# Calculus plots with Makie


{{< include ../_common_code.qmd >}}

The [Makie.jl webpage](https://github.com/JuliaPlots/Makie.jl) 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.


:::{.callout-note}
## Examples and tutorials
`Makie` is a sophisticated plotting package, and capable of an enormous range of plots (cf. [examples](https://makie.juliaplots.org/stable/examples/plotting_functions/).) `Makie` also has numerous [tutorials](https://makie.juliaplots.org/stable/tutorials/) to learn from. These are far more extensive than what is described herein, as this section focuses just on the graphics from calculus.

:::

## 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:


```{julia}
using GLMakie
import LinearAlgebra: norm
```

The `Makie` developers have workarounds for the delayed time to first plot, but without utilizing these the time to load the package is lengthy.


## 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`:


```{julia}
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.


### `Point2`, `Point3`


When learning about points on the Cartesian plane, a "`t`"-chart is often produced:


```
x | y
-----
1 | 2
2 | 3
3 | 2
```

The `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:


```{julia}
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:


```{julia}
r(t) = [sin(t), cos(t)]
```

Then broadcasting values gives a vector of vectors, each identified with a point:


```{julia}
ts = [1,2,3]
r.(ts)
```

We can broadcast `Point2` over this to create a vector of `Point` objects:


```{julia}
pts = Point2.(r.(ts))
```

These then can be plotted directly:


```{julia}
scatter(pts)
```

The plotting of points in three dimensions is essentially the same, save the use of `Point3` instead of `Point2`.


```{julia}
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.


```{julia}
unzip(vs) = Tuple([vs[j][i] for j in eachindex(vs)] for i in eachindex(vs[1]))
```

:::{.callout-note}
## Note
In the `CalculusWithJulia` package, `unzip` is implemented using `SplitApplyCombine.invert`.

:::

We might have then:


```{julia}
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))`.)


### Attributes


A point is drawn with a "marker" with a certain size and color. These attributes can be adjusted, as in the following:


```{julia}
scatter(xs, ys;
        marker=[:x,:cross, :circle], markersize=25,
        color=:blue)
```

Marker attributes include


  * `marker` a symbol, shape.
  * `marker_offset` offset coordinates
  * `markersize` size (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.


## 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.


### 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:


```{julia}
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:


```{julia}
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:


```{julia}
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:


```{julia}
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:


```{julia}
xs = range(0, 2pi, length=100)
lines(xs, sin.(xs))
lines!(xs, cos.(xs))
current_figure()
```

:::{.callout-note}
## 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:


```{julia}
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`.


```{julia}
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](https://discourse.julialang.org/t/how-to-plot-step-functions-x-correctly-in-julia/84087/5) post by user `@rafael.guerra`, shows how to plot a step function (`floor`) using `NaN`s to create line breaks. The marker colors set for `scatter!` use `:white` to match the background color.


```{julia}
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()
```

### 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:


```{julia}
xs = 1:5
pts = Point2.(xs, xs)
scatter(pts)
annotation!(pts;
            text = "Point " .* string.(xs),
            fontsize = 30 .- 5*xs)

current_figure()
```

The graphic shows that `fontsize` adjusts the displayed size.


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. This example is modified from the documentation and utilizes some interval functions to draw an arrow with an arc:

```{julia}
g(x) = cos(6x) * exp(x)
xs = 0:0.01:4

_, ax, _ = lines(xs, g.(xs); axis = (; xgridvisible = false, ygridvisible = false))

annotation!(ax, 1, 20, 2.1, g(2.1),
    text = "A relative maximum",
    path = Ann.Paths.Arc(0.3),
    style = Ann.Styles.LineArrow(),
    labelspace = :data
)

current_figure()
```


#### 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:


```{julia}
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:


```{julia}
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:


```{julia}
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:


```{julia}
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:


```{julia}
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.


### 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:


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

Or


```{julia}
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:


```{julia}
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:


```{julia}
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`:


```{julia}
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:


```{julia}
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


  * `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


## 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:


```{julia}
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
```

Here we see how `peaks` can be visualized over the region $[-5,5]\times[-5,5]$:


```{julia}
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.


```{julia}
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:


```{julia}
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:


```{julia}
wireframe(xs, ys, peaks.(xs, ys'); linewidth=5)
```

The `surface` call triangulates the frame and fills in the shading:


```{julia}
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:


```{julia}
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
```

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:


```{julia}
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:


```{julia}
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


```{julia}
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:


```{julia}
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()
```

## 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:


```{julia}
xs = ys = range(-5, 5, length=100)
contour(xs, ys, peaks)
```

The default of $5$ levels can be adjusted using the `levels` keyword:


```{julia}
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`:


```{julia}
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`:


```{julia}
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:


```{julia}
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:


```{julia}
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()
```

To add the contour, 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!`:


```{julia}
ezs = extrema(zs)
zmin, zmax = floor(first(ezs)), ceil(last(ezs))
contour!(ax3, xs, ys, zs;
         levels = 15, linewidth = 2,
         transformation = (:xy, zmin))
zlims!(zmin, zmax)
current_figure()
```

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.


### 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 by the following:


```{julia}
f(x,y,z) = x^2 + y^2 + z^2
xs = ys = zs = range(-3, 3, length=100)

contour(xs, ys, zs, f)
```

### 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]`.


```{julia}
f(x,y) = x^3 + x^2 + x + 1 - x*y        # solve x^3 + x^2 + x + 1 = x*y
xs = range(-5, 5, length=100)
ys = range(-10, 10, length=100)

contour(xs, ys, f.(xs, ys'); levels=[0])
```

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:


```{julia}
#| eval: false
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 `countour` with the attribute `levels=[0]`. We will illustrate this technique.

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 to below. We don't illustrate it, as it *currently* doesn't work with the latest version of `Makie`.

To begin, we plot a sphere implicitly as a solution to $F(x,y,z) = x^2 + y^2 + z^2 - 1 = 0$>


```{julia}
f(x,y,z) = x^2 + y^2 + z^2 - 1
xs = ys = zs = range(-3/2, 3/2, 100)
contour(xs, ys, zs, f; levels=[0])
```



Here we visualize an intersection of a sphere with another figure:


```{julia}
r₂(x) = sum(x.^2) - 2 # a sphere
r₄(x) = sum(x.^4) - 1
ϕ(x,y,z) = (x,y,z)

xs = ys = zs = range(-2, 2, 100)
contour(xs, ys, zs, r₂∘ϕ; levels = [0], colormap=:RdBu)
contour!(xs, ys, zs, r₄∘ϕ; levels = [0], colormap=:viridis)
current_figure()
```

This example comes from [Wikipedia](https://en.wikipedia.org/wiki/Implicit_surface) showing an implicit surface of genus $2$:


```{julia}
f(x,y,z) = 2y*(y^2 -3x^2)*(1-z^2) + (x^2 +y^2)^2 - (9z^2-1)*(1-z^2)
xs = ys = zs = range(-5/2, 5/2, 100)
contour(xs, ys, zs, f; levels=[0], colormap=:RdBu)

```

(This figure does not render well though, as the hole is not shown.)


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." That is:


```{julia}
function cassini(λ, ps = ((1,0,0), (-1, 0, 0)))
    n = length(ps)
    x -> prod(norm(x .- p) for p ∈ ps) - λ^n
end
xs = ys = zs = range(-2, 2, 100)
contour(xs, ys, zs, cassini(0.80) ∘ ϕ; levels=[0], colormap=:RdBu)
```

## 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:


```{julia}
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
```

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:


```{julia}
arrows(pts, dus)
```

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):


```{julia}
dvs = dus ./ norm.(dus)
arrows(pts, dvs)
```

The rotational pattern becomes much clearer now.


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 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)`:


```{julia}
f(x, y) = [y, -x]
g(xs) = Point2(f(xs...))

streamplot(g, -5..5, -5..5)
```

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


:::{.callout-note}
## Note
The calling pattern of `streamplot` is different than other functions, such as `surface`, in that the function comes first.

:::

## Layoutables and Observables


### Layoutables


`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.


:::{.callout-note}
## Note
The Layout [Tutorial](https://makie.juliaplots.org/stable/tutorials/layout-tutorial/) has *much* more detail on this subject.

:::

The basic plotting commands, like `lines`, return a `FigureAxisPlot` object. For laying out our own graphic, we manage the figure and axes manually. The commands below create a figure, then assign axes to portions of the figure:


```{julia}
F = Figure()
af   = F[2,1:2]   = Axis(F)
afp  = F[3,1:end] = Axis(F)
afpp = F[4,:]     = Axis(F)
```

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:


```{julia}
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:


```{julia}
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:


```{julia}
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.


```{julia}
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.
""");
```

Finally we display the figure:


```{julia}
F
```

### Observables


The basic components of a plot in `Makie` can be updated [interactively](https://makie.juliaplots.org/stable/documentation/nodes/index.html#observables_interaction). `Makie` uses the `Observables` package which allows complicated interactions to be modeled quite naturally. In the following we give a simple example.


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`.


```{julia}
f(x) = sqrt(x)
c = 1
xs = 0..3
h = Observable(3/2)

points = lift(h) 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(xs, f)
lines!(points)
current_figure()
```

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:


```{julia}
h[] = 1/4
current_figure()
```

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.


```{julia}
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)
current_figure()
```

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