11.3: Other Function Plots

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Here's some preliminary commands to run if they haven't been yet.

using Pkg; Pkg.add("Plots")

   Updating registry at `/srv/julia/pkg/registries/General`
  Resolving package versions...
Updating `~/Project.toml`
  [91a5bcdd] + Plots v1.14.0
Updating `~/Manifest.toml`
  [6e34b625] + Bzip2_jll v1.0.6+5
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using Plots

Parametric Plots

Recall that a parametric curve is a set of points in the \(xy\)-plane given by \( (x(t),y(t)\) for functions \(x(t)\) and \(y(t)\). The variable \(t\) is called the parameter. A classic example is the circle that can be written as

\[x(t) = \cos t, \qquad y(t) = \sin t \nonumber \]

To plot the circle using this form, enter

plot(t->cos(t),t->sin(t),0,2*pi,legend=false)

UndefVarError: plot not defined

Stacktrace:
 [1] top-level scope at In[1]:1
 [2] include_string(::Function, ::Module, ::String, ::String) at ./loading.jl:1091

and note that the legend is turned off, since with one curve, it doesn’t make much sense. Notice that this should be a circle, but it looks like an ellipse due to the aspect ratio. If one instead adds the aspect_ratio=:equal option, as in

plot(t->cos(t),t->sin(t),0,2*pi,aspect_ratio=:equal, legend=false)

Exercise

Produce a plot of the curve \( x(t) = t^3-t, y(t)=t^2 \) for \(-2 \leq t \leq 2\).

# insert your code here

Implicit Curves

An implicit curve is the set of points such that \(f (x, y) = 0\) (or any constant) and a circle is the classic example. For example, \(x^2+y^2=1\). Although there are others ways of doing this, we will use some algebra to write the circle as \( f(x,y) = x^2+y^2-1 \).

We can plot this with the contourfunction for example

contour(-1.05:0.05:1.05, -1.05:0.05:1.05, (x,y) -> x^2+y^2-1, levels=[0], aspect_ratio = :equal, legend = false)

Note that again, we have used the option aspect_ratio = :equal to ensure that the circle looks like a circle. The resulting plot is exactly the same as the circle above.

The following example is a bit more visually interesting. This is the function \( f(x,y) = sin(x+y)-cos(xy)+1 \)

contour(-10.1:0.1:10.1, -10.1:0.1:10.1, (x,y) -> sin(x+y)-cos(x*y)+1, levels=[0], aspect_ratio = :equal, colorbar_entry = false)

Exercise

A cardiod is a 2D curve that looks a bit like a heart. It can be represented by an implicit curve with the equation

\[ (x^2+y^2)^2 +4ax(x^2+y^2) -4a^2y^2 = 0 \nonumber \]

Plot the cardiod with \(a=1\) using the code block below

# insert your code here

Surface Plots

If we have a function of 2 variables, a surface plot is nice to use. For example, if we have the function

\[f(x,y) = e^{-0.1(x^2+y^2)} \label{3dbell}\]

and we want to plot it from -3 to 3 in both directions, if we define

f(x,y)=exp(-0.1*(x^2+y^2))
x = y = range(-5, stop = 5, length = 40)

-5.0:0.2564102564102564:5.0

and then plot with

surface(x,y, f, legend = false)

UndefVarError: surface not defined

Stacktrace:
 [1] top-level scope at In[2]:1
 [2] include_string(::Function, ::Module, ::String, ::String) at ./loading.jl:1091

Exercise

Produce a surface plot of the function \(f(x,y)=\sin x \cos y\) for \(0\leq x \leq 2\pi, 0\leq y \leq 2\pi\) using the code block below

# insert your code here

Heat Maps

A heat map is a 2-dimensional version of a surface plot in which the height of each value is given a color. The following produces a heat map of the function in \ref{3dbell}

heatmap(x,y,f)

UndefVarError: heatmap not defined

Stacktrace:
 [1] top-level scope at In[2]:1
 [2] include_string(::Function, ::Module, ::String, ::String) at ./loading.jl:1091

Exercise

Produce a surface plot of the function \(f(x,y)=\sin x \cos y\) for \(0\leq x \leq 2\pi, 0\leq y \leq 2\pi\) using the code block below

# Insert your code here