussed in"Options for Graphics".

Specifying the scaling of sound amplitudes.

While it is often convenient to use the settingPlayRange->Automatic, you should realize thatPlay may run significantly faster if you give an explicitPlayRange specification, so it does not have to derive one.

Playing sounds programmatically.

ASound object in output is typically formatted as a button which contains a visualization of the sound and which plays the sound when pressed. Sounds can be played without the need for user intervention or producing output by usingEmitSound. In fact, the internal implementation ofSound buttons usesEmitSound when the button is pressed.

The internal structure ofSound objects is discussed in"The Representation of Sound".