Circle

1. Equation: \[ (x-x_0)^2+(y-y_0)^2=r^2 \]
2. Parametrization: \[ \underline{r}(t) = \begin{pmatrix} x(t)\\y(t) \end{pmatrix} = r \begin{pmatrix} \cos{t}\\\sin{t} \end{pmatrix}, ~0 \leq t <2 \pi \]
3. Unit vectors: \[ \underline{e}= \underline{e}(t)= \begin{pmatrix} \cos{t}\\\sin{t} \end{pmatrix} = -\underline{n}, \qquad \underline{f}= \underline{f}(t)= \begin{pmatrix} -\sin{t}\\\cos{t} \end{pmatrix} = \underline{t} \]
4. Scalar Products: \[ \underline{e}\cdot \underline{e} = \underline{f}\cdot \underline{f} =1, \qquad \underline{e}\cdot \underline{f}=0 \]
5. Hat vectors: \[ \underline{ \widehat e}=\underline{f}, \qquad \underline{\widehat f}=-\underline{e} \]
6. Derivatives: \[ \underline{e}'=\underline{f}, \qquad \underline{f}'=-\underline{e} \]

\tikzmath{\R=3;}
\tikzmath{\RR=\R*1.2;}
\tikzmath{\AngleA=30;}
\tikzmath{\AngleB=75;}

\coordinate (O) at (0,0);
\coordinate (A) at (\AngleA:\R);
\coordinate (B) at (\AngleB:\R);

\draw (O) circle(\R);

\draw[-latex] (O) -- (0:\RR) node[right] {$x$};
\draw[-latex] (O) -- (90:\RR) node[above] {$y$};
\draw[-latex] (O) -- (A);
\draw[-latex] (O) -- (B);

\node at (\AngleA:\RR) {$\underline{e}(t_1)$};
\node at (\AngleB:\RR) {$\underline{e}(t_2)$};

\tkzMarkAngle(A,O,B);
\tkzLabelAngle[pos=1.5](A,O,B) {\small{$\theta=t_2-t_1$}};

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Glob Path: Images/*.svg

\tikzmath{\R=3;}
\tikzmath{\RR=\R*1.2;}
\tikzmath{\AngleA=30;}
\tikzmath{\AngleB=75;}

\coordinate (O) at (0,0);
\coordinate (e) at (\T:\R);
\coordinate (f) at (\T+90:\R);

\draw (O) circle(\R);

\draw[-latex] (O) -- (0:\RR) node[right] {$x$};
\draw[-latex] (O) -- (90:\RR) node[above] {$y$};
\draw[-latex] (O) -- (e);
\draw[-latex] (O) -- (f);

\node at (\T:\RR) {$\underline{e}$};
\node at (\T+90:\RR) {$\underline{f}$};

\tkzMarkRightAngle(e,O,f);