Prove that an arc segment of an ellipse cannot be similar to an arc segment of an ellipse with different eccentricity. I would like to know if there is any proof to this: Prove that an arc segment of an ellipse cannot be similar to an arc segment of an ellipse with different eccentricity.
I specifically exclude circles, which can be considered ellipses with coincident focal points.
Motivation: this seems obvious to me, and maybe to many others, I'm curious if there's any formal proof of it mathematically. I should add: I may be wrong and this may not actually be the case. I would like a formal proof of the *negation, if that's the case.
 A: This is true of any conic section, but let this proceed with two similar ellipse segments, $S_1$ and $S_2$, segments of ellipses having corresponding eccentricity $e_1$ and $e_2$.
Let $S_2$ be dilated by the ratio of similitude, so that its image is $S_3$, congruent to $S_1$. The dilation does not change the eccentricity, so $S_3$ is still a segment of an ellipse having eccentricity $e_2$.
Ellipse segments $S_1$ and $S_3$ are congruent. Therefore their corresponding eccentricities, $e_1$ and $e_2$, are equal.
See the Conica of Apollonius, where Book 6, Proposition 6 states that if any segment of a conic section can be fitted to a segment of another, then the entire sections are congruent. That goes for congruent segments. Extending it to similar segments is only a matter of scaling.
A: All parabolas are similar. An arc of one parabola can be made to coincide with another parabola arc by three transformations/ geometric  operations viz., Zoom,translation and rotation in the plane.
This happens because only a single constant is involved in its $(x,y)$ description equation, for example in its polar form
$$ p/r = (1- \cos \theta) \tag 1$$
For  an ellipse two constants occur. Zoom,translation and rotation cannot make a rigid arc to be placed inside another arc where eccentricity $e$ is now introduced:
$$ p/r = (1- e \cos \theta) \tag2$$
For such a match to take effect for an ellipse a fourth transformation/operation is necessary and that is.. changing of Aspect Ratio after zoom magnification/ uniform reduction or dilation  in the plane.
In other words for a matching between two given elliptic arcs as mentioned in the question to take place, these  four unique operations have be determined and applied.
Geometric similarity occurs  when $p/r$ has the same value at a given $\theta$.
This can occur if and only if the value of eccentricity $e$ as a constant is same for either ellipse (or conic section)... and that completes the proof for ellipse and any curve described by two parameters.
A: Experimental aspect using radius of curvature
For circle,
$$\rho=a \implies \frac{d\rho}{ds}=0$$
For logarithmic spiral $r=ae^{\theta \cot \beta}$ which is self-similar,
$$\rho=a\csc \beta+s\cot \beta \implies \frac{d\rho}{ds}=\cot \beta$$
For general conics
\begin{align}
  0 &= F(x,y) \\
  &\equiv
  \begin{pmatrix}
    x & y & 1
  \end{pmatrix}
  \begin{pmatrix}
    a & h & g \\
    h & b & f \\
    g & f & c
  \end{pmatrix}
  \begin{pmatrix}
    x \\ y \\ 1
  \end{pmatrix} \\
  \begin{pmatrix}
    u \\ v
  \end{pmatrix} &=
  \begin{pmatrix}
    a & h \\
    h & b
  \end{pmatrix}
  \begin{pmatrix}
    x \\ y
  \end{pmatrix}+
  \begin{pmatrix}
    g \\ f
  \end{pmatrix} \\
  \Delta &= \det
    \begin{pmatrix}
    a & h & g \\
    h & b & f \\
    g & f & c
  \end{pmatrix} \\
  \kappa &= \frac{\Delta}{(u^2+v^2)^{3/2}} \\
  \kappa' &= \frac{3\Delta}{(u^2+v^2)^3} [hu^2+(b-a)uv-hv^2] \\
  \frac{d\rho}{ds} &= -\frac{\kappa'}{\kappa^2} \\
  &= -\frac{3}{\Delta} [hu^2+(b-a)uv-hv^2]
\end{align}
which is closely related to the evolute.
Observations without proofs.

*

*The conics are similar by varying $c$ only.


*Testing with CAS, $\dfrac{d\rho}{ds}$ is invariant for the same relative position of similar conics.


*Similar conics implying similar evolutes.


*Up to scale, evolute varies as eccentricity.


*Conics with different eccentricities $e_i>0$ are locally similar only at vertices where $\kappa'=0$.


*Osculating conics should have the same $\kappa$, $\kappa'$ and so on, but they are usually not (locally) similar.
