Is it possible to join a set of points using multiple circular arcs to get a smooth curve?

Question

I have a set of points on a plane, and I want to join these points using a circular arc between consecutive points such that the final curve I get is smooth (no sharp edges).

Is this possible? If so, how can I find that arc?

Answer 1

First I will rewrite the problem in a more precise terms.

Given $n>1$ points in the plane, $A_1$, $\ldots$, $A_n$, prove or disprove the existence of $n-1$ circular arcs $A_1A_2$, $\ldots$, $A_{n-1}A_n$ such that $A_{k-1}A_k$ is tangent to $A_kA_{k+1}$ for every $2\le k\le n-1$.

Draw any arc between $A_1$ and $A_2$ (the center $C_1$ must be on the perpendicular bisector of $\overline{A_1A_2}$). Now, draw the line $C_1A_2$. This line shall intersect the perpendicular bisector of $A_2A_3$ at $C_2$. This is the center of the second arc. If $C_1A_2$ and the perpendicular bisector of $A_2A_3$ are parallel, $C_2$ will be the middle point of $A_2A_3$.

Can you continue?