It is well documented that certain string-art patterns generate quadratic Bezier curves: let $x, y_1, y_2$ be three points in $\mathbb{E}^2$, consider the family of line segments joining $x + (1-t) (y_1 - x)$ to $x + t (y_2 - x)$ for $t\in [0,1]$, its envelope forms a quadratic Bezier curve (and hence a parabola). Furthermore, at the parameter $t$, the intersection of the line segment with the parabolic curve is at $$ (1-t)[x + (1-t)(y_1 - x)] + t[x + t(y_2 - x)] $$ the linear interpolant with parameter $t$ on the line segment.
It is also not hard to verify that given the same three points, considering the family of line segments joining $x + e^t(y_1 - x)$ to $x + e^{-t}(y_2 - x)$, its envelope forms a hyperbola. The intersection of the line segments with the hyperbola always occur at the midpoint.
In both cases the construction changes appropriately under affine transformations of $\mathbb{E}^2$, and in the case $x, y_1, y_2$ being colinear, recovers the degenerate conics of the appropriate types.
Given that the parabolas can be thought of as a limit of hyperbolas.
Question 1: Is there a way to take a "limit" of the construction above for the hyperbolas to get the one for the parabola?
Questions 2: Can this be taken further to include elliptical arcs as part of the family of constructions?