Can an arc from a circle be made a straight line? I want to write an algorithm that morphs or transforms an arc of a circle to a straight line joining both ends of the arc. I would like to understand the maths behind this such a transformation. How can this be done?
If $x(t)$ is a parametrization of your circle arc (see the other answers for how to find one), and $y(t)$ a parametrization of the straight line segment joining the endpoints of the arc, then

$$F_\lambda(t)=(1-\lambda)x(t)+\lambda y(t)$$

is the parametrizations for the circle arc when $\lambda=0$, the parametrization of the line segment when $\lambda=1$, and some intermediate form when $\lambda\in(0,1)$ is something in between. This technique is called linear interpolation and is one way (out of infinitely many) to achieve such a blend.

So draw $F_\lambda(t)$ for $\lambda=0$ and then slowly increase the value of $\lambda$ and repeatedly daw the resulting curve.
We can depict bending of  circular arcs of increasing radius. The math is elementary trig. Angle subtended at arc center goes on increasing when arc bends more. It is straight for infinite radius of curvature ($R$, $r$) at beginning..

When arc starts with horizontal tangent, the tangential rotation, arc length and radius respectively are $(\phi, s,R) .$

$$ \phi = s/R$$

$$  \frac{dx}{ds}= \cos \phi,\, \frac{dy}{ds}= \sin \phi $$

integrate,

$$ x= x_0 + R \sin (s/R),\, y= y_0 + R (1-\cos (s/R))\, ; $$

that gives co-ordinates as a function of distance $s$ from fixed point $ (x_0,y_0), $ the fixity point for any radius R. It can be animated. If only the endpoint is desired, we let $s=L_{max}$.



Lmax = 100.;
ParametricPlot[{r Sin[L/r], r (1 - Cos[L/r])}, {r, 20, 100}, {L, 
  Lmax/10, Lmax}, Mesh -&gt; {30, 20}, GridLines -&gt; Automatic]