ch8_tikz_plot.tex

\chapter{Plotting with TikZ (Part I)}

\paragraph{Introduction} This chapter introduces the usage of the \textbf{TikZ} engine to draw various mathematical plots and diagrams in \LaTeX{}. This only means to be a brief guide, and readers should consult the \textbf{PGF Manual} \href{https://tikz.dev/pgfplots/}{https://tikz.dev/pgfplots/} for full details.

\section{Basic Drawing Syntax}

\subsection{Coordinates and Nodes} 

\paragraph{Cartesian Coordinates}
To create a \textbf{TikZ}\index{TikZ} plot, we first have to import the \texttt{pgfplots}\index{pgfplots@\texttt{pgfplots}} package and initialize a \texttt{tikzpicture}\index{tikzpicture@\texttt{tikzpicture}} environment. We will start by specifying \textit{coordinates}\index{Coordinate} and labeling that point on the plot as a \textit{node}\index{Node}. A simple example is given in Figure \ref{fig:coordnodes} below, and the corresponding code is
\begin{lstlisting}
\begin{tikzpicture}
\draw[help lines] (0,0) grid (4,3);
\coordinate (A) at (2,1);
\coordinate (B) at (3,3);
\node at (A) {\Large $(2,1)$};
\node at (B) {\footnotesize Another point}; % equivalently, directly use \node at (3,3) {\footnotesize Another point};
\end{tikzpicture}    
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \draw[help lines] (0,0) grid (4,3);
    \coordinate (A) at (2,1);
    \coordinate (B) at (3,3);
    \node at (A) {\Large $(2,1)$};
    \node at (B) {\footnotesize Another point};
    \end{tikzpicture}
    \caption{Simple Cartesian coordinates as nodes in TikZ.}
    \label{fig:coordnodes}
\end{figure}
The \texttt{\textbackslash draw[help lines]} command sketches helper grids for refining the positioning. The \texttt{\textbackslash coordinate (<name>) at (<coordinates>)}\index{coordinate@\texttt{\textbackslash coordinate}} syntax marks the coordinates of a point internally for later use. Here we use the simplest \textit{Cartesian}\index{Cartesian Coordinates} $xy$-coordinates. The \texttt{\textbackslash node at (<coordinates>) \{<text>\}}\index{node@\texttt{\textbackslash node}} command then puts a node, possibly with some text, at the corresponding position.

\paragraph{Polar Coordinates}
Another common type of coordinates is the \textit{polar}\index{Polar Coordinates} coordinates, whose expression is
\texttt{(angle:radius)} where \texttt{angle} is relative to the positive $x$-axis. This is illustrated in the following Figure \ref{fig:coordpolar}:
\begin{lstlisting}
\begin{tikzpicture}
\draw[help lines] (0,0) grid (4,3);
\coordinate (O) at (0,0);
\coordinate[label=above:$A$] (A) at (30:4);
\node at (O) [below left] {$O$}; % below left can be replaced by anchor=north east
\node at (A) [circle,fill,inner sep=2pt] {};
\end{tikzpicture}
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \draw[help lines] (0,0) grid (4,3);
    \coordinate (O) at (0,0);
    \coordinate[label=above:$A$] (A) at (30:4);
    \node at (O) [below left] {$O$};
    \node at (A) [circle,fill,inner sep=2pt] {};
    \end{tikzpicture}
    \caption{Defining a point in TikZ using polar form instead.}
    \label{fig:coordpolar}
\end{figure}
Point $A$ is then positioned at $(4\cos{30^\circ},4\sin{30^\circ}) = (2\sqrt{3},2)$.

There are also some other new things. We can place the node text $O$ below and to the left of the origin coordinates by adding \texttt{[below left]} (as you may have guessed, there are also \texttt{above}, \texttt{right}, and their combinations) before it. However, notice that it will also displace the node. Another labeling method is to provide the \texttt{[label=<position>:<text>]}\index{label@\texttt{label}} option when calling \texttt{\textbackslash coordinate}, which has been applied to point $A$. Then, we can make a dot to denote the point by using \texttt{\textbackslash node} with the set of options \texttt{[circle,fill,inner sep=2pt]}\index{inner sep@\texttt{inner sep}} so it fills a small circle with size $2$ pt.

\subsection{Drawing Paths}

\paragraph{Straight Lines}
Given some coordinates, a natural next step is to connect them with \textit{curves}. We will deal with the simplest case of \textit{straight lines}\index{Line} first. The basic \textit{path}\index{Path} syntax is \verb|(coordinates) -- (coordinates)|\index{\texttt{-{}-}}, and can be stacked as we like. This is demonstrated in Figure \ref{fig:path1} on the next page.
\begin{lstlisting}
\begin{tikzpicture}
\draw[help lines] (-3,-3) grid (3,3);
\coordinate[label=$A$] (A) at (2,1);
\coordinate[label=$B$] (B) at (-2,0);
\coordinate[label=below:$C$] (C) at (1,-1);
\draw[blue,dashed] (-1,-3) -- node[midway,sloped]{Cut} (3,3);
\path[red,draw] (A) -- (B) -- (C) -- cycle;
\end{tikzpicture}    
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \draw[help lines] (-3,-3) grid (3,3);
    \coordinate[label=$A$] (A) at (2,1);
    \coordinate[label=$B$] (B) at (-2,0);
    \coordinate[label=below:$C$] (C) at (1,-1);
    \draw[blue,dashed] (-1,-3) -- node[midway,sloped,above]{Cut} (3,3);
    \path[red,draw] (A) -- (B) -- (C) -- cycle;
    \end{tikzpicture}
    \caption{Drawing a line and a closed path as a triangle.}
    \label{fig:path1}
\end{figure}
We have two possible methods to draw a line. The first one is to just use the \texttt{\textbackslash draw}\index{draw@\texttt{\textbackslash draw}} command, whereas the second one is more verbose and uses the \texttt{\textbackslash path}\index{path@\texttt{\textbackslash path}} command combined with the \texttt{draw} option. For the triangle, the \texttt{cycle}\index{cycle@\texttt{cycle}} alias tells the path to travel back to the initial point. Other takeaways are that we can supply color (\texttt{red}, \texttt{blue}) and line style (\texttt{dashed}\index{dashed@\texttt{dashed}}, \texttt{dotted}\index{dotted@\texttt{dotted}}) when drawing the path, and we can put a label over the line by adding the \texttt{node} syntax after the \verb|--| part with the options \texttt{midway}\index{midway@\texttt{midway}} (or \texttt{pos=0.5}\index{pos@\texttt{pos}}, to put it in the middle) and \texttt{sloped}\index{sloped@\texttt{sloped}} (sloped with respect to the line).

\paragraph{Relative Coordinates}
Sometimes it is more convenient to specify coordinates relative to the previous one when constructing a path. This is done by adding the incremental \verb|++|\index{\texttt{++}} after \verb|--|. Figure \ref{fig:relativecoords} below is an illustrative example.
\begin{lstlisting}
\begin{tikzpicture}
\draw[help lines] (-3,-3) grid (3,3);
\coordinate[label=below right:$O$] (O) at (0,0);
\draw[Green, line width=1.5] (O) -- (2,1) coordinate[at end](A) --++ (-1.5,1) coordinate[at end](B) --++ (-3,-3) coordinate[at end](C) --++ (-30:2) coordinate[at end](D) -- cycle;
\node[right] at (A) {$A$}; \node[above] at (B) {$B$}; \node[left] at (C) {$C$};\node[below] at (D) {$D$};
\end{tikzpicture}    
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \draw[help lines] (-3,-3) grid (3,3);
    \coordinate[label=below right:$O$] (O) at (0,0);
    \draw[Green, line width=1.5] (O) -- (2,1) coordinate[at end](A) --++ (-1.5,1) coordinate[at end](B) --++ (-3,-3) coordinate[at end](C) --++ (-30:2) coordinate[at end](D) -- cycle;
    \node[right] at (A) {$A$}; \node[above] at (B) {$B$}; \node[left] at (C) {$C$};\node[below] at (D) {$D$};
    \end{tikzpicture}
    \caption{Connecting a path with relative coordinates.}
    \label{fig:relativecoords}
\end{figure}
where we have used \textit{relative coordinates}\index{Relative Coordinates}: Cartesian for the second and third segments, and polar for the fourth segment. We may append the \texttt{coordinate[at end]}\index{coordinate@\texttt{coordinate}} construct (can be omitted) at each step to remember the last coordinates for subsequent labeling. In addition, we can supply the \texttt{line width}\index{line width@\texttt{line width}} parameter (may be substituted by short keywords like \texttt{thin}, \texttt{thick}, etc.), which is self-explanatory.

\paragraph{Fill}
Apart from drawing lines, we may also want to fill the area bounded by them. This is done by either the \texttt{\textbackslash fill}\index{fill@\texttt{\textbackslash fill}} (or \texttt{\textbackslash filldraw}\index{filldraw@\texttt{\textbackslash filldraw}}) command or appending the \texttt{fill=<color>}\index{fill@\texttt{fill}} option to the \texttt{draw} command. This is demonstrated by Figure \ref{fig:filldraw} on the next page.
\begin{lstlisting}
\begin{tikzpicture}
\draw[help lines] (-4,-4) grid (4,4);
\coordinate[label={[xshift=7]$A$}] (A) at (3,-1);
\coordinate[label=$B$] (B) at (2,2);
\coordinate[label={[yshift=-3]below:$C$}] (C) at (-1,-4);
\coordinate[label={[xshift=-7]$D$}] (D) at (-3,-1);
\coordinate[label=$E$] (E) at (-2,3);
\draw[Orange, line width=2, rounded corners, fill=Gray, fill opacity=0.5] (A) \foreach \P in {B,...,E} { -- (\P)} -- cycle; % equivalent to \draw (A) -- (B) -- (C) -- (D) -- (E) -- cycle;
\end{tikzpicture}
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \draw[help lines] (-4,-4) grid (4,4);
    \coordinate[label={[xshift=7]$A$}] (A) at (3,-1);
    \coordinate[label=$B$] (B) at (2,2);
    \coordinate[label={[yshift=-3]below:$C$}] (C) at (-1,-4);
    \coordinate[label={[xshift=-7]$D$}] (D) at (-3,-1);
    \coordinate[label=$E$] (E) at (-2,3);
    \draw[Orange, line width=2, rounded corners, fill=Gray, fill opacity=0.5] (A) \foreach \P in {B,...,E} {-- (\P)} -- cycle; % equivalent to \draw (A) -- (B) -- (C) -- (D) -- (E) -- cycle;
    \end{tikzpicture}
    \caption{Filling the area enclosed by a path with color.}
    \label{fig:filldraw}
\end{figure}
We can specify the fill color opacity with the \texttt{fill opacity}\index{opacity@\texttt{opacity}} option. Notice that we have utilized the PGF for-loop functionality to simplify chaining the path. Finally, we have added the \texttt{xshift}\index{xshift@\texttt{xshift}} and \texttt{yshift}\index{yshift@\texttt{yshift}} parameters to fine-tune the positioning of labels, and the effect of \texttt{rounded corners} should not be hard to see.

\begin{exercisebox}
\begin{Exercise}
\phantomsection%
\label{exer:star}%
Try to draw and fill a star shape using TikZ. An example is given below as Figure \ref{fig:star}.
\end{Exercise}
\end{exercisebox}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \draw[line width=1.5, fill=Yellow] (-18:2) \foreach \d in {0,72,144,216} {-- (\d+18:4) -- (\d+54:2)} -- (306:4) -- cycle;
    \node at (0,0) {\Huge ;)};
    \end{tikzpicture}
    \caption{The example star for Exercise \ref{exer:star}.}
    \label{fig:star}
\end{figure}
\setboolean{firstanswerofthechapter}{true}
\begin{Answer}
\begin{lstlisting}
\begin{tikzpicture}
\draw[line width=1.5, fill=Yellow] (-18:2) \foreach \d in {0,72,144,216} {-- (\d+18:4) -- (\d+54:2)} -- (306:4) -- cycle;
\node at (0,0) {\Huge ;)};
\end{tikzpicture}
\end{lstlisting}
\end{Answer}
\setboolean{firstanswerofthechapter}{false}

\subsection{Shapes}

\paragraph{Rectangles, Circles, Ellipses}
Often, we have to draw some simple shapes like rectangles, circles, and ellipses. In TikZ, it is easily done by writing exactly \texttt{rectangle}\index{rectangle@\texttt{rectangle}}, \texttt{circle}\index{circle@\texttt{circle}}, and \texttt{ellipse}\index{ellipse@\texttt{ellipse}} with the appropriate dimensions after them. This is illustrated in Figure \ref{fig:shape1} below.
\begin{lstlisting}
\begin{tikzpicture}
\draw[help lines] (-2,-3) grid (5,5);
\coordinate[label=$O$] (O) at (0,0);
\draw[Blue] (O) circle (1.5); % at origin with radius = 1.5
\coordinate (A) at (3,2);
\draw[Red] (A) ellipse (1 and 2); % with x/y-axis = 1 and 2
\draw[Green] (1.5,-1.5) rectangle (3.5,-2.5); % two opposite vertices
\end{tikzpicture}
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \draw[help lines] (-3,-3) grid (5,5);
    \coordinate[label=$O$] (O) at (0,0);
    \draw[Blue] (O) circle (1.5); % at origin with radius = 1.5
    \coordinate (A) at (3,2);
    \draw[Red] (A) ellipse (1 and 2); % with x/y-axis = 1 and 2
    \draw[Green] (1.5,-1.5) rectangle (3.5,-2.5); % two opposite vertices
    \end{tikzpicture}
    \caption{Drawing various simple shapes in Tikz.}
    \label{fig:shape1}
\end{figure}
There are also other shapes like \texttt{parabola}.

\paragraph{Rotation}
Another useful functionality is to rotate lines and shapes. We can use either \texttt{rotate=<degree>}\index{rotate@\texttt{rotate}} or the more advanced \texttt{rotate around={<degree>:\allowbreak<about\_coordinates>}}\index{rotate around@\texttt{rotate around}} to achieve that (the former is a special case of the latter with the reference coordinates determined implicitly, usually the origin). Their difference is demonstrated in Figure \ref{fig:rotate}.
\begin{lstlisting}
\begin{tikzpicture}
\draw[help lines] (-3,-3) grid (5,5);
\coordinate[label=$O$] (O) at (0,0);
\coordinate[label=$A$] (A) at (3,0);
\draw[Gray] (A) ellipse (2 and 1);
\draw[Red, dashed, rotate=45] (A) ellipse (2 and 1);
\draw[Blue, dashed, rotate around={60:(O)}] ([rotate around={60:(O)}]A) ellipse (2 and 1); % an extra rotate around is needed in front of A
\draw[Blue, dashed, rotate=60] (1,0) -- (5,0);
\draw[Red, dashed, rotate around={45:(A)}] (1,0) -- (5,0);
\end{tikzpicture}    
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \draw[help lines] (-3,-3) grid (5,5);
    \coordinate[label=$O$] (O) at (0,0);
    \coordinate[label=$A$] (A) at (3,0);
    \draw[Gray] (A) ellipse (2 and 1);
    \draw[Red, dashed, rotate=45] (A) ellipse (2 and 1);
    \draw[Blue, dashed, rotate around={60:(O)}] ([rotate around={60:(O)}]A) ellipse (2 and 1); % an extra rotate around is needed in front of A
    \draw[Blue, dashed, rotate=60] (1,0) -- (5,0);
    \draw[Red, dashed, rotate around={45:(A)}] (1,0) -- (5,0);
    \end{tikzpicture}
    \caption{Two different kinds of coordinate rotation in Tikz.}
    \label{fig:rotate}
\end{figure}

\paragraph{Clipping}
Sometimes we may want to fill a limited area within a shape clipped by some other shape. This can be done by the \texttt{clip}\index{clip@\texttt{clip}} construct. Here we draw a Venn diagram as an illustrative example in Figure \ref{fig:clipping}.
\begin{lstlisting}
\begin{tikzpicture}
\draw[Red] (0,0) circle (2) node{A};
\begin{scope}
\clip (0,0) circle (2);
\fill[Green!25] (2.5,0) circle (2);
\end{scope}
\draw[Blue] (2.5,0) circle (2) node{B};
\end{tikzpicture}
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \draw[Red] (0,0) circle (2) node{A};
    \begin{scope}
    \clip (0,0) circle (2);
    \fill[Green!25] (2.5,0) circle (2);
    \end{scope}
    \draw[Blue] (2.5,0) circle (2) node{B};
    \end{tikzpicture}
    \caption{A Venn diagram created by clipping.}
    \label{fig:clipping}
\end{figure}
Be aware that clipping is cumulative, and we will have to limit its effect within a local \texttt{scope}\index{scope@\texttt{scope}} so that the blue circle to the right can be drawn without being clipped wrongly.

\paragraph{Perpendicular Lines}
A convenient feature in TikZ is to draw a line perpendicular to another line without the need to do the manual calculation by loading the extra TikZ library \texttt{calc}\index{calc@\texttt{calc}} with
\begin{lstlisting}
\usetikzlibrary{calc}    
\end{lstlisting}
The intersection point for that perpendicular line will then be automatically computed by it along the lines of \texttt{\$(P)!(Q)!(R)\$}. This is showcased in Figure \ref{fig:perp} below.
\begin{lstlisting}
\begin{tikzpicture}
\coordinate[label={below:$O$}] (O) at (0,0); 
\node at (O) [circle,fill,inner sep=1pt] {};
\coordinate[label=$A$] (A) at (-1,2);
\coordinate[label=$B$] (B) at (4,1);
\coordinate[label=$C$] (C) at ($(A)!(O)!(B)$); % here!
\draw (A) -- (B);
\draw[dashed] (O) -- (C);
\draw let \p1 = ($(C)-(O)$) in (O) circle ({veclen(\x1,\y1)});
\end{tikzpicture}
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \coordinate[label={below:$O$}] (O) at (0,0); 
    \node at (O) [circle,fill,inner sep=1pt] {};
    \coordinate[label=$A$] (A) at (-1,2);
    \coordinate[label=$B$] (B) at (4,1);
    \coordinate[label=$C$] (C) at ($(A)!(O)!(B)$);
    \draw (A) -- (B);
    \draw[dashed] (O) -- (C);
    \draw let \p1 = ($(C)-(O)$) in (O) circle ({veclen(\x1,\y1)});
    \end{tikzpicture}
    \caption{Demonstration of drawing perpendicular lines, in addition to calculating the distance between two coordinates.}
    \label{fig:perp}
\end{figure}
We further use the \texttt{calc} library with its \texttt{let \ldots in}\index{let in@\texttt{let in}} syntax (using \texttt{\textbackslash p1} to denote the displacement vector resulting from the \texttt{\$\$} calculation and \texttt{\textbackslash x1,\textbackslash y1} for its $x$/$y$-component), and the \texttt{veclen}\index{veclen@\texttt{veclen}} function to compute its length, a.k.a.\ the radius of the circle tangent to the line.

Alternatively, the special cases of vertical/horizontal perpendicular lines can be done by the \texttt{|-} and \texttt{-|} syntax. They are demonstrated in Figure \ref{fig:perpq} above.
\begin{lstlisting}
\begin{tikzpicture}
\draw (0,0) |- (2,2);
\draw[dashed] (0,0) -| (1,1);
\end{tikzpicture}
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \draw (0,0) |- (2,2);
    \draw[dashed] (0,0) -| (1,1);
    \end{tikzpicture}
    \caption{Quick shortcuts for making perpendicular lines.}
    \label{fig:perpq}
\end{figure}

\paragraph{Angles}
For geometry purposes, we often need to label \textit{angles}\index{Angle}, like in a triangle or polygon. The \texttt{angles}\index{angles@\texttt{angles}} TikZ library is exactly made for this. Similar to above, we import it via writing
\begin{lstlisting}
\usetikzlibrary{angles, quotes}    
\end{lstlisting}
and then we can draw angles as some \texttt{pic} (refer to Section \ref{sec:stylepic} later) with the construct in the form of \verb|{angle = A--B--C}|\index{angle@\texttt{angle}}, demonstrated in the following code for Figure \ref{fig:righttrig}:
\begin{lstlisting}
\begin{tikzpicture}
\coordinate[label={left:$A$}] (A) at (-1,0);
\coordinate[label={below right:$B$}] (B) at (3,0);
\coordinate[label=$C$] (C) at (3,3);
\draw (A) -- (B) -- (C) -- cycle;
\pic [draw,angle radius=10] {right angle = A--B--C};
\pic [draw,"$\theta$",angle radius=15,angle eccentricity=1.5] {angle = B--A--C};
\end{tikzpicture}
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \coordinate[label={left:$A$}] (A) at (-1,0);
    \coordinate[label={below right:$B$}] (B) at (3,0);
    \coordinate[label=$C$] (C) at (3,3);
    \draw (A) -- (B) -- (C) -- cycle;
    \pic [draw,angle radius=10] {right angle = A--B--C};
    \pic [draw,"$\theta$",angle radius=15,angle eccentricity=1.5] {angle = B--A--C};
    \end{tikzpicture}
    \caption{Drawing a right-angled triangle with the angles labeled.}
    \label{fig:righttrig}
\end{figure}
The \texttt{angle radius} option controls the extent of the angle marking, and \texttt{angle eccentricity} determines the distance of the angle and its label ($\theta$ in this example). 

\paragraph{Arcs}
Although drawing circles is not a rare task, sometimes we will need to draw just an \textit{arc}\index{Arc}. It is not hard to do so in TikZ with the \texttt{arc}\index{arc@\texttt{arc}} shape, the syntax of which is
\begin{lstlisting}
\draw (x,y) arc (start_angle:stop_angle:radius);
\end{lstlisting}
The arc will start from point \texttt{(x,y)} (it is also possible to use polar coordinates) as a part of the arc with a starting angle, stopping angle, and radius as indicated by the subsequent input values. An example is shown in Figure \ref{fig:arcs} above.
\begin{lstlisting}
\begin{tikzpicture}
\coordinate[label={left:$O$}] (O) at (0,0);
\coordinate[label={right:$A$}] (A) at (4,0);
\draw (O) -- (A) arc (0:30:4) --++ (-150:1.5) arc (30:210:0.5) coordinate[at end] (B) -- cycle;
\draw[dashed] (B) --++ (30:1);
\pic[draw,"$30^\circ$",angle radius=20,angle eccentricity=1.75] {angle = A--O--B};
\end{tikzpicture}
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \coordinate[label={left:$O$}] (O) at (0,0);
    \coordinate[label={right:$A$}] (A) at (4,0);
    \draw (O) -- (A) arc (0:30:4) --++ (-150:1.5) arc (30:210:0.5) coordinate[at end] (B) -- cycle;
    \draw[dashed] (B) --++ (30:1);
    \pic[draw,"$30^\circ$",angle radius=20,angle eccentricity=1.75] {angle = A--O--B};
    \end{tikzpicture}
    \caption{Drawing multiple arcs in one diagram involving relative coordinates.}
    \label{fig:arcs}
\end{figure}

\section{Advanced Controls on Paths}

\subsection{Curves}

\paragraph{Bézier Control Curves}
Up until now, we have been drawing only straight lines or segments. A reasonable expectation is to go one step further and construct curved paths. In TikZ, it is implemented as \textit{Bézier control curves}\index{Curve}\index{Bézier Control Curve} that take one or two control points, with either one of the following syntaxes:
\begin{lstlisting}
\draw <starting_coords> .. controls <control_coords> .. <end_coords>;
\draw <starting_coords> .. controls <control_coords_1> and <control_coords_2> .. <end_coords>;
\end{lstlisting}
A schematic diagram is given as Figure \ref{fig:curves1} above, and the code to produce that example is
\begin{lstlisting}
\begin{tikzpicture}
\coordinate[label={below:$A$}] (A) at (-1,0);
\coordinate[label=$B$] (B) at (3,1);
\coordinate[label={[Gray]$C_1$}] (C1) at (0,2);
\coordinate[label={[Gray]left:$C_2$}] (C2) at (2,0);
\draw (A) .. controls (C1) and (C2) .. (B);
\draw[dashed, Gray] (A) -- (C1);
\draw[dashed, Gray] (B) -- (C2);
\end{tikzpicture}    
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \coordinate[label={below:$A$}] (A) at (-1,0);
    \coordinate[label=$B$] (B) at (3,1);
    \coordinate[label={[Gray]$C_1$}] (C1) at (0,2);
    \coordinate[label={[Gray]left:$C_2$}] (C2) at (2,0);
    \draw (A) .. controls (C1) and (C2) .. (B);
    \draw[dashed, Gray] (A) -- (C1);
    \draw[dashed, Gray] (B) -- (C2);
    \end{tikzpicture}
    \caption{The anatomy of a Bézier control curve.}
    \label{fig:curves1}
\end{figure}

\paragraph{In and Out Angles}
An alternative way to draw a curve is to use the \texttt{to}\index{to@\texttt{to}} operation plus the \texttt{in=<degree>, out=<degree>} construct. It is not difficult to guess that the \texttt{in}\index{in@\texttt{in}} and \texttt{out}\index{out@\texttt{out}} options represent the direction of the incoming/outgoing ray as angles (relative to the $x$-axis). This is demonstrated in Figure \ref{fig:curves2} below.
\begin{lstlisting}
\begin{tikzpicture}
\coordinate[label={below:$A$}] (A) at (-1,-1);
\coordinate[label={below right:$B$}] (B) at (2,1);
\coordinate[label=$C$] (C) at (3,-2);
\draw (A) to[in=165, out=120] (B) to [in=225, out=-90] (C);
\draw[dashed, Gray] (A) --++ (120:2) coordinate[at end] (Aa);
\draw[dashed, Gray] (A) --++ (0:2) coordinate[at end] (X1);
\pic[draw,"$120^\circ$",Gray,angle radius=15,angle eccentricity=1.75] {angle = X1--A--Aa};
\draw[dashed, Gray] (B) --++ (165:2) coordinate[at end] (Ba);
\draw[dashed, Gray] (B) --++ (0:2) coordinate[at end] (X2);
\pic[draw,"$165^\circ$",Gray,angle radius=15,angle eccentricity=1.75] {angle = X2--B--Ba};
\end{tikzpicture}    
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \coordinate[label={below:$A$}] (A) at (-1,-1);
    \coordinate[label={below right:$B$}] (B) at (2,1);
    \coordinate[label=$C$] (C) at (3,-2);
    \draw (A) to[in=165, out=120] (B) to [in=225, out=-90] (C);
    \draw[dashed, Gray] (A) --++ (120:2) coordinate[at end] (Aa);
    \draw[dashed, Gray] (A) --++ (0:2) coordinate[at end] (X1);
    \pic[draw,"$120^\circ$",Gray,angle radius=15,angle eccentricity=1.75] {angle = X1--A--Aa};
    \draw[dashed, Gray] (B) --++ (165:2) coordinate[at end] (Ba);
    \draw[dashed, Gray] (B) --++ (0:2) coordinate[at end] (X2);
    \pic[draw,"$165^\circ$",Gray,angle radius=15,angle eccentricity=1.75] {angle = X2--B--Ba};
    \end{tikzpicture}
    \caption{Another way to draw control curves specified by angles.}
    \label{fig:curves2}
\end{figure}

\paragraph{Intersection}
It is handy if we can mark the intersection point(s) of two different curves. This can be delegated to the TikZ library \texttt{intersections}\index{intersections@\texttt{intersections}}. To use it, we need to give a name to those curves with the \texttt{name path}\index{name path@\texttt{name path}} option, and then we can invoke the \texttt{name intersections}\index{name intersections@\texttt{name intersections}} option that refers to the intersection points as \texttt{(intersection-<number>)}. For instance, Figure \ref{fig:intersect} below can be produced by
\begin{lstlisting}
\begin{tikzpicture}
\draw[name path=myellipse, rotate=30] (0,0) ellipse (2 and 1);
\draw[name path=mycurve] (-2,1) to[in=120, out=-45] (2,0) to[in=-90, out=-60] (0.5,-0.5);
\fill[Red, name intersections={of=myellipse and mycurve}]
    (intersection-1) circle (2pt) node[above]{1}
    (intersection-2) circle (2pt) node[right]{2}
    (intersection-3) circle (2pt) node[below]{3};
\end{tikzpicture}    
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \draw[name path=myellipse, rotate=30] (0,0) ellipse (2 and 1);
    \draw[name path=mycurve] (-2,1) to[in=120, out=-45] (2,0) to[in=-90, out=-60] (0.5,-0.5);
    \fill[Red, name intersections={of=myellipse and mycurve}]
    (intersection-1) circle (2pt) node[above]{1}
    (intersection-2) circle (2pt) node[right]{2}
    (intersection-3) circle (2pt) node[below]{3};
    \end{tikzpicture}
    \caption{Labeling intersection points between an ellipse and an arbitrary curve.}
    \label{fig:intersect}
\end{figure}

\begin{exercisebox}
\begin{Exercise}
\phantomsection%
\label{exer:arcgeo}%
Try to reproduce the (essence of) geometry in Figure \ref{fig:arcgeo}. The \verb|shorten >=<length>|\index{shorten@\texttt{shorten}} (the space is needed!) option may be useful.
\end{Exercise}
\end{exercisebox}
\begin{Answer}
\begin{lstlisting}
\begin{tikzpicture}
\coordinate[label={left:$O$}] (O) at (0,0);
\node[circle,fill,inner sep=1pt] at (O) {};
\draw[rotate=25, name path=arc] (4,0) arc (0:50:4);
\draw[rotate=50, dashed, name path=line] (3.8,2) -- (3.8,-2);
\path[name intersections={of=arc and line}] coordinate[label={above:$A$}] (A) at (intersection-1) coordinate[label={right:$B$}] (B) at (intersection-2);
\node[circle,fill,inner sep=1pt] at (A) {};
\node[circle,fill,inner sep=1pt] at (B) {};
\coordinate[label={[Red]below:$C$}] (C) at ($(A)!(O)!(B)$);
\draw[shorten >=-1cm] (O) -- (C);
\draw[dashed] (A) -- (O) -- (B);
\node[Red,circle,fill,inner sep=1pt] at (C) {};
\pic [draw,"$\phi$",angle radius=20,angle eccentricity=1.75] {angle = C--O--A};
\pic [draw,"$\phi$",angle radius=20,angle eccentricity=1.75] {angle = B--O--C};
\pic [draw,angle radius=8] {right angle = A--C--O};
\end{tikzpicture}        
\end{lstlisting}
\end{Answer}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \coordinate[label={left:$O$}] (O) at (0,0);
    \node[circle,fill,inner sep=1pt] at (O) {};
    \draw[rotate=25, name path=arc] (4,0) arc (0:50:4);
    \draw[rotate=50, dashed, name path=line] (3.8,2) -- (3.8,-2);
    \path[name intersections={of=arc and line}] coordinate[label={above:$A$}] (A) at (intersection-1) coordinate[label={right:$B$}] (B) at (intersection-2);
    \node[circle,fill,inner sep=1pt] at (A) {};
    \node[circle,fill,inner sep=1pt] at (B) {};
    \coordinate[label={[Red]below:$C$}] (C) at ($(A)!(O)!(B)$);
    \draw[shorten >=-1cm] (O) -- (C);
    \draw[dashed] (A) -- (O) -- (B);
    \node[Red,circle,fill,inner sep=1pt] at (C) {};
    \pic [draw,"$\phi$",angle radius=20,angle eccentricity=1.75] {angle = C--O--A};
    \pic [draw,"$\phi$",angle radius=20,angle eccentricity=1.75] {angle = B--O--C};
    \pic [draw,angle radius=8] {right angle = A--C--O};
    \end{tikzpicture}
    \caption{The "Bow" diagram for Exercise \ref{exer:arcgeo}.}
    \label{fig:arcgeo}
\end{figure}

\subsection{Decorations}

\paragraph{Decorations/Morphing}
An interesting effect that can be applied to curves is \textit{decorations}\index{Decoration} (or morphing), generating variations along them. This is done by loading the TikZ library \texttt{decorations.pathmorphing}\index{decorations.pathmorphing@\texttt{decorations.pathmorphing}}. The simplest usage is via \texttt{decorate, decoration=<shape>}\index{decorate@\texttt{decorate}}\index{decoration@\texttt{decoration}} that applies the morphing to the entire path, illustrated by Figure \ref{fig:deco1}.
\begin{lstlisting}
\begin{tikzpicture}
\coordinate (O) at (0,0);
\draw[Blue,line width=1.5] (O) circle (2.5);
\draw[Red,decorate,decoration={zigzag,segment length=2ex,amplitude=0.5em}] (O) circle (2.5);
\end{tikzpicture}
\end{lstlisting}
Notice that we have also passed some other keys to adjust the look of the zigzagging line.
\begin{figure}
    \centering
    \begin{tikzpicture}
    \coordinate (O) at (0,0);
    \draw[Blue,line width=1.5] (O) circle (2.5);
    \draw[Red,decorate,decoration={zigzag,segment length=2ex,amplitude=0.5em}] (O) circle (2.5);
    \end{tikzpicture}
    \caption{A circle decorated by the zigzag effect.}
    \label{fig:deco1}
\end{figure}

\paragraph{Decorating Subpaths}
The previous syntax will decorate the entire path. If we want to apply the effect only on some parts of it, then we can put the \texttt{decorate} statement to enclose each of them correspondingly. Figure \ref{fig:deco2} is shown below as an example.
\begin{lstlisting}
\begin{tikzpicture}
\draw decorate[decoration=saw] {(0,0) -- (2,1)} -- (4,-1) decorate[decoration={coil,aspect=1.5}] {-- (6,0)};
\end{tikzpicture}    
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \draw decorate[decoration=saw] {(0,0) -- (2,1)} -- (4,-1) decorate[decoration={coil,aspect=1.5}] {-- (6,0)};
    \end{tikzpicture}
    \caption{Different decorations on individual segments.}
    \label{fig:deco2}
\end{figure}

\paragraph{Positioning and Extent of Decorations}
Furthermore, we can fine-tune the positioning, as well as the starting/ending points of a decoration. This is achieved by supplying the \texttt{raise} (displacement across the path), \texttt{pre length} (starts after), and \texttt{post length} (ends before) options, demonstrated by Figure \ref{fig:deco3} above.
\begin{lstlisting}
\begin{tikzpicture}
\draw[decoration={pre length=9mm,post length=12mm,raise=-3mm,crosses}] decorate{(0,0) -- (5,1)}; % requires the extra library decorations.shapes too for the cross symbols
\end{tikzpicture}    
\end{lstlisting}
\begin{figure}
\centering
\begin{tikzpicture}
\draw[decoration={pre length=9mm,post length=12mm,raise=-3mm,crosses}] decorate{(0,0) -- (5,1)};
\end{tikzpicture}
\caption{Fine-tuning a decoration of crosses.}
\label{fig:deco3}
\end{figure}
There are many more possible choices for decorations; Unfortunately, we don't have the scope to include all of them.

\subsection{Arrows}

\paragraph{Arrow Tips}
To draw arrows, we need to load the \texttt{arrows.meta}\index{arrows.meta@\texttt{arrows.meta}} TikZ library. Then we can specify the type of arrow tip(s) during a \texttt{\textbackslash draw} command. The syntax is easier to understand by directly looking at the examples in Figure \ref{fig:arrow1}.
\begin{lstlisting}
\begin{tikzpicture}
\draw[->] (0,0) --++ (3,0);
\draw[Blue, >-{>[length=3ex, width=2ex]}] (0,-1) --++ (3,0);
\draw[Red, -{Stealth[scale=3, angle'=90]}] (0,-2) --++ (3,0);
\draw[Green, {Latex}-{Latex[]Latex[reversed]}] (0,-3) to[out=-15,in=-165]++ (3,0);
\end{tikzpicture}
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \draw[->] (0,0) --++ (3,0);
    \draw[Blue, >-{>[length=3ex, width=2ex]}] (0,-1) --++ (3,0);
    \draw[Red, -{Stealth[scale=3, angle'=90]}] (0,-2) --++ (3,0);
    \draw[Green, {Latex}-{Latex[]Latex[reversed]}] (0,-3) to[out=-15,in=-165]++ (3,0);
    \end{tikzpicture}
    \caption{Different types of arrow tips and related options.}
    \label{fig:arrow1}
\end{figure}
The two most frequently used named arrow tips are \texttt{Stealth}\index{Stealth@\texttt{Stealth}} and \texttt{Latex}\index{Latex@\texttt{Latex}}. Aside from \texttt{length}, \texttt{width}, \texttt{scale}, and \texttt{angle'} (remember the \texttt{'}!), there are many more keys like \texttt{inset}, \texttt{slant}, \texttt{left}, and \texttt{right}, etc.

It is also possible to set the global arrow style using \texttt{\textbackslash tikzset}\index{tikzset@\texttt{\textbackslash tikzset}}, like \texttt{\textbackslash tikzset\{\allowbreak>=\{Stealth\}\}}, which changes the type for all arrow tips to \texttt{Stealth}. Or, we can do it for an individual path by moving \texttt{>=\{<arrow\_type>\}} inside the corresponding \texttt{\textbackslash draw} option.

\paragraph{Arrow in the Middle}
More often than not, we would like to put the arrow in the middle of a line. We can utilize the \texttt{decorations.markings}\index{decorations.markings@\texttt{decorations.markings}} TikZ library for that. An example is given by Figure \ref{fig:arrow2} below.
\begin{lstlisting}
\begin{tikzpicture}
\draw[postaction={decorate}, decoration={markings,
      mark=at position 0.35 with {\arrow{Latex[Red,scale=2]}},
      mark=at position 0.5 with {\arrow{Stealth[Blue,scale=2.5]}},
      mark=at position 0.8 with {\arrow{Latex[Green,scale=3]}}}] 
      (0,1) .. controls (1,-3) and (3,1) .. (5,-2);
\end{tikzpicture}    
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \draw[postaction={decorate}, decoration={markings,
    mark=at position 0.35 with {\arrow{Latex[Red,scale=2]}},
    mark=at position 0.5 with {\arrow{Stealth[Blue,scale=2.5]}},
    mark=at position 0.8 with {\arrow{Latex[Green,scale=3]}},
    }] (0,1) .. controls (1,-3) and (3,1) .. (5,-2);
    \end{tikzpicture}
    \caption{Marking multiple arrow tips along a curve.}
    \label{fig:arrow2}
\end{figure}
We need to first supply the \texttt{markings}\index{markings@\texttt{markings}} keyword to the decoration option, then we can add the arrow marks as \texttt{\textbackslash arrow\{<arrow\_type>\}} at the corresponding relative positions along the curve. A new thing is that the \texttt{decorate} keyword is now placed in the \texttt{postaction}\index{postaction@\texttt{postaction}} option, which indicates that the decorations are laid on the original curve that will be kept.

\paragraph{Edges} As an extra note, a handy functionality is to draw branching paths using \texttt{edge}\index{edge@\texttt{edge}} while staying on the main path afterwards. This is demonstrated by Figure \ref{fig:edge} above.
\begin{lstlisting}
\begin{tikzpicture}
\coordinate (O) at (0,0);
\draw (O) --++ (0,2) edge[->] ++ (1,0) edge[dashed] ++ (-1,0) [-{Circle[fill=Green]}] --++ (0,1);
\end{tikzpicture}    
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \coordinate (O) at (0,0);
    \draw (O) --++ (0,2) edge[->] ++ (1,0) edge[dashed] ++ (-1,0) [-{Circle[fill=Green]}] --++ (0,1);
    \end{tikzpicture}
    \caption{Branching arrows from an intermediate point.}
    \label{fig:edge}
\end{figure}

\begin{exercisebox}
\begin{Exercise}
\phantomsection%
\label{exer:pendulum}%
Draw the pendulum diagram in Figure \ref{fig:pendulum}.
\end{Exercise}
\begin{Exercise}
\phantomsection%
\label{exer:closedint}%
Try to reproduce the closed integration path in Figure \ref{fig:closedint}.
\end{Exercise}
\end{exercisebox}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \coordinate (O) at (0,0);
    \draw (O) --++ (-110:4) node (A) {};
    \draw[Red, thick, -Stealth] (A) --++ (0,-1.5) node[below left]{$mg$}; 
    \draw[blue, thick, -Stealth] (A) --++ (70:1.4) node[above left]{$T$};
    \draw[fill=Green!20] (-110:4) circle (0.5) node{$m$};
    \draw[dashed] (O) --++ (-90:4) node (B) {};
    \draw[dashed, Gray, <->] (-120:5) arc (-120:-60:5);
    \pic[draw,"$\theta$",angle radius=20,angle eccentricity=1.6] {angle=A--O--B};
    \end{tikzpicture}
    \caption{The pendulum schematic for Exercise \ref{exer:pendulum}.}
    \label{fig:pendulum}
\end{figure}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \coordinate[label={below left:$O$}] (O) at (0,0);
    \draw[-Latex, thick] (O) -- (5,0) node[right]{$x$};
    \draw[-Latex, thick] (O) -- (0,5) node[above]{$y$};
    \coordinate[label={left:$A$}] (A) at (1,1);
    \coordinate[label=$B$] (B) at (4,4);
    \draw[Green, very thick, postaction={decorate}, decoration={markings,
    mark=at position 0.25 with {\arrow{Latex}},
    mark=at position 0.75 with {\arrow{Latex}}}] (A) .. controls (2,4) and (3,3) .. (B) .. controls (5,0) and (3,2) .. (A);
    \end{tikzpicture}
    \caption{The integration path for Exercise \ref{exer:closedint}.}
    \label{fig:closedint}
\end{figure}
\begin{Answer}[ref=exer:pendulum]
\begin{lstlisting}
\begin{tikzpicture}
\coordinate (O) at (0,0);
\draw (O) --++ (-110:4) node (A) {};
\draw[Red, thick, -Stealth] (A) --++ (0,-1.5) node[below left]{$mg$}; 
\draw[blue, thick, -Stealth] (A) --++ (70:1.4) node[above left]{$T$};
\draw[fill=Green!20] (-110:4) circle (0.5) node{$m$};
\draw[dashed] (O) --++ (-90:4) node (B) {};
\draw[dashed, Gray, <->] (-120:5) arc (-120:-60:5);
\pic[draw,"$\theta$",angle radius=20,angle eccentricity=1.6] {angle=A--O--B};
\end{tikzpicture}
\end{lstlisting}
\end{Answer}
\begin{Answer}
\begin{lstlisting}
\begin{tikzpicture}
\coordinate[label={below left:$O$}] (O) at (0,0);
\draw[-Latex, thick] (O) -- (5,0) node[right]{$x$};
\draw[-Latex, thick] (O) -- (0,5) node[above]{$y$};
\coordinate[label={left:$A$}] (A) at (1,1);
\coordinate[label=$B$] (B) at (4,4);
\draw[Green, very thick, postaction={decorate}, decoration={markings,
    mark=at position 0.25 with {\arrow{Latex}},
    mark=at position 0.75 with {\arrow{Latex}}}] (A) .. controls (2,4) and (3,3) .. (B) .. controls (5,0) and (3,2) .. (A);
\end{tikzpicture} 
\end{lstlisting}
\end{Answer}

\section{Styles and Pics}
\label{sec:stylepic}

\paragraph{Styles and Nodes}
It is convenient if we can repeatedly apply some \textit{style}\index{Style} to similar objects in TikZ. This can be done by providing the name and attributes of that style at the beginning of the \texttt{tikzpicture} using the \texttt{/.style}\index{\texttt{/.}}\index{style@\texttt{/.style}} syntax. Figure \ref{fig:stylenode} below demonstrates the usage.
\begin{lstlisting}
\begin{tikzpicture}[myrectangle/.style={rectangle, minimum width=3cm, minimum height=1.5cm, draw=black, fill=Mahogany!20}]
\node[myrectangle] (myA) at (0,0) {$A$}; % using style defined above
\node[rectangle, minimum width=3cm, minimum height=1.5cm, draw=black, 
fill=RoyalBlue!20] (myB) at (4,-3) {$B$}; % equivalent syntax except the fill color
\draw (myA.east) -- (myB.north);
\end{tikzpicture}
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}[myrectangle/.style={rectangle, minimum width=3cm, minimum height=1.5cm, draw=black, fill=Mahogany!20}]
    \node[myrectangle] (myA) at (0,0) {$A$}; % using style defined above, 
    \node[rectangle, minimum width=3cm, minimum height=1.5cm, draw=black, 
    fill=RoyalBlue!20] (myB) at (4,-3) {$B$}; % equivalent syntax except the fill color
    \draw (myA.east) -- (myB.north);
    \end{tikzpicture}
    \caption{Two styled rectangle nodes.}
    \label{fig:stylenode}
\end{figure}
Here we have created two nodes that are in the shape of a rectangle. The \texttt{minimum width}\index{minimum width@\texttt{minimum width}} and \texttt{minimum height}\index{minimum height@\texttt{minimum height}} keys work exactly as their names suggest and maintain the extent of the rectangles beyond the node text. We additionally draw a line connecting the two nodes with the \textit{anchors}\index{Anchor} (at where the two ends of the line are fixed) stated as directions (optional).

\paragraph{Pics - Small Pictures}
Similarly, it will be handy if we can insert and reuse the same piece of drawing that is needed many times. This is known as a \texttt{pic}\index{pic@\texttt{pic}} (small picture) in TikZ, and we have been using this feature for labeling angles. To define a \texttt{pic}, we do it like it is a style by \texttt{<pic\_name>/.pic=\{<drawing\_code>\}}. Then we can append the \texttt{pic} after a path. This is illustrated by the damper defined for the mechanical system (Reference: \href{https://tex.stackexchange.com/questions/476076/drawing-mechanical-systems-mass-damper-spring-in-latex}{\TeX{} StackExchange 476076}) in the subsequent Figure \ref{fig:mechanic}.
\begin{lstlisting}
\begin{tikzpicture}
    % Styles
    [dampic/.pic={
    \fill[pgftransparent!0] (-0.1,-0.3) rectangle (0.3,0.3);
    \draw (-0.3,0.3) -| (0.3,-0.3) -- (-0.3,-0.3);
    \draw[line width=1mm] (-0.1,-0.3) -- (-0.1,0.3);},
    spring/.style={decorate,decoration={zigzag,pre length=0.4cm,post length=0.4cm,segment length=2mm,amplitude=3mm}},
    mass/.style={rectangle,minimum height=1.6cm, minimum width=2.4cm, draw=black, top color=brown!30, bottom color=brown!50},
    ground/.style={fill,pattern=north east lines,draw=none}]
% Drawing
\node[mass] (mass1) at (0,0) {$m$};
\node[mass] (mass2) at (4,0) {$m$};
\draw ($(mass1.east)+(0,0.5cm)$) -- ($(mass2.west)+(0,0.5cm)$) pic[midway,sloped]{dampic};
\draw[spring] ($(mass1.east)-(0,0.5cm)$) -- ($(mass2.west)-(0,0.5cm)$);
\node[ground,minimum width=3mm,minimum height=2.5cm] (g1) at (-3,0) {};
\draw (g1.north east) -- (g1.south east);
\draw ($(mass1.west)+(0,0.5cm)$) -- ($(g1.east)+(0,0.5cm)$) pic[midway,sloped]{dampic};
\draw[spring] ($(mass1.west)-(0,0.5cm)$) -- ($(g1.east)-(0,0.5cm)$);
\end{tikzpicture}
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}[dampic/.pic={
    \fill[pgftransparent!0] (-0.1,-0.3) rectangle (0.3,0.3);
    \draw (-0.3,0.3) -| (0.3,-0.3) -- (-0.3,-0.3);
    \draw[line width=1mm] (-0.1,-0.3) -- (-0.1,0.3);},
    spring/.style={decorate,decoration={zigzag,pre length=0.4cm,post length=0.4cm,segment length=2mm,amplitude=3mm}},
    mass/.style={rectangle,minimum height=1.6cm, minimum width=2.4cm, draw=black, top color=brown!30, bottom color=brown!50},
    ground/.style={fill,pattern=north east lines,draw=none}]
    \node[mass] (mass1) at (0,0) {$m$};
    \node[mass] (mass2) at (4,0) {$m$};
    \draw ($(mass1.east)+(0,0.5cm)$) -- ($(mass2.west)+(0,0.5cm)$) pic[midway,sloped]{dampic};
    \draw[spring] ($(mass1.east)-(0,0.5cm)$) -- ($(mass2.west)-(0,0.5cm)$);
    \node[ground,minimum width=3mm,minimum height=2.5cm] (g1) at (-3,0) {};
    \draw (g1.north east) -- (g1.south east);
    \draw ($(mass1.west)+(0,0.5cm)$) -- ($(g1.east)+(0,0.5cm)$) pic[midway,sloped]{dampic};
    \draw[spring] ($(mass1.west)-(0,0.5cm)$) -- ($(g1.east)-(0,0.5cm)$);
    \end{tikzpicture}
    \caption{A schematic for a mechanical mass-spring-damper system.}
    \label{fig:mechanic}
\end{figure}
There are some other points worth mentioning. The special \texttt{pgftransparent!0} color code is an alias equivalent to the white color, and covers the original path under it. The drawing style for the masses specifies \texttt{top color} and \texttt{bottom color} to produce \textit{color gradient}\index{Color Gradient}. Then, we sketch the outline of the damper. We also need to load the \texttt{patterns}\index{patterns@\texttt{patterns}} TikZ library to produce the hatching lines (\texttt{pattern=north east lines}\index{pattern@\texttt{pattern}}) for the ground style. Also, we have used the \texttt{\$\$}\index{\texttt{\$\$}} syntax to carry out coordinate calculations in deriving the starting/ending points of the connecting lines.

\paragraph{Styles for Every Node/Path}
An even more convenient shortcut is to assign the same style for every node/path (of the same kind) at once. This is unsurprisingly done by providing the \texttt{every node}/\texttt{every path}\index{every@\texttt{every}} name, and is readily showcased in Figure \ref{fig:everynode} below.
\begin{lstlisting}
\begin{tikzpicture}[every node/.style={circle,black,solid,draw=black,fill=Red!20,minimum size=30pt}, 
    every rectangle node/.style={black,solid,draw=black,fill=Blue!20,minimum height=15pt, minimum width=30pt},
    every path/.style={Green,dashed}]
\node (A) at (0,0) {$A$}; 
\node (B) at (2,-1) {$B$};
\node[rectangle] (C) at (-0.5,-2) {$C$};
\node (D) at (3,-3) {$D$};
\draw (A) -- (B) -- (C) -- (D);
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}[every node/.style={circle,black,solid,draw=black,fill=Red!20,minimum size=30pt}, 
    every rectangle node/.style={black,solid,draw=black,fill=Blue!20,minimum height=15pt, minimum width=30pt},
    every path/.style={Green,dashed}]
    \node (A) at (0,0) {$A$}; 
    \node (B) at (2,-1) {$B$};
    \node[rectangle] (C) at (-0.5,-2) {$C$};
    \node (D) at (3,-3) {$D$};
    \draw (A) -- (B) -- (C) -- (D);
    \end{tikzpicture}
    \caption{Reusing styles for every node and path.}
    \label{fig:everynode}
\end{figure}
Notice that \texttt{every path} also affects node texts and lines, and we have to override them in \texttt{every node} for it to work as intended.

\paragraph{Path Building}
We can further break down a path into the corresponding components (lines, curves, etc.) and execute certain code for each of them every time they occur. This is accomplished by the \texttt{show path construction}\index{show path construction@\texttt{show path construction}} keyword for \texttt{decoration}, demonstrated by the contour integration plot in Figure \ref{fig:complexcontour} as follows.
\begin{lstlisting}
\begin{tikzpicture}
% Defining parameters
\newcommand*{\gap}{0.3}
\newcommand*{\bigradius}{3}
\newcommand*{\littleradius}{0.7}
% Drawing
\draw[very thick,-Latex] (-1.15*\bigradius, 0) -- (1.15*\bigradius, 0);
\draw[very thick, decorate, decoration={zigzag, segment length=0.5cm, amplitude=0.2cm}] (0, 0) -- (1.15*\bigradius, 0);
\draw[very thick,-Latex] (0, -1.15*\bigradius) -- (0, 1.15*\bigradius);
% The red contour
\draw[red, thick, postaction=decorate, decoration={show path construction, curveto code={
    \draw[decorate, decoration={markings, mark=at position 0.75 with {\arrow{Latex[Red,scale=1.25]}}}] 
    (\tikzinputsegmentfirst) .. controls (\tikzinputsegmentsupporta) and (\tikzinputsegmentsupportb) .. (\tikzinputsegmentlast);}, % no need to change this unless you know what you are doing
    lineto code={
    \draw[decorate, decoration={markings, mark=at position 0.65 with {\arrow{Latex[Red,scale=1.25]}}}] 
    (\tikzinputsegmentfirst) -- (\tikzinputsegmentlast);} % don't change this too
    }]  
let
\n1 = {asin(\gap/2/\bigradius)},
\n2 = {asin(\gap/2/\littleradius)}
in (\n1:\bigradius) arc (\n1:360-\n1:\bigradius) node[pos=0.4,below right]{$\Gamma$} -- (-\n2:\littleradius) arc (-\n2:-360+\n2:\littleradius) -- (\n1:\bigradius);
\end{tikzpicture}
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \newcommand*{\gap}{0.3}
    \newcommand*{\bigradius}{3}
    \newcommand*{\littleradius}{0.7}
    \draw[very thick,-Latex] (-1.15*\bigradius, 0) -- (1.15*\bigradius, 0);
    \draw[very thick, decorate, decoration={zigzag, segment length=0.5cm, amplitude=0.2cm}] (0, 0) -- (1.15*\bigradius, 0);
    \draw[very thick,-Latex] (0, -1.15*\bigradius) -- (0, 1.15*\bigradius);
    \draw[red, thick, postaction=decorate, decoration={show path construction, curveto code={
    \draw[decorate, decoration={markings, mark=at position 0.75 with {\arrow{Latex[Red,scale=1.25]}}}] 
    (\tikzinputsegmentfirst) .. controls (\tikzinputsegmentsupporta) and (\tikzinputsegmentsupportb) .. (\tikzinputsegmentlast);},
    lineto code={
    \draw[decorate, decoration={markings, mark=at position 0.65 with {\arrow{Latex[Red,scale=1.25]}}}] 
    (\tikzinputsegmentfirst) -- (\tikzinputsegmentlast);}
    }]  
    let
    \n1 = {asin(\gap/2/\bigradius)},
    \n2 = {asin(\gap/2/\littleradius)}
    in (\n1:\bigradius) arc (\n1:360-\n1:\bigradius) node[pos=0.4,below right]{$\Gamma$} -- (-\n2:\littleradius) arc (-\n2:-360+\n2:\littleradius) -- (\n1:\bigradius);
    \end{tikzpicture}
    \caption{A standard complex contour integration path with a branch cut over the positive $x$-axis.}
    \label{fig:complexcontour}
\end{figure}
The code is a bit complex, and we will have it explained step by step. As their names suggest, the \texttt{curveto code}/\texttt{lineto code} part will be applied to every curve/line. The main purpose of involving \texttt{\textbackslash tikzinputsegmentfirst}, \texttt{\textbackslash tikzinputsegmentlast}, \texttt{\textbackslash tikzinputsegmentsupporta}, in addition to \texttt{\textbackslash tikzinputsegmentsupportb}, is to replicate the curve/line internally, which can be left untouched as a template. Then the \texttt{decorate} and \texttt{markings} parts are the same as previously to put an arrow mark along each replicated path segment. Finally, to draw the actual contour, we use \texttt{\textbackslash newcommand*} and the \texttt{let \ldots in} syntax to store lengths as variables and compute the coordinates for the turning points (copying \href{https://tex.stackexchange.com/questions/103176/drawing-complex-integration}{\TeX{} StackExchange 103176}).

\section{Plotting Functions}

\paragraph{Axis Settings}
To plot a function or \textit{graph}\index{Graph} in TikZ, we have to configure an \texttt{axis}\index{axis@\texttt{axis}} scope first. There are many options to customize an axis, some of which are showcased in the two examples in Figure \ref{fig:axis} as follows.
\begin{lstlisting}
\begin{tikzpicture}
\begin{axis}[xlabel=$x$, ylabel=$y$, xmin=-1, xmax=5, ymin=-10, ymax=10, ytick={-7.5,-2.5,2.5,7.5}, minor x tick num=4, grid=major, axis lines=center, axis line style={line width=1.2pt}, width=\textwidth]
    
\end{axis}
\end{tikzpicture}
\begin{axis}[ymode=log, xlabel=$t$, ylabel=Concentration, x label style={at={(axis description cs:1.1,0.2)}}, minor ytick={10^(0.1), 10^(0.3)}, minor x tick num=1, major grid style={line width=.5pt,draw=black!66}, grid=both, enlarge x limits={0.4,upper}, width=\textwidth]

\end{axis}
\end{lstlisting}
\begin{figure}
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[xlabel=$x$, ylabel=$y$, xmin=-1, xmax=5, ymin=-10, ymax=10, ytick={-7.5,-2.5,2.5,7.5}, minor x tick num=4, grid=major, axis lines=center, axis line style={line width=1.2pt}, width=\textwidth]
    
\end{axis}
\end{tikzpicture}
\vspace{0.5cm}
\caption{A typical axis.}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\begin{tikzpicture}
\begin{axis}[ymode=log, xlabel=$t$, ylabel=Concentration, x label style={at={(axis description cs:1.1,0.2)}}, minor ytick={10^(0.1), 10^(0.3)}, minor x tick num=1, major grid style={line width=.5pt,draw=black!66}, grid=both, enlarge x limits={0.4,upper}, width=\textwidth]

\end{axis}
\end{tikzpicture}
\caption{A semi-logarithmic axis.}
\end{subfigure}
\caption{Two example TikZ axes.}
\label{fig:axis}
\end{figure}
Most of the options are not hard to comprehend, except \texttt{axis description cs:}\index{axis description cs@\texttt{axis description cs}}, which refers to the position relative to the axis frame so that we can adjust the label position, and \texttt{enlarge limits}\index{enlarge limits@\texttt{enlarge limits}}, which extends the axis limits by the amount indicated. The biggest difference between the two example axes is perhaps the appearance of the axis lines, produced by the option \texttt{axis lines=center}\index{axis lines@\texttt{axis lines}} in the first one.

\paragraph{Plotting Simple Functions}
Of course, an axis is nothing if there are no data or functions plotted on it. To add points or graphs on it, we can use the \texttt{\textbackslash addplot}\index{addplot@\texttt{\textbackslash addplot}} command. For points, we may add a list of \texttt{coordinates} after that, and use the \texttt{only marks}\index{only marks@\texttt{only marks}} keyword to draw only the dots; while for functions, we can simply enter the expression in PGF format. We can control the \texttt{domain}\index{domain@\texttt{domain}} of \texttt{x} and the number of points (\texttt{samples}\index{samples@\texttt{samples}}) used in drawing. Both of these are demonstrated in Figure \ref{fig:halflife} on the next page.
\begin{lstlisting}
\begin{tikzpicture}
\begin{axis}[xlabel=$T$, ylabel=$N$, title=Half-life Experiment, xmin=0, xmax=3, ymin=0, ymax=1, enlargelimits=0.1, legend pos=south west, width=0.9\textwidth, height=0.6\textwidth]
\addplot[Red, only marks]
coordinates {
(0,1) (0.2,0.86) (0.6,0.67) (0.9,0.53)
(1.4,0.37) (1.7,0.29) (2.5,0.17)};
\addplot[blue, domain=0:4, samples=100]{(1/2)^(x)} node[pos=0.6,above,sloped] {$N = (1/2)^T$};
\legend{Measurements, Theoretical}
\end{axis}   
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \begin{axis}[xlabel=$T$, ylabel=$N$, title=Half-life Experiment, xmin=0, xmax=3, ymin=0, ymax=1, enlargelimits=0.1, legend pos=south west, width=0.9\textwidth, height=0.6\textwidth]
    \addplot[Red, only marks]
    coordinates {
    (0,1) (0.2,0.86) (0.6,0.67) (0.9,0.53)
    (1.4,0.37) (1.7,0.29) (2.5,0.17) 
    };
    \addplot[blue, domain=0:4, samples=100]{(1/2)^(x)} node[pos=0.6,above,sloped] {$N = (1/2)^T$};
    \legend{Measurements, Theoretical}
    \end{axis}
    \end{tikzpicture}
\caption{The half-life process as an example plot.}
\label{fig:halflife}
\end{figure}
We have also adjusted the size of the figure and made a legend with \texttt{\textbackslash legend}\index{legend@\texttt{\textbackslash legend}}.

\begin{exercisebox}
\begin{Exercise}
\phantomsection%
\label{exer:axisex}%
Try to reproduce the following plot in Figure \ref{fig:axisex}. Notice that to draw with usual TikZ commands but now inside an axis, we can call the axis coordinate system with the syntax \texttt{axis cs:<coords>}\index{axis cs@\texttt{axis cs}}.
\end{Exercise}
\end{exercisebox}
\begin{Answer}
\begin{lstlisting}
\begin{tikzpicture}
\begin{axis}[xlabel=$x$, ylabel=$y$, axis lines=center, xmin=0, xmax=10, ymin=0, ymax=4, width=0.7\textwidth, height=0.4\textwidth, clip=false]
\addplot[blue, domain=3:10, samples=100] {2+sin(deg(pi*(x-3)))} node[pos=0.65,above]{$y=2+\sin(\pi (x-3))$};
\addplot[Red, domain=0:3] {1} node[midway,above]{$y=1$};
\draw[Green,dashed] (axis cs:3,0) -- (axis cs:3,4) node[right]{$x=3$};
\draw[Green, fill=Green] (axis cs:3,3) circle[radius=3pt] node[left]{$(3,3)$};
\draw[blue] (axis cs:3,2) circle[radius=3pt];
\draw[Red] (axis cs:3,1) circle[radius=3pt];
\end{axis}
\end{tikzpicture}
\end{lstlisting}    
\end{Answer}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \begin{axis}[xlabel=$x$, ylabel=$y$, axis lines=center, xmin=0, xmax=10, ymin=0, ymax=4, width=0.7\textwidth, height=0.4\textwidth, clip=false]
    \addplot[blue, domain=3:10, samples=100] {2+sin(deg(pi*(x-3)))} node[pos=0.65,above]{$y=2+\sin(\pi (x-3))$};
    \addplot[Red, domain=0:3] {1} node[midway,above]{$y=1$};
    \draw[Green,dashed] (axis cs:3,0) -- (axis cs:3,4) node[right]{$x=3$};
    \draw[Green, fill=Green] (axis cs:3,3) circle[radius=3pt] node[left]{$(3,3)$};
    \draw[blue] (axis cs:3,2) circle[radius=3pt];
    \draw[Red] (axis cs:3,1) circle[radius=3pt];
    \end{axis}
    \end{tikzpicture}
    \caption{The plot for Exercise \ref{exer:axisex}.}
    \label{fig:axisex}
\end{figure}

\paragraph{Parametric Equations}
A natural generalization of \texttt{\textbackslash addplot} is to draw \textit{parametric curves}\index{Parametric Curve} where the command now treats \texttt{x} as the parameter and accepts two equations. We will borrow the famous cycloid to illustrate the usage in Figure \ref{fig:cycloid}.
\begin{lstlisting}
\begin{tikzpicture}
\newcommand*{\ap}{0.7}
\begin{axis}[xlabel=$x$, ylabel=$y$, axis lines=center, xmin=0, xmax=8, ymin=0, ymax=1.7, width=0.8\textwidth, height=0.25\textwidth]
\addplot[very thick, domain=0:5*pi, samples=100] ({\ap*(x - sin(deg(x))},{\ap*(1 - cos(deg(x)))});
\node[align=left] at (axis cs:2.5,0.5) {$x = a(t-\sin(t))$,\\ $y = a(1-\cos(t))$};
\end{axis}
\end{tikzpicture}
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \newcommand*{\ap}{0.7}
    \begin{axis}[xlabel=$x$, ylabel=$y$, axis lines=center, xmin=0, xmax=8, ymin=0, ymax=1.7, width=0.8\textwidth, height=0.25\textwidth]
    \addplot[very thick, domain=0:5*pi, samples=100] ({\ap*(x - sin(deg(x))},{\ap*(1 - cos(deg(x)))});
    \node[align=left] at (axis cs:2.5,0.5) {$x = a(t-\sin(t))$,\\ $y = a(1-\cos(t))$};
    \end{axis}
    \end{tikzpicture}
    \caption{A parametric cycloid graph.}
    \label{fig:cycloid}
\end{figure}

\paragraph{Vector Fields}
The final topic to introduce in this chapter is about drawing a \textit{vector field}\index{Vector Field} using the \texttt{\textbackslash addplot3}\index{addplot3@\texttt{\textbackslash addplot3}} function (used for 3D plotting, more in the next chapter) with the \texttt{quiver}\index{quiver@\texttt{quiver}} option. This is illustrated in Figure \ref{fig:clockvec} here, with the vector field defined by $(y/\sqrt{x^2+y^2}, -x/\sqrt{x^2+y^2})$.
\begin{lstlisting}
\begin{tikzpicture}
\begin{axis}[xlabel=$x$, ylabel=$y$, xmin=-3, xmax=3, ymin=-3, ymax=3, view={0}{90}]
\addplot3[blue, domain=-2.5:2.5, quiver={u={y/(x^2+y^2)^0.5}, v={-x/(x^2+y^2)^0.5}, scale arrows=0.3}, -stealth, samples=20] {0};
\end{axis}
\end{tikzpicture}
\end{lstlisting}
\begin{figure}
    \centering
    \begin{tikzpicture}
    \begin{axis}[xlabel=$x$, ylabel=$y$, xmin=-3, xmax=3, ymin=-3, ymax=3, view={0}{90}]
    \addplot3[blue, domain=-2.5:2.5, quiver={u={y/(x^2+y^2)^0.5}, v={-x/(x^2+y^2)^0.5}, scale arrows=0.3}, -stealth, samples=20] {0};
    \end{axis}
    \end{tikzpicture}
    \caption{A clockwise rotational vector field.}
    \label{fig:clockvec}
\end{figure}
The \texttt{u} and \texttt{v} keys represent the velocities along the $x$/$y$-axes, and we have set the scale, type, and density for the arrows. Finally, \texttt{view=\{0\}\{90\}} is needed for the \texttt{axis} since we have invoked the 3D plotting method and need to reset the camera to look downward overhead.