TITLE: What is the form of curvature that is invariant under rotations and uniform scaling
QUESTION [8 upvotes]: This is a followup to this question, where I learned that curvature is invariant to rotations. 
I have learned of a version of curvature that is invariant under affine transformations.
I am wondering if there a is a form of curvature between the two. Invariant under uniform scaling and rotation but not all affine transformations?

REPLY [5 votes]: I don't know if this would suit you, but one thing you can consider (much more naive than
the notion of affine curvature) is to fix a point P_0 on your curve, and then consider the
function on the curve given by sending a point P to the quantity
curvature(P)/curvature(P_0)  .
This is a kind of relative curvature, where you measure how much everything is curving
in comparison to the curvature at P_0, and is invariant under scaling and rotation.