# If there any plane curve whose critical points' curvature are invariant by linear transformation?

I was studying if the curvature of $$f$$: $$f(x) = \frac{ax}{b+x}$$

can have the critical points located at the same vertical than this other $$g$$ curve:

$$g(x) = \frac{ax}{b+x} + cx = f(x) + cx$$

Then I've realised that $$g(x)$$ actually is a linear transformation of $$f(x)$$, and I thought that maybe this linear transformation (a shear) would not alter the vertical position of the critical points of the curvature just by intuition. Sadly, I was wrong... But a more broad question came to my mind, and I think it can be worthy to share it with you:

What is the curve that keeps at least one coordinate fixed of the critical points of its curvature invariant under a shear linear transformation?

Extra1: and for other linear transformations?

Extra2: what if we want to keep both coordinates of the critical point fixed?

I know, very broad question, maybe. But I don't even know where I should start!