# How to parametrize circular paths?

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 $x_2, y_2, \phi_2$?

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