How to parametrize circular paths?

Question

How can I parametrize path (1) and (2) in this picture?

If the part of path (1) between point $c$ and point $b$ is : $x(R_1,\phi)=R_1\cos\phi$ and $y(R_1,\phi)=R_1sin\phi$ for $R_1\in[c,b], \phi\in[0,2\pi]$

How can this curve connect the part between $a$ and $c$?

Part of path (2) between point $a$ and $d$: $x(R_2,\phi)=R_2\cos\phi$ and $y(R_2,\phi)=R_2\sin\phi$ for $R_2\in[a,d], \phi\in[0,2\pi]$

Same here, how can this curve connect the path between $d$ and $b$?

Btw, should I use diffrent subscripts for the path (1) and (2)? I.e. $x_1, y_1, \phi_1$ and $x_2, y_2, \phi_2$?

Answer 1

A standard way of parametrizing a straight line from $A$ to $B$ is: $$ (1-t)A+tB, \quad t\in [0,1] $$

Answer 2

If, for path (1) we let $\theta_1$ represent the angle of segment $ca$ and $\theta_2$ the angle of segment $bd$ then the path can be represented parametrically by

\begin{align} x(t)&=\begin{cases}[(1-t)R_1+tR_2]\cos\theta_1&\text{ for }0\le t<1\\R_1\cos[(2-t)\theta_1+(t-1)\theta_2]&\text{ for }1\le t\le2 \end{cases}\\ y(t)&=\begin{cases}[(1-t)R_1+tR_2]\sin\theta_1&\text{ for }0\le t<1\\R_1\sin[(2-t)\theta_1+(t-1)\theta_2]&\text{ for }1\le t\le2 \end{cases} \end{align}

For path (2) you could use

\begin{align} x(t)&=\begin{cases}R_2\cos[(1-t)\theta_1+t\theta_2]&\text{ for }0\le t<1\\ [(2-t)R_2+(t-1)R_1]\cos\theta_2&\text{ for }1\le t\le2 \end{cases}\\ y(t)&=\begin{cases}R_2\sin[(1-t)\theta_1+t\theta_2]&\text{ for }0\le t<1\\ [(2-t)R_2+(t-1)R_1]\sin\theta_2&\text{ for }1\le t\le2 \end{cases} \end{align}