Drawing the circle for a cycloid curve in Tikz

Question

Unfortunately I have yet to figure out how to get things done in Tikz and I wanted to ask if someone could help me doing this picture in TeX-Code (Tikz preferably). I think I could manage to get the circle done but whenever it comes to angles I just screw up massively and I can't figure it out.

i would highly appreciate any help. ty very much

Answer 1

UPDATE: Removed the calc library and added an animation.

 \documentclass[tikz,border=4pt]{standalone}
 \usetikzlibrary{intersections}
 \makeatletter % from https://tex.stackexchange.com/a/20864/121799
 \newcommand{\tikzAngleOfLine}{\tikz@AngleOfLine}
    \def\tikz@AngleOfLine(#1)(#2)#3{
    \pgfmathanglebetweenpoints{
        \pgfpointanchor{#1}{center}}{
        \pgfpointanchor{#2}{center}}
    \pgfmathsetmacro{#3}{\pgfmathresult}
  }
 \makeatother 
 \begin{document}
 \begin{tikzpicture}[scale=5,font=\sffamily]
  \draw[-latex] (0,-0.2) -- (0,2.2);
  \draw[-latex] (-0.1,0) -- (2.4,0);
  \coordinate[label=45:{$O$}] (O) at (0,0);
  \coordinate[label=45:{$M$}] (M) at (1.1,1);
  \coordinate[label=45:{$T$}] (T) at (M|-O);
  \draw[name path=circle] (M) circle (1);
  \draw[name path=cycloid,green!50!black] (O) plot[variable=\x,domain=0:5*pi/6] ({\x-sin(\x*180/pi)}
  ,{1-cos(\x*180/pi)});
  \path [name intersections={of=cycloid and circle,by={P,Q}}];
  \coordinate[label=45:{$R$}] (R) at (P-|M);
  \tikzAngleOfLine(M)(R){\AngleStart}
  \tikzAngleOfLine(M)(P){\AngleEnd}
  \filldraw[red,fill opacity=0.3]  (M) -- (M)+(\AngleStart:4pt)
   arc (\AngleStart:\AngleEnd:4pt) node[pos=0.3,above,opacity=1]{$t$} -- (M)  --cycle;
  \filldraw[green,fill opacity=0.3]  (R) -- ++(-4pt,0) arc (180:90:4pt) -- (R) -- cycle;
  \draw[name path=line] (T) -- (M) -- (P) node[above,midway,sloped]{$r$} -- (R);
  \node[label=above:{$P$}] at (P){};
  \foreach \X in {O,T,M,P,R}
  {\draw[fill=blue] (\X) circle (0.4pt);}
 \end{tikzpicture}
 \end{document}

\documentclass{article}

\usepackage{animate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage[active,tightpage]{preview}
\makeatletter
\def\@anim@@newframe{\@ifstar\@anim@newframe\@anim@newframe}
\def\@anim@newframe{\end{preview}\begin{preview}}
\renewenvironment{animateinline}[2][]{%
  \let\newframe\@anim@@newframe%
  \let\multiframe\@anim@multiframe%
  \begin{preview}}{%
  \end{preview}}
\makeatother
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \usepackage{tikz}
 \usetikzlibrary{intersections}
 \makeatletter % from https://tex.stackexchange.com/a/20864/121799
 \newcommand{\tikzAngleOfLine}{\tikz@AngleOfLine}
    \def\tikz@AngleOfLine(#1)(#2)#3{
    \pgfmathanglebetweenpoints{
        \pgfpointanchor{#1}{center}}{
        \pgfpointanchor{#2}{center}}
    \pgfmathsetmacro{#3}{\pgfmathresult}
  }
 \makeatother 
 \begin{document}
 \begin{animateinline}[autoplay,loop]{2}
 \multiframe{62}{i=1+1}{\pgfmathsetmacro{\X}{\i/10}
 \begin{tikzpicture}[scale=2,font=\sffamily]
  \draw[-latex] (0,-0.2) -- (0,2.2);
  \draw[-latex] (-1,0) -- (7.2,0);
  \path[use as bounding box] (-1,-0.4) rectangle (7.4,2.5);
  \coordinate[label=45:{$O$}] (O) at (0,0);
  \coordinate[label=45:{$M$}] (M) at (\X,1);
  \coordinate[label=45:{$T$}] (T) at (M|-O);
  \draw[name path=circle] (M) circle (1);
  \draw[name path=cycloid,green!50!black] (O) plot[variable=\x,domain=0:0.01+\X] ({\x-sin(\x*180/pi)}
  ,{1-cos(\x*180/pi)});
  % \Disc discriminates between the cases of one and two intersections
  \pgfmathtruncatemacro{\Disc}{ifthenelse(\i>31,1,0)}
  \ifnum\Disc=0
  \path [name intersections={of=cycloid and circle,by={P}}];
  \else
  \path [name intersections={of=cycloid and circle,by={Q,P}}];
  \fi
  \coordinate[label=45:{$R$}] (R) at (P-|M);
  \tikzAngleOfLine(M)(R){\AngleStart}
  \tikzAngleOfLine(M)(P){\AngleEnd}
  \filldraw[red,fill opacity=0.3]  (M) -- (M)+(\AngleStart:4pt)
   arc (\AngleStart:\AngleEnd:4pt) node[pos=0.3,above,opacity=1]{$t$} -- (M)  --cycle;
  \tikzAngleOfLine(R)(P){\AngleStart}
  \tikzAngleOfLine(R)(M){\AngleEnd}
  \ifnum\Disc=0
  \filldraw[green,fill opacity=0.3]  (R) -- ++(-4pt,0) arc (180:\AngleEnd:4pt) -- (R) -- cycle;
  \else
  \filldraw[green,fill opacity=0.3]  (R) -- ++(4pt,0) arc (0:\AngleEnd:4pt) -- (R) -- cycle;
  \fi
  \draw[name path=line] (T) -- (M) -- (P) node[above,midway,sloped]{$r$} -- (R)
  --cycle;
  \node[label=above:{$P$}] at (P){};
  \foreach \X in {O,T,M,P,R}
  {\draw[fill=blue] (\X) circle (1pt);}
 \end{tikzpicture}}
 \end{animateinline}
 \end{document}

Using this trick one can create an animated gif.

Answer 2

\documentclass[dvipsnames,tikz,margin=5mm]{standalone}
\usepackage{luatex85}
\usetikzlibrary{%
    calc,
    backgrounds,
    quotes,
    angles,
    }


% 3 couches de dessin
\pgfdeclarelayer{background}
\pgfdeclarelayer{foreground}
\pgfsetlayers{background,main,foreground}


\tikzset{% Angle droit
    angle droit/.style={very thin},
    angle droit size/.store in=\AngleDtLgth,
    angle droit size=5pt,
    pics/.cd,
    angle droit/.style args={#1-#2-#3}{%
        code = {%
        \draw[angle droit] ($(#2)!\AngleDtLgth!(#1)$)
        --($(#2)!2!($($(#2)!\AngleDtLgth!(#1)$)!.5!($(#2)!\AngleDtLgth!(#3)$)$) $)
        --($(#2)!\AngleDtLgth!(#3)$) ; ;
        }
    }}


\begin{document}

\foreach \x [count=\n from 0] in {0,5,...,360} {%

\begin{tikzpicture}

\draw[-stealth] (-1.1,0) -- (7.5,0) ;
\draw[-stealth] (0,-.5)--(0,2.5) ;
\coordinate (O) at (0,0) ;

\begin{scope}[shift={(\x/180*3.14159,1)}]

\draw[blue, semithick] circle (1) ;

\coordinate (A\n) at (-\x-90:1) ;

\draw[thin] (0,0) coordinate (M) -- (A\n) ;

\draw[red,smooth] (A0) \foreach \p in {0,...,\n} {--(A\p)} ;

\draw[thin,-stealth] (0,-1) coordinate (A) 
    -- (0,1.5) coordinate (B) ;

\draw[thin,dashed] (A\n) -- ($(A)!(A\n)!(B)$) coordinate (R);

\begin{pgfonlayer}{background}
\draw pic[%
    "\tiny\x",
    draw=green,
    fill=green!15,
    angle eccentricity=2,
    angle radius=.2cm
    ] {angle=A\n--M--A} ;
\end{pgfonlayer}

\pgfmathsetmacro{\Rangle}{ifthenelse(\x<180,"0","180")}

\foreach \Coor/\Text/\Pos in 
    {O/$O$/225,
    M/$M$/0,
    A/$T$/-45,
    A\n/$P$/225-\x,
    R/$R$/\Rangle%
    } {%
    \draw[fill=blue] (\Coor) circle (1pt);
    \node[shift=(\Pos:6pt),anchor=center] at (\Coor) {\tiny\Text} ;
    }

\path pic[angle droit size=4pt] {angle droit=A\n-R-M} ;


\fill[red] (A\n) circle (1pt) ;

\end{scope}

\end{tikzpicture}}

\end{document}