59 Calculus plots with Makie

The Makie.jl webpage says

From the Japanese word Maki-e, which is a technique to sprinkle lacquer with gold and silver powder. Data is basically the gold and silver of our age, so let’s spread it out beautifully on the screen!

Makie itself is a metapackage for a rich ecosystem. We show how to use the interface provided by the CairoMakie backend to produce the familiar graphics of calculus.

Examples and tutorials

Makie is a sophisticated plotting package, and capable of an enormous range of plots (cf. examples.) Makie also has numerous tutorials to learn from. These are far more extensive than what is described herein, as this section focuses just on the graphics from calculus.

59.1 Figures

Makie draws graphics onto a canvas termed a “scene” in the Makie documentation. A scene is an implementation detail, the basic (non-mutating) plotting commands described below return a FigureAxisPlot object, a compound object that combines a figure, an axes, and a plot object. We also briefly show the details of constructing a separate figure and axis. The show method for figures displays the resulting graphic.

For Makie there are the different backends for different types of canvases. In the following, we have used CairoMakie.

We begin by loading the main package and the norm function from the standard LinearAlgebra package:

using CairoMakie
import LinearAlgebra: norm

The package load time as of recent version of Makie is quite reasonable for a complicated project. (The time to first plot is around five seconds on a typical machine.)

59.2 Points (`scatter`)

The task of plotting the points, say \((1,2)\), \((2,3)\), \((3,2)\) can be done different ways.

We can plot two vectors holding the x and y coordinates, e.g. [1,2,3] and [2,3,2].
Using a tuple to represent a point, we can plot a vector of tuples, e.g. [(1,2), (2,3), (3,2)].
More idiomatically, using a Point2 object to represent a point, a vector of point objects can be plotted, e.g. [Point2(1,2), Point2(2,3), Point2(3,2)].

The Point2 function also accepts a vector or tuple for input to describe the point. There is also Point3 for 3-d plotting. Makie uses a GPU, when present, to accelerate the graphic rendering. GPUs employ 32-bit numbers. Julia uses an f0 to indicate 32-bit floating points. Hence the alternate types Point2f0 to store 2D points as 32-bit numbers and Points3f0 to store 3D points as 32-bit numbers are seen in the documentation for Makie.

It is not so difficult to convert between these storage formats. For example starting with

xs = [1, 2, 3]
ys = [2, 3, 2]

3-element Vector{Int64}:
 2
 3
 2

We can convert to a vector of tuples using broadcasting or zip:

tuple.(xs, ys), collect(zip(xs, ys))

([(1, 2), (2, 3), (3, 2)], [(1, 2), (2, 3), (3, 2)])

A vector of points can be generated similarly:

Point2.(xs, ys), Point2.(zip(xs, ys))

(Point{2, Int64}[[1, 2], [2, 3], [3, 2]], Point{2, Int64}[[1, 2], [2, 3], [3, 2]])

The unzip function from the CalculusWithJulia package can reverse this direction:

using CalculusWithJulia: unzip # just SplitApplyCombine.invert
unzip(Point2.(xs, ys))

([1, 2, 3], [2, 3, 2])

We illustrate in Figure 59.1 where we generate a vector of points using these two functions:

r(t) = [cos(t), sin(t)]
h(t) = (cos(t), sin(t), t)
ts = range(0, 2pi, 25)

0.0:0.2617993877991494:6.283185307179586

scatter(Point2.(r.(ts)))
scatter(Point3.(h.(ts)))

Figure 59.1: A scatter plot in 2d and 3d generated by broadscasting `Point2` over a vector of 2 component vectors and `Point3` over a vector of 3 component tuples.

59.2.1 Attributes of a marker

A point is drawn with a “marker” with a certain size and color. These attributes can be adjusted, as in Figure 59.2.

Figure 59.2: Different marker attributes

Marker attributes include

marker a symbol, shape
markersize size (radius pixels) of marker
marker_offset offset coordinates
color to adjust color

A single value will be repeated. A vector of values of a matching size will specify the attribute on a per point basis.

59.3 Curves

A visualized curve in calculus is comprised of line segments. The lines command of Makie will render a curve by connecting a series of points with straight-line segments. By taking a sufficient number of points the connect-the-dot figure can appear curved.

59.3.1 Plots of univariate functions

The basic plot of univariate calculus is the graph of a function \(f\) over an interval \([a,b]\). This is implemented using a familiar strategy: produce a series of representative values between \(a\) and \(b\); produce the corresponding \(f(x)\) values; plot these as points and connect the points with straight lines.

To create regular values between a and b typically the range function or the range operator (a:h:b) are employed. The related LinRange function is also an option.

For example:

Figure 59.3: Use of `lines` to generate a curve

Makie has a recipe that allows the y position to be just the function—lines(xs, f) would have generated Figure 59.3 as well.

Makie also will read the interval notation of IntervalSets and select its own set of intermediate points, so this command would also render the same plot as that of Figure 59.3: lines(a..b, f).

As with scatter, lines can also be drawn using a vector of points. (Though the advantage isn’t clear here, this will be useful when the points are generated in different manners.) In this case, this approach would involve a command like lines(Point2.(xs, f.(xs))).

As with scatter, lines returns an object that produces a graphic when displayed.

When a y value is NaN or infinite, the connecting lines are not drawn:

Figure 59.4: An `NaN` value in the `y` vector has no connecting line segment

As with other plotting packages, this is useful to represent discontinuous functions, such as what occurs at a vertical asymptote or a step function.

Adding to a figure (`lines!`, `scatter!`, …)

To add or modify a figure can be done using a mutating version of a plotting primitive, such as lines! or scatter!. The names follow Julia’s convention of using an ! to indicate that a function modifies an argument, in this case the underlying figure.

Here is one way to show two plots at once:

Figure 59.5: Using `lines!` to add a layer

Current figure

The current_figure call is needed to have the figure display, as the returned value of lines! is not a figure object. (Figure objects display when shown as the output of a cell.)

We will see soon how to modify the line attributes so that the curves can be distinguished.

Figure 59.6 shows the construction details to produce the graphic.

Figure 59.6: Using `scatter!` to show the points used when creating a dot-to-dot plot

As an example, Figure 59.7 shows how to add the tangent line to a graph. The slope of the tangent line being computed by ForwardDiff.derivative.

Figure 59.7: Adding a tangent line to a curve at a point

This example¹ shows how to plot a step function (floor) using NaNs to create line breaks. The marker colors set for scatter! use :white to match the background color.

Figure 59.8: Using layers to plot a step function

59.3.2 Text (`annotations`)

Text can be placed at a point, as a marker is. To place text, the desired text and a position need to be specified along with any adjustments to the default attributes.

For example Figure 59.9 show an annotation and the use of fontsize to adjust the displayed text size.

Figure 59.9: Using `annotation!` to add text within a graphic

Attributes for text, among many others, include:

align Specify the text alignment through (:pos, :pos), where :pos can be :left, :center, or :right.
fontsize the font point size for the text
font to indicate the desired font

Annotations with an arrow can be useful to highlight a feature of a graph. The code to produce Figure 59.10 is modified from the documentation of Makie; it utilizes some interval functions to draw an arrow with an arc.²

Figure 59.10: Use of `path` and `style` attributes of `annotation!` to draw an arrow

Line attributes

In a previous example, we added the argument color=:blue to the lines! call. This was to set an attribute for the line being drawn. Lines have other attributes that allow different ones to be distinguished, as above where colors indicate the different graphs.

Other attributes can be seen from the help page for lines, and include:

color set with a symbol, as above, or a string
label a label for the line to display in a legend
linestyle available styles are set by a symbol, one of :dash, :dot, :dashdot, or :dashdotdot.
linewidth width of line
transparency the alpha value, a number between \(0\) and \(1\), smaller numbers for more transparent.

Simple legends

A simple legend displaying labels given to each curve can be produced by axislegend. For example:

xs = 0..pi
lines(xs, x -> sin(x^2), label="sin(x^2)")
lines!(xs, x -> sin(x)^2, label = "sin(x)^2")
axislegend()

current_figure()

Later, we will see how to control the placement of a legend within a figure.

Titles, axis labels, axis ticks

The basic plots we have seen are of type FigureAxisPlot. The “axis” part controls attributes of the plot such as titles, labels, tick positions, etc. These values can be set in different manners. On construction we can pass values to a named argument axis using a named tuple.

For example:

Figure 59.11: Passing `title`, `xlabel`, and `ylabel` values to the underlying axis

To access the axis element of a plot after the plot is constructed, values can be assigned to the axis property of the FigureAxisPlot object. For example:

Figure 59.12: One way to access the underlying axis after construction. (The more systematic way is to produce a `Figure` and construct an `Axis` object to modify.)

The ticks are most easily set as a collection of values. Above, the MultiplesTicks function was used to label with multiples of \(\pi\).

Later we will discuss how Makie allows for subsequent modification of several parts of the plot (not just the ticks) including the data.

Figure size, \(x\) and \(y\) limits

As just mentioned, the basic plots we have seen are of type FigureAxisPlot. The “figure” part can be used to adjust the background color or the size. As with attributes for the axis, these too can be passed to a simple constructor:

Figure 59.13: Adjust `size` value for the enclosing `Figure` object

The ; in the tuple passed to figure is one way to create a named tuple with a single element.

To set the limits of the graph there are shorthand functions xlims!, ylims!, and zlims!. This might prove useful if vertical asymptotes are encountered, as in the code to produce Figure 59.14.

Figure 59.14: Adjusting the range of possible `y` values with `ylims!`

Adjusting the y limits still leaves an artifact due to the vertical asymptote at \(0\) having different values from the left and the right.

59.3.3 Plots of parametric functions

A space curve is a plot of a function \(f:R^2 \rightarrow R\) or \(f:R^3 \rightarrow R\).

To construct a curve from a set of points, we have a similar pattern in both \(2\) and \(3\) dimensions:

r(t) = [sin(2t), cos(3t)]
ts = range(0, 2pi, length=200)
pts = Point2.(r.(ts))  # or (Point2∘r).(ts)
lines(pts)
r(t) = [sin(2t), cos(3t), t]
ts = range(0, 2pi, length=200)
pts = Point3.(r.(ts))
lines(pts)

Figure 59.15: Two and three dimensional parametric plots

Alternatively, vectors of the \(x\), \(y\), and \(z\) components can be produced and then plotted using the pattern lines(xs, ys) or lines(xs, ys, zs). For example, using unzip, as above, we might have done the prior example with lines(unzip(r.(ts))...).

Aspect ratio

A simple plot of a parametrically defined circle will show an ellipse, as the aspect ratio of the \(x\) and \(y\) axis is not \(1\). To enforce this, we can pass a value of aspect=1 to the underlying “Axis” object. Figure 59.16 provides an example.

Figure 59.16: Passing `aspect=1` to the axis make the `x` and `y`-axis scales equal

Tangent vectors (`arrows`)

A tangent vector along a curve can be drawn quite easily using the arrows function. There are different interfaces for arrows, but we show the one which uses a vector of positions and a vector of “vectors”. For the latter, we utilize the derivative function from ForwardDiff. In 3 dimensions the differences are minor, as seen in the code to produce Figure 59.17.

r(t) = [sin(t), cos(t)] # vector, not tuple
ts = range(0, 4pi, length=200)
lines(Point2.(r.(ts)))

nts = 0:pi/4:2pi
us = r.(nts)
dus = derivative.(r, nts)

arrows2d!(Point2.(us), Point2.(dus))

current_figure()
r(t) = [sin(t), cos(t), t] # vector, not tuple
ts = range(0, 4pi, length=200)
lines(Point3.(r.(ts)))

nts = 0:pi/2:(4pi-pi/2)
us = r.(nts)
dus = derivative.(r, nts)

arrows3d!(Point3.(us), Point3.(dus))

current_figure()

Figure 59.17: Plot of tangent lines in both two and three dimensions

Arrow attributes

Attributes for arrows include

arrowsize to adjust the size
lengthscale to scale the size
arrowcolor to set the color
arrowhead to adjust the head
arrowtail to adjust the tail

59.4 Surfaces

Plots of surfaces in \(3\) dimensions are useful to help understand the behavior of multivariate functions. There are a few common visualizations.

Surfaces defined through \(z=f(x,y)\)

The “peaks” function defined below has a few prominent peaks:

function peaks(x, y)
    p = 3*(1-x)^2*exp(-x^2 - (y+1)^2)
    p -= 10(x/5-x^3-y^5)*exp(-x^2-y^2)
    p -= 1/3*exp(-(x+1)^2-y^2)
    p
end

peaks (generic function with 1 method)

Figure 59.18 shows how peaks can be visualized over the region \([-5,5]\times[-5,5]\):

Figure 59.18: Surface plot produced by `surface(xs, ys, f)`

The calling pattern surface(xs, ys, f) implies a rectangular grid over the \(x\)-\(y\) plane defined by xs and ys with \(z\) values given by \(f(x,y)\).

Alternatively a “matrix” of \(z\) values can be specified. For a function f, this is conveniently generated by the pattern f.(xs, ys'), the ' being important to get a matrix of all \(x\)-\(y\) pairs through Julia’s broadcasting syntax.

zs = peaks.(xs, ys')
surface(xs, ys, zs);

Figure 59.19: Surface generated by surface(xs, ys, zs) where zs is a matrix of values

Example: surface graph construction

To see how a surface graph is constructed, the points \((x,y,f(x,y))\) are plotted over the grid and displayed.

In this examplpe, we downsample to illustrate.

Figure 59.20: Points used in downsampled graphic

The points in Figure 59.20 are then connected. The wireframe function illustrates just the frame in Figure 59.21.

Figure 59.21: Wireframe used in downsampled graphic

The surface call triangulates the frame and fills in the shading.

Figure 59.22: The `surface` shading used in the downsampled graphic

Parametrically defined surfaces

A surface may be parametrically defined through a function \(r(u,v) = (x(u,v), y(u,v), z(u,v))\). For example, the surface generated by \(z=f(x,y)\) is of the form with \(r(u,v) = (u,v,f(u,v))\).

The surface function and the wireframe function can be used to display such surfaces. In previous usages, the x and y values were vectors from which a 2-dimensional grid is formed. For parametric surfaces, a grid for the x and y values must be generated. This function will do so:

function parametric_grid(us, vs, r)
    n,m = length(us), length(vs)
    xs, ys, zs = zeros(n,m), zeros(n,m), zeros(n,m)
    for (i, uᵢ) in pairs(us)
        for (j, vⱼ) in pairs(vs)
            x,y,z = r(uᵢ, vⱼ)
            xs[i,j] = x
            ys[i,j] = y
            zs[i,j] = z
        end
    end
    (xs, ys, zs)
end

parametric_grid (generic function with 1 method)

With the data suitably massaged, we can directly plot either a surface or wireframe plot.

As an aside, The above can be done more campactly with nested list comprehensions:

xs, ys, zs = [[pt[i] for pt in r.(us, vs')] for i in 1:3]

Or using the unzip function directly after broadcasting:

xs, ys, zs = unzip(r.(us, vs'))

For example, a sphere can be parameterized by \(r(u,v) = (\sin(u)\cos(v), \sin(u)\sin(v), \cos(u))\) and visualized through these commands to produce Figure 59.23.

Figure 59.23: Part of sphere plotted using a parametric description of the data

A surface of revolution for \(g(u)\) revolved about the \(z\) axis can be visualized through the commands to produce Figure 59.24.

Figure 59.24: Surface of revolution formed by revolving `g` around the `z` axis.

A torus with big radius \(2\) and inner radius \(1/2\) is visualized in Figure 59.25.

Figure 59.25: Surface of torus plotted as a parametrically defined surface

A Möbius strip is produced in Figure 59.26.

Figure 59.26: A Möbius strip can be parameterized and displayed

59.5 Contour plots (`contour`, `contourf`, `heatmap`)

For a function \(z = f(x,y)\) an alternative to a surface plot, is a contour plot. That is, for different values of \(c\) the level curves \(f(x,y)=c\) are drawn.

For a function \(f(x,y)\), the syntax for generating a contour plot follows that for surface.

For example, using the peaks function, previously defined, we have a contour plot over the region \([-5,5]\times[-5,5]\) is generated through contour(xs, ys, peaks). A figure is shown in Figure 59.27`.

The default of \(5\) levels can be adjusted using the contour function’s levels keyword. Figure 59.28 show the peaks function with levels = 20. The levels argument can also specify precisely what levels are to be drawn.

Figure 59.28: Contour plot of `peaks` with the display of 20 levels specified

The contour graph makes identification of peaks and valleys easy as the limits of patterns of nested contour lines.

A filled contour plot is produced by contourf, as seen in Figure 59.29.

Figure 59.29: The `contourf` command produces filled contour plots

A related, but alternative visualization, using color to represent magnitude is a heatmap, produced by the heatmap function. The calling syntax is similar to contour and surface. Figure 59.30 shows peaks and valleys through “hotspots” on the graph.

Figure 59.30: Heatmap of `peaks` function

Example

The MakieGallery package includes an example of a surface plot with both a wireframe and 2D contour graph added. It is replicated here using the peaks function scaled by \(5\).

The function and domain to plot are described by:

xs = ys = range(-5, 5, length=51)
zs = peaks.(xs, ys') / 5;

The zs were generated, as wireframe does not provide the interface for passing a function.

The surface and wireframe graphics are produced as follows. In the following we manually create the figure and axis object (using Figure and Axis as shown). We do this to set the viewing angle through the elevation argument to the axis object. We plot onto this axis in producing Figure 59.31 and then display the Figure object.

Figure 59.31: Surface and wireframe for the `peaks` function

Next, we add a contour graph to Figure 59.31 to produce Figure 59.32. A simple call via contour!(scene, xs, ys, zs) will place the contour at the \(z=0\) level which will make it hard to read. Rather, placing at the “bottom” of the figure is desirable.

To identify that the minimum value, is identified (and rounded) and the argument transformation = (:xy, zmin) is passed to contour!:

Figure 59.32: Surface, wireframe, and contour for the `peaks` function

The transformation plot attribute sets the “plane” (one of :xy, :yz, or :xz) at a location, in this example zmin.

The manual construction of a figure and an axis object will be further discussed later.

59.5.1 Three dimensional contour plots

The contour function can also plot \(3\)-dimensional contour plots. Concentric spheres, contours of \(x^2 + y^2 + z^2 = c\) for \(c > 0\) are presented in Figure 59.33.

Figure 59.33: Three dimensional contour plot

59.5.2 Implicitly defined curves and surfaces

Suppose \(f\) is a scalar-valued function. If f takes two variables for its input, then the equation \(f(x,y) = 0\) implicitly defines \(y\) as a function of \(x\); \(y\) can be visualized locally with a curve. If \(f\) takes three variables for its input, then the equation \(f(x,y,z)=0\) implicitly defines \(z\) as a function of \(x\) and \(y\); \(z\) can be visualized locally with a surface.

Implicitly defined curves

The graph of an equation is the collection of all \((x,y)\) values satisfying the equation. This is more general than the graph of a function, which can be viewed as the graph of the equation \(y=f(x)\). An equation in \(x\)-\(y\) can be graphed if the set of solutions to a related equation \(f(x,y)=0\) can be identified, as one can move all terms to one side of an equation and define \(f\) as the rule of the side with the terms. The implicit function theorem ensures that under some conditions, locally near a point \((x, y)\), the value \(y\) can be represented as a function of \(x\). So, the graph of the equation \(f(x,y)=0\) can be produced by stitching together these local function representations.

The contour graph can produce these graphs by setting the levels argument to [0].

Figure 59.34: Using `contour` to graph an implicitly defined function

The implicitPlots.jl function uses the Contour package along with a Plots recipe to plot such graphs. Here we see how to use Makie in a similar manner:

import Contour

function implicit_plot(xs, ys, f; kwargs...)
    fig = Figure()
    ax = Axis(fig[1,1])
    implicit_plot!(ax, xs, ys, f; kwargs...)
    fig
end

function implicit_plot!(ax, xs, ys, f; kwargs...)
    z = [f(x, y) for x in xs, y in ys]
    cs = Contour.contour(collect(xs), collect(ys), z, 0.0)
    ls = Contour.lines(cs)

    isempty(ls) && error("empty")

    for l ∈ ls
        us, vs = Contour.coordinates(l)
        lines!(ax, us, vs; kwargs...)
    end

end

Implicitly defined surfaces, \(F(x,y,z)=0\)

To plot the equation \(F(x,y,z)=0\), for \(F\) a scalar-valued function, again the implicit function theorem says that, under conditions, near any solution \((x,y,z)\), \(z\) can be represented as a function of \(x\) and \(y\), so the graph will look like surfaces stitched together.

With Makie, many implicitly defined surfaces can be adequately represented using contour with the attribute levels=[0]. We will illustrate this technique.

GLMakie

The CairoMakie backend does not handle these next few plots, so we use GLMakie.contour in the following after importing with:

import GLMakie

To begin, we plot a sphere implicitly as a solution to \(F(x,y,z) = x^2 + y^2 + z^2 - 1 = 0\) in Figure 59.35.

Figure 59.35: Three dimensional implicitly defined surface plotted with `contour`

Figure 59.36 visualizes an intersection of a sphere with another figure. To show the different surfaces, different colormaps are chosen.

Figure 59.36: Two implicitly defined surfaces in three dimensions disambiguated through different colormaps

Figure 59.37 presents an example from Wikipedia showing an implicit surface of genus \(2\).

Figure 59.37: Implicit surface of a genus 2 function. This figure does not render well though, as the hole is not displayed.

The Implicit3DPlotting package takes an approach like ImplicitPlots to represent these surfaces. It replaces the Contour package computation with a \(3\)-dimensional alternative provided through the Meshing and GeometryBasics packages. This package has a plot_implicit_surface function that does something similar as just illustrated, but handles the “hole” not shown in Figure 59.37.

The plot_implicit_surface takes a function of a single argument, so we wrap f within splat which takes that single argument and “splats” them so f can be used. The result appears in Figure 59.38.

Figure 59.38: Implicit surface of a genus 2 function. This surface, rendered with the `Implicit3DPlotting` package, shows the holes present in the surface.

For one last example from Wikipedia, we have the Cassini oval which “can be defined as the point set for which the product of the distances to \(n\) given points is constant.”

Figure 59.39: Cassini oval implicitly defined

59.6 Vector fields. Visualizations of \(f:R^2 \rightarrow R^2\)

The vector field \(f(x,y) = \langle y, -x \rangle\) can be visualized as a set of vectors, \(f(x,y)\), positioned at a grid. These arrows can be visualized with the arrows function. The arrows function is passed a vector of points for the anchors and a vector of points representing the vectors.

We can generate these on a regular grid through:

f(x, y) = [y, -x]
xs = ys = -5:5
pts = vec(Point2.(xs, ys'))
dus = vec(Point2.(f.(xs, ys')));
first(pts), first(dus)  # show an example

([-5, -5], [-5, 5])

Broadcasting over (xs, ys') ensures each pair of possible values is encountered. The vec call reshapes an array into a vector.

Calling arrows on the prepared data produces the graphic:

Figure 59.40: Vectorfield plot generated by `arrows2d`. Modification is needed.

The grid seems rotated at first glance; but is also confusing. This is due to the length of the vectors as the \((x,y)\) values get farther from the origin. Plotting the normalized values (each will have length \(1\)) can be done easily using norm (which is found in the standard LinearAlgebra library):

Figure 59.41: Vectorfield plot generated by `arrows2d` after modification

The rotational pattern in Figure 59.41 is much clearer than from Figure 59.40.

The streamplot function also illustrates this phenomenon. This implements an “algorithm [that] puts an arrow somewhere and extends the streamline in both directions from there. Then, it chooses a new position (from the remaining ones), repeating the exercise until the streamline gets blocked, from which on a new starting point, the process repeats.”

The streamplot function expects a Point not a pair of values, so we adjust f slightly and call the function using the pattern streamplot(g, xs, ys):

Figure 59.42: Graph produced by `streamplot`

(We used interval notation to set the viewing range, a range could also be used.)

Note

The calling pattern of streamplot is different than other functions, such as surface, in that the function comes first.

59.7 Layoutables

The FigureAxisPlot comprises an enclosing figure and one or more axes. These can be constructed directly through a pattern like the following:

Figure 59.43: Basic pattern to construct a figure and an axis to layer on

Makie makes it really easy to piece together figures from individual plots. To illustrate, we create a graphic consisting of a plot of a function, its derivative, and its second derivative. In our graphic, we also leave space for a label.

Note

The Layout Tutorial has much more detail on this subject.

For laying out our own composite graphic, we manage the figure and axes manually. The commands below create a figure, then assign axes to certain portions of the figure:

F = Figure()
#af   = F[2,1:2]   = Axis(F)
#afp  = F[3,1:end] = Axis(F)
#afpp = F[4,:]     = Axis(F)
af   = Axis(F[2,1:2])
afp  = Axis(F[3,1:end])
afpp = Axis(F[4,:])

Makie.Axis with 0 plots:

The axes are named af, afp and afpp, as they will hold the respective graphs. The key here is the use of matrix notation to layout the graphic in a grid. The first one is row 2 and columns 1 through 2; the second row 3 and again all columns, the third is row 4 and all columns.

In this figure, we want the \(x\)-axis for each of the three graphics to be linked. This command ensures that:

linkxaxes!(af, afp, afpp);

By linking axes, if one is updated, say through xlims!, the others will be as well.

We now plot our functions. The key here is the mutating form of lines! takes an axis object to mutate as its first argument:

f(x) = 8x^4 - 8x^2 + 1
fp(x) = 32x^3 - 16x
fpp(x) = 96x^2 - 16

xs = -1..1
lines!(af,   xs, f)
lines!(afp,  xs, fp)
lines!(afp,  xs, zero, color=:blue)
lines!(afpp, xs, fpp)
lines!(afpp, xs, zero, color=:blue);

We can give title information to each axis on construction or after construction, through commands like:

af.title = "f"
afp.title = "fp"
afpp.title = "fpp";

Finally, we add a label in the first row, but for illustration purposes, only use the first column.

Label(F[1,1], """
Plots of f and its first and second derivatives.
When the first derivative is zero, the function
f has relative extrema. When the second derivative
is zero, the function f has an inflection point.
""");

We display the figure in Figure 59.44

Figure 59.44: Figure containing multiple axes

59.8 Observables

This needs updating

As over v"0.24" of Makie, there is an alternative to using Observables. These notes need updating to reflect that change.

The basic components of a plot in Makie can be updated interactively. Historically Makie used the Observables package which allows complicated interactions to be modeled quite naturally. In the following we give a simple example, though newer versions of Makie rely on a different mechanism.

In Makie, an Observable is a structure that allows its value to be updated, similar to an array. When changed, observables can trigger an event. Observables can rely on other observables, so events can be cascaded.

This simple example shows how an observable h can be used to create a collection of points representing a secant line. The figure shows the value for h=3/2.

Figure 59.45: Illustration of using an `Observable`

We can update the value of h using setindex! notation (square brackets). For example, to see that the secant line is a good approximation to the tangent line as \(h \rightarrow 0\) we can set h to be 1/4 and replot in Figure 59.46.

Figure 59.46: Same plot as in Figure 59.45 with `h'` value updated

The line h[] = 1/4 updated h which then updated points (a points is lifted up from h) which updated the graphic. (In these notes, we replot to see the change, but in an interactive session, the current displayed figure would be updated; no replotting would be necessary.)

Finally, this example shows how to add a slider to adjust the value of h with a mouse. The slider object is positioned along with a label using the grid reference, as before.

let
f(x) = sqrt(x)
c = 1
xs = 0..3

F = Figure()
ax = Axis(F[1,1:2])
h = Slider(F[2,2], range = 0.01:0.01:1.5, startvalue = 1.5)
Label(F[2,1], "Adjust slider to change `h`";
      justification = :left)

points = lift(h.value) do h
    xs = [0,c,c+h,3]
    tl = x-> f(c) + (f(c+h)-f(c))/h * (x-c)
    [Point2(x, tl(x)) for x ∈ xs]
end

lines!(ax, xs, f)
lines!(ax, points)
scatter!(ax, points; markersize=10)
    current_figure()
end

A slider added in position F[2,2] and connected through lift

The slider value is “lifted” by its value component, as shown. Otherwise, the above is fairly similar to just using an observable for h.

This example is modified from a discourse post by user @rafael.guerra.↩︎
This example annotates the underlying Axis object, extracted using tuple destructuring. A more direct approach of creating a Figure object and then an Axis object will be illustrated later.↩︎