Straightening an arc OR curving a line segment to compare their respective lengths. With no available means for their ready measurement, how can it be geometrically determined whether an arc and a line segment, both of unknown lengths, are equal or not lengthwise?
In a similar vein, how can an arc be mapped such that it is straightened and it's true length preserved? OR, vice versa, how to map a line segment to a curve such that its length is maintained?
I have a project that is stalled because I cannot find a solution myself despite repeated experiments and googling without success until I am near blue in the face.
I found two supposed methods that were heavily mathematical and beyond my level of comprehension, and one was also coordinate based and I am not well versed in coordinate geometry.
I have opted not to include my geometric 'scratchings' because they are particularly crude and messy and are far from being a working solution. I have included a figure of a simpler example from which I can likely extrapolate the methods from any solutions you may provide here. I want to map or transform (straighten) the red arc and compare its length with (or against) that of the blue line segment. (NB:  The red arc and the distance between the 2 large red dots are of equal length, but assume that this is not known). I have added some auxiliary lines and marked the half- and quarter-points of the arc to show my partial efforts and thinking.

 A: Let the side length of the square be $s$, so its diagonal is equal to $s\sqrt{2}$.
Then, notice that the line is equal to $2$ of the square's diagonals, so its length is $2s\sqrt{2}$.
Now, note that the arc has a radius equal to the square's diagonal, or $s\sqrt{2}$. Then, if the arc were equal to a full circle, its length would be $2\pi r = 2\pi s\sqrt{2}$.
However, since the arc is only a quarter of a full circle, we have to divide by $4$. Thus, the length of the arc is $\frac{\pi s\sqrt{2}}{2}$.
Thus, $\boxed{\text{The line has length }2s\sqrt{2}\approx 2.828s\text{, and the arc has length }\frac{\pi s\sqrt{2}}{2}\approx 2.221s.\ \ }$
For the general case with an arbitrary line and arc, you can first determine the length of the line by using the Distance Formula on its endpoints.
Next, given $3$ points on the arc (two endpoints and another random point), find the center of the arc by finding the intersection of the perpendicular bisectors of $2$ pairs of the points. Once given the center, find the radius of the arc using the Distance Formula.
Now, find the angle between the $2$ radii intersecting the endpoints. You can do this by converting the radii to vectors, taking their dot product, dividing by the radius squared, and taking the inverse cosine. (You might have to add $180^{\circ}$ or so.)
Finally, given the angle subtended by the arc (in radians) and its radius, multiply the two to get the length of the arc.
Thus, you can determine the length of the line given its endpoints, and you can determine the length of the arc given the endpoints and $1$ other point on the arc.
