Eccentricity going to zero -- Geometric definition conic Given a straight line $D$, a point $F\notin D$ and a positive real number $e$, a conic is a subset of ${\cal P}_2$ defined as:
$$
\mathcal{C}(e,F,D) = \{M\in {\cal P}_2,\, d(F,M)=e\,d(M,D) \}
$$
where ${\cal P}_2$ is the Euclidean plane and $d$ is the Euclidean distance.

It is well  known that when $e$ tends to $0$, ${\cal C}(e,F,D)$ tends to a circle and I would like to see this from this definition without taking any algebraic computations.
(we can show this fact from algebraic computations computing the distances in some coordinates system but I would like to avoid this approach.)
I see that $e,F,D$ are not independent in the sense that if $e$ tends to $0$ then $M$ has to tend to $F$ and we do not obtain a circle so there should be some connections between $e$, $D$, $F$. But then it means that the previous definition is not "well-posed". Moreover a conic being defined by a polynomial equation of degree less than 2 (so 5 coefficients seen as parameters), there should not have connection between $e$ (1 parameter), $F$ (2 parameters), and $D$ (2 parameters)... What do I miss here ?
A second point assume there is a connection between $e,F,D$ and I have the feeling that when $e$ tends to $0$ then $D$ as to be seen as a point at infinity. This framework should be related to projective geometry and I would like some developments in this direction so that I can deduce the circle as a limit. An idea is that since we have a point to infinity, the object has to be invariant by any rotation. Moreover it is a convex set so it is a circle... I need to put some maths on this idea and I think projective geometry and the study of a group of transformations acting on the set should lead to find the symmetries. 

Keep $D$ fixed and reduce $e$ so that $M$ gets closer and closer to $F$. At the same time, $d(M,D)$ becomes more and more constant and the locus tends to a circle of radius $e\,d(F,D)$.
If you want the radius to be $R$, dilate the whole figure by $\dfrac{R}{e\,d(F,D)}$. For eccentricity $0$, you get a perfect circle with a line at infinity.
