Cycloid: Rolling Circle on Line

1. Parametrization, $r \mathcal{C}$: \[ \underline{r}(t) = \begin{pmatrix} r(t+\sin{t})\\r(1-\cos{t}) \end{pmatrix} = r\left( t\underline{\mathbf{i}} + \underline{\mathbf{j}} + \underline{\mathbf{q}}(t) \right), ~t \in \mathbb{R} \]
2. Unit Vectors: \[ \underline{p}(t) = \begin{pmatrix} -\cos{t}\\-\sin{t} \end{pmatrix}, \qquad \underline{q}(t) = \begin{pmatrix} \sin{t}\\-\cos{t} \end{pmatrix} \]
   
3. Properties:
   
4. The Evolute of a Cycloid is again a Cycloid
5. Cycloid:

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6. TikZ:

   TiKZ Listing: Cycloid.tikz.tex. PDF   PNG   SVG   ZIP*   %Uses \tikzmath{\r=2;} %\R{\t} \newcommand{\R}[2] { \Q{#1}{Q} \coordinate (#2) at ($#1*(i)+(j)+(Q)$); \coordinate (#2) at ($\r*(#2)$); } %Rolling center %\RC{\t} \newcommand{\RC}[2] { \coordinate (#2) at (\r*#1,\r); } %\dR{\t} \newcommand{\dR}[2] { \P{#1}{P} \coordinate (#2) at ($(i)+(P)$); \coordinate (#2) at ($\r*(#2)$); } %\V{\t}{\varname} \newcommand{\V}[2] { \tikzmath{#2=1+cos(deg(#1));} \tikzmath{#2=\r*sqrt(2*#2);} } %Velocity Normal %\dN{\t}{\coordname} \newcommand{\dRN}[2] { \Q{#1}{Q} \coordinate (#2) at ($(j)+(Q)$); \coordinate (#2) at ($\r*(#2)$); } %\ddR{\t}{\coordname} \newcommand{\ddR}[2] { \Q{#1}{Q} \coordinate (#2) at ($-(Q)$); \coordinate (#2) at ($\r*(#2)$); } %\V{\t}{\varname} \newcommand{\Det}[2] { \tikzmath{#2=(cos(deg(#1))+1);} \tikzmath{#2=-\r*\r*#2;} } %\Evolute{\t}{\coordname} \newcommand{\Evolute}[2] { \Q{#1}{Q} \tikzmath{\cost=cos(deg(#1));} %\tikzmath{\sint=sin(deg(#1));} %%\tikzmath{\rphi=1+\cost;} %%\tikzmath{\rphi=-\rphi/(\cost+1);} \coordinate (#2) at ($#1*(i)-(j)-(Q)$); \coordinate (#2) at ($\r*(#2)$); }