11.3: Other Function Plots
- Page ID
- 63930
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Here's some preliminary commands to run if they haven't been yet.
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
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
Exercise
Produce a plot of the curve \( x(t) = t^3-t, y(t)=t^2 \) for \(-2 \leq t \leq 2\).
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
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 \( f(x,y) = sin(x+y)-cos(xy)+1 \)
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
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
and then plot with
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
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}
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