Polygon / Any shape invariant for comparison or fiting For my personal curiosity, I was wondering which would be simplest algorithmic way to compare two shapes to say whether they are the same or not. After some researches, I found out that there are many graphical/visual tools that rely on either probabilities or neural networks, either of which didn't satisfy me.
I pursued my researches following the idea that there should be a mathematic formula or teaching to get an "invariant", meaning (for me) a footprint or a mathematical "checksum" of any shape no matter the zoom or the rotation. That would allow me to decide, for instance, whether two jigsaw pieces are the same, no matter size or rotation.
What are the mathematical tools that can be interesting to study/resolve for this? Is this the kind of ellipsis perimeter issue that still has to be solved by mathematicians around the world? Is this topology only?
In the end, this would allow to always "handle" a shape in the same way, so that it can be compared or processed.
 A: You are perhaps asking about isometrically invariant relationship between (geodesic) curvature and arc length variables $(kg,s),$ from the first fundamental form of classical surface theory. Yes, ellipse like oval shapes can have curvatures related to their boundary arc lengths.
To obtain definition of ovals we can have general relations among them
$f(k_g,s, \phi)=0$ particularly trig relations of $\phi$.
Two typical  quantities  when linked by a single constant $a$ defines scale as:
$$ \kappa_g = s/a^2$$
forms a natural or intrinsic equation of a clothoid/Cornu's spiral which decides shape that is invariant.
When integrated with certain boundary/ initial conditions BC/IC, it is this set of boundary/initial conditions that decide  scale and orientation of any particular Cornu spiral, defining the varying  rotation.
When there are sudden changes in slope or curvature like in an arbitrary polygon, discrete interval forms define particular function dependencies. ation
However differential equation of contour for pieces to fit together in a jigsaw puzzle is a more important and different tiling problem.

