using GLMakie
import LinearAlgebra: norm68 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 GLMakie backend to produce the familiar graphics of calculus.
Examples and tutorials
68.1 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. The show method for these objects display the figure.
For Makie there are the GLMakie, WGLMakie, and CairoMakie backends for different types of canvases. In the following, we have used GLMakie. WGLMakie is useful for incorporating Makie plots into web-based technologies.
We begin by loading the main package and the norm function from the standard LinearAlgebra package:
The Makie developers have workarounds for the delayed time to first plot, but without utilizing these the time to load the package is lengthy.
68.2 Points (scatter)
The task of plotting the points, say \((1,2)\), \((2,3)\), \((3,2)\) can be done different ways. Most plotting packages, and Makie is no exception, allow the following: form vectors of the \(x\) and \(y\) values then plot those with scatter:
xs = [1,2,3]
ys = [2,3,2]
scatter(xs, ys)
The scatter function creates and returns an object, which when displayed shows the plot.
68.2.1 Point2, Point3
When learning about points on the Cartesian plane, a “t”-chart is often produced:
x | y
-----
1 | 2
2 | 3
3 | 2The scatter usage above used the columns. The rows are associated with the points, and these too can be used to produce the same graphic. Rather than make vectors of \(x\) and \(y\) (and optionally \(z\)) coordinates, it is more idiomatic to create a vector of “points.” Makie utilizes a Point type to store a 2 or 3 dimensional point. The Point2 and Point3 constructors will be utilized.
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.
We can plot a vector of points in as direct manner as vectors of their coordinates:
pts = [Point2(1,2), Point2(2,3), Point2(3,2)]
scatter(pts)A typical usage is to generate points from some vector-valued function. Say we have a parameterized function r taking \(R\) into \(R^2\) defined by:
r(t) = [sin(t), cos(t)]r (generic function with 1 method)Then broadcasting values gives a vector of vectors, each identified with a point:
ts = [1,2,3]
r.(ts)3-element Vector{Vector{Float64}}:
[0.8414709848078965, 0.5403023058681398]
[0.9092974268256817, -0.4161468365471424]
[0.1411200080598672, -0.9899924966004454]We can broadcast Point2 over this to create a vector of Point objects:
pts = Point2.(r.(ts))3-element Vector{Point{2, Float64}}:
[0.8414709848078965, 0.5403023058681398]
[0.9092974268256817, -0.4161468365471424]
[0.1411200080598672, -0.9899924966004454]These then can be plotted directly:
scatter(pts)
The plotting of points in three dimensions is essentially the same, save the use of Point3 instead of Point2.
r(t) = [sin(t), cos(t), t]
ts = range(0, 4pi, length=100)
pts = Point3.(r.(ts))
scatter(pts; markersize=5)
To plot points generated in terms of vectors of coordinates, the component vectors must be created. The “t”-table shows how, simply loop over each column and add the corresponding \(x\) or \(y\) (or \(z\)) value. This utility function does exactly that, returning the vectors in a tuple.
unzip(vs) = Tuple([vs[j][i] for j in eachindex(vs)] for i in eachindex(vs[1]))unzip (generic function with 1 method)
Note
In the CalculusWithJulia package, unzip is implemented using SplitApplyCombine.invert.
We might have then:
scatter(unzip(r.(ts))...; markersize=5)where splatting is used to specify the xs, ys, and zs to scatter.
(Compare to scatter(Point3.(r.(ts))) or scatter(Point3∘r).(ts)).)
68.2.2 Attributes
A point is drawn with a “marker” with a certain size and color. These attributes can be adjusted, as in the following:
scatter(xs, ys;
marker=[:x,:cross, :circle], markersize=25,
color=:blue)
Marker attributes include
markera symbol, shape.marker_offsetoffset coordinatesmarkersizesize (radius pixels) of marker
A single value will be repeated. A vector of values of a matching size will specify the attribute on a per point basis.
68.3 Curves
The curves of calculus are lines. 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.
68.3.1 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:
f(x) = sin(x)
a, b = 0, 2pi
xs = range(a, b, length=250)
lines(xs, f.(xs))
Makie also will read the interval notation of IntervalSets and select its own set of intermediate points:
lines(a..b, f)
As with scatter, lines returns an object that produces a graphic when displayed.
As with scatter, lines can can also be drawn using a vector of points:
pts = [Point2(x, f(x)) for x ∈ xs]
lines(pts)(Though the advantage isn’t clear here, this will be useful when the points are generated in different manners.)
When a y value is NaN or infinite, the connecting lines are not drawn:
xs = 1:5
ys = [1,2,NaN, 4, 5]
lines(xs, ys)
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 scene 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:
xs = range(0, 2pi, length=100)
lines(xs, sin.(xs))
lines!(xs, cos.(xs))
current_figure()
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.
The following shows the construction details in the graphic:
xs = range(0, 2pi, length=10)
lines(xs, sin.(xs))
scatter!(xs, sin.(xs);
markersize=10)
current_figure()
As an example, this shows how to add the tangent line to a graph. The slope of the tangent line being computed by ForwardDiff.derivative.
import ForwardDiff
f(x) = x^x
a, b= 0, 2
c = 0.5
xs = range(a, b, length=200)
tl(x) = f(c) + ForwardDiff.derivative(f, c) * (x-c)
lines(xs, f.(xs))
lines!(xs, tl.(xs), color=:blue)
current_figure()
This example, modified from a discourse post by user @rafael.guerra, 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.
x = -5:5
δ = 5eps() # for rounding purposes; our interval is [i,i+1) ≈ [i, i+1-δ]
xx = Float64[]
for i ∈ x[1:end-1]
append!(xx, (i, i+1 - δ, NaN))
end
yy = floor.(xx)
lines(xx, yy)
scatter!(xx, yy, color=repeat([:black, :white, :white], length(xx)÷3))
current_figure()
68.3.2 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:
xs = 1:5
pts = Point2.(xs, xs)
scatter(pts)
annotations!("Point " .* string.(xs), pts;
fontsize = 50 .- 2*xs,
rotation = 2pi ./ xs)
current_figure()
The graphic shows that fontsize adjusts the displayed size and rotation adjusts the orientation. (The graphic also shows a need to manually override the limits of the y axis, as the Point 5 is chopped off; the ylims! function to do so will be shown later.)
Attributes for text, among many others, include:
alignSpecify the text alignment through(:pos, :pos), where:poscan be:left,:center, or:right.rotationto indicate how the text is to be rotatedfontsizethe font point size for the textfontto indicate the desired font
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:
colorset with a symbol, as above, or a stringlabela label for the line to display in a legendlinestyleavailable styles are set by a symbol, one of:dash,:dot,:dashdot, or:dashdotdot.linewidthwidth of linetransparencythealphavalue, 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:
xs = 0..2pi
lines(xs, sin;
axis=(title="Plot of sin(x)", xlabel="x", ylabel="sin(x)")
)
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:
xs = 0..2pi
p = lines(xs, sin;
axis=(title="Plot of sin(x)", xlabel="x", ylabel="sin(x)")
)
p.axis.xticks = MultiplesTicks(5, pi, "π") # label 5 times using `pi`
current_figure()
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 resolution, \(x\) 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 resolution. As with attributes for the axis, these too can be passed to a simple constructor:
lines(xs, sin;
axis=(title="Plot of sin(x)", xlabel="x", ylabel="sin(x)"),
figure=(;resolution=(300, 300))
)
The ; in the tuple passed to figure is one way to create a named tuple with a single element. Alternatively, (resolution=(300,300), ) – with a trailing comma – could have been used.
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 this example:
f(x) = 1/x
a,b = -1, 1
xs = range(-1, 1, length=200)
lines(xs, f.(xs))
ylims!(-10, 10)
current_figure()
This still leaves the artifact due to the vertical asymptote at \(0\) having different values from the left and the right.
68.3.3 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)
Or
r(t) = [sin(2t), cos(3t), t]
ts = range(0, 2pi, length=200)
pts = Point3.(r.(ts))
lines(pts)
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:
xs, ys, zs = unzip(r.(ts))
lines(xs, ys, zs)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. For example:
ts = range(0, 2pi, length=100)
xs, ys = sin.(ts), cos.(ts)
lines(xs, ys; axis=(; aspect = 1))
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:
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 = ForwardDiff.derivative.(r, nts)
arrows!(Point2.(us), Point2.(dus))
current_figure()
In 3 dimensions the differences are minor:
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 = ForwardDiff.derivative.(r, nts)
arrows!(Point3.(us), Point3.(dus))
current_figure()
Arrow attributes
Attributes for arrows include
arrowsizeto adjust the sizelengthscaleto scale the sizearrowcolorto set the colorarrowheadto adjust the headarrowtailto adjust the tail
68.4 Surfaces
Plots of surfaces in \(3\) dimensions are useful to help understand the behavior of multivariate functions.
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
endpeaks (generic function with 1 method)Here we see how peaks can be visualized over the region \([-5,5]\times[-5,5]\):
xs = ys = range(-5, 5, length=25)
surface(xs, ys, peaks)
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)To see how this graph is constructed, the points \((x,y,f(x,y))\) are plotted over the grid and displayed.
Here we downsample to illustrate:
xs = ys = range(-5, 5, length=5)
pts = [Point3(x, y, peaks(x,y)) for x in xs for y in ys]
scatter(pts, markersize=25)
These points are then connected. The wireframe function illustrates just the frame:
wireframe(xs, ys, peaks.(xs, ys'); linewidth=5)
The surface call triangulates the frame and fills in the shading:
surface!(xs, ys, peaks.(xs, ys'))
current_figure()
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)
endparametric_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:
r(u,v) = [sin(u)*cos(v), sin(u)*sin(v), cos(u)]
us = range(0, pi, length=25)
vs = range(0, pi/2, length=25)
xs, ys, zs = parametric_grid(us, vs, r)
surface(xs, ys, zs)
wireframe!(xs, ys, zs)
current_figure()
A surface of revolution for \(g(u)\) revolved about the \(z\) axis can be visualized through:
g(u) = u^2 * exp(-u)
r(u,v) = (g(u)*sin(v), g(u)*cos(v), u)
us = range(0, 3, length=10)
vs = range(0, 2pi, length=10)
xs, ys, zs = parametric_grid(us, vs, r)
surface(xs, ys, zs)
wireframe!(xs, ys, zs)
current_figure()
A torus with big radius \(2\) and inner radius \(1/2\) can be visualized as follows
r1, r2 = 2, 1/2
r(u,v) = ((r1 + r2*cos(v))*cos(u), (r1 + r2*cos(v))*sin(u), r2*sin(v))
us = vs = range(0, 2pi, length=25)
xs, ys, zs = parametric_grid(us, vs, r)
surface(xs, ys, zs)
wireframe!(xs, ys, zs)
current_figure()
A Möbius strip can be produced with:
ws = range(-1/4, 1/4, length=8)
thetas = range(0, 2pi, length=30)
r(w, θ) = ((1+w*cos(θ/2))*cos(θ), (1+w*cos(θ/2))*sin(θ), w*sin(θ/2))
xs, ys, zs = parametric_grid(ws, thetas, r)
surface(xs, ys, zs)
wireframe!(xs, ys, zs)
current_figure()
68.5 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:
xs = ys = range(-5, 5, length=100)
contour(xs, ys, peaks)
The default of \(5\) levels can be adjusted using the levels keyword:
contour(xs, ys, peaks; levels = 20)
The levels argument can also specify precisely what levels are to be drawn.
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:
contourf(xs, ys, peaks)
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:
heatmap(xs, ys, peaks)
This graph shows peaks and valleys through “hotspots” on the graph.
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 are produced as follows. Here we manually create the figure and axis object so that we can set the viewing angle through the elevation argument to the axis object:
fig = Figure()
ax3 = Axis3(fig[1,1];
elevation=pi/9, azimuth=pi/16)
surface!(ax3, xs, ys, zs)
wireframe!(ax3, xs, ys, zs;
overdraw = true, transparency = true,
color = (:black, 0.1))
current_figure()