Parametric equation of a curve: a line in a circle transform to a curve in an ellipse Giving the circle with a line segment inside, if the circle was stretched into an ellipse, what is the parametric equation of the parabolic curve (I assume) transformed from the line segment?
circle to ellipse
I want the curve as in shown in the pic. Scaling axis only give the line segment with a new slope. I could draw the curve by interpolated the line according to the displacement field inside the circle, but I don't know the parametric equation of the curve. Could anyone give me some hints how to determine the equation in trigonometric way?

How do you define your  stretch transformation? In any transformation a relation between variables before and after transformation is defined. 
Setting a variable in one system to a constant we can map the curve in the other system. Like eg polar to cartesian coordinates $ r=a \rightarrow x^2+y^2= a^2. $
I could not figure out transfomation relations in your question. Are any lengths conserved?
Setting aside a full approach if we proceed just with the relation you gave that automatically defines and additionally could take care of its relation to the circle:
$$ x = a \cos \theta + c,$$
by converting to polar coordinates we get directly (no parametrization) the curve Conchoid of Nichomedes:
$$ r  \cos \theta   = a \cos \theta + c \,;  \quad r= a + c \cdot \sec \theta, $$
where the unit circle is added for comparision. For the curve graph $ (a=1, c= 0.3 ) .$ 
As stated above the strectch definition has still to be looked into.

