Tranformations of Curves A while ago I asked a question similar to this, but looking back, I think I would have to further clarify. Please excuse how I ask this question, as I am very new to the Math Stack Exchange.
Suppose I have a curve, with one endpoint on the origin of the x-y axis, and the other at some other point on the x-axis. In other words, the two endpoints of the curve are on (0,0) and some (x,0)
If I were to change the position of the latter endpoint (x,0), how do other points on curve f(x) change with respect to the change of (x,0)? Intuitively, if I have a string, and I stretch an endpoint to some other position, how does the original string change?
Moreover, if I were to move that endpoint in a specific path (represented as a function), what would the paths of the other points on the curve be?
Looking at the problem, I thought about representing the function as a series of points, to which I would draw respective lines through each point, and would then calculate the motion of the other points with respect to the change of the latter endpoint. I did this for a 'one-point system', which is merely a point connected with 2 lines, with these lines connecting towards the two endpoints listed above (I would draw a diagram but I do not know how to on this software). Doing this, I found an equation, although it was huge. I then did this for two-point systems, and I soon realized that the solutions would be much bigger than I imagined.
I am sure that if I continued to work on this problem, I would arrive at somewhat of a verdict. However, the people on this Stack Exchange are much more well-trained in mathematics than I am. And so my question is this: Does a solution currently exist to this problem? If so, what is it? If not, how would I further proceed with the method I used above? How would you all solve the problem?
Please note that the new, transformed curve is subject to the previous curve's arc length (the original and new curve have the same length). The question I seek to find is how the original curve changes into its new form.
Again, please excuse any difficulties with how I have written this post. Hopefully, you all do not mind.
 A: When arc length can change for bent profiles:
The geometric property of the set  that does not change should be identified or defined. There should be some basis for transformation. The property can be a subset constant but also a super-set variable in order to accommodate a longer arc between two $x-$ axis points.
Basically it is a physics principle defining Bending and Stretching resulting in an ode but not just some  mathematical manipulation.
Whether or how much curvature/torsion/strain property should change must a priori be determined through a property of the curve which in fact represents a phenomenon.
Here I chose a circle for simplicity and computational convenience using physics/mechanics Euler relation between changing curvature and bending moment of a beam ..
A Parabolic arc, Catenary, Cornu's spiral.. any curve can well be taken if we know its differential equation and what physical situation causes it.
For example  all circles in the following example can have constant radii of curvature but we can choose smaller circles or bigger ones in separate sets with a different property.
The intrinsic differential  equation of a circle radius $R$ in the plane  is
$$ \frac{1}{R}=\frac{d \phi}{ds} $$
When constant radius is changed, an ordinary differential equation for the superset of line points is obtained by one more differentiation with respect to arc.
$$ \frac{d^2 \phi}{ds^2} =\phi^{''}(s)=0 $$
In the following Mathematica program arc lengths of circles were chosen for "stretched arc lengths" $(8.9,11.25) $ units for $x$ segment lengths $(7,8,9)$ in the manner you wanted.

When same arc length ($5$ units) is bent
between two fixed points $ \{(0,0)-(2,0)\} $ with constants p =(0.8,1.3,1.4)  for (blue,red,magenta) curves respectively, when we take intrinsic curvature rate property as :
$$ \phi^{''}(s)= p \cos s.  $$

Hope  the above is useful for further numerical work.
