Parametric equation of a curve: a line in a circle transform to a curve in an ellipse

Question

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?

Answer 1

The stretch in question is a linear transformation, so lines go to lines.

Moreover, horizontal lines go to horizontal lines, and vertical lines go to vertical lines.

The line $x=c\;$goes to the line $x=c'$, where $c'$ satisfies $$\frac{c'}{a}=\frac{c}{r}$$ hence the new line is $x=c\bigl({\large{\frac{a}{r}}}\bigr)$.

Answer 2

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.