Q:

Can we build a tool that measure curvature on an arc?

I know some biologist that are taking curvature measurements on photos with softwares to understand the variation on beak morphology. 
I'm wondering if there is a way to build a mechanical tool (or if one exist already) to measure the curvature of a circle's (or ellipse) arc. The arc is not always of the same length, so it would need to be adjustable. 
There is a constrain (see the grey line): 

I can't fit a big calliper on the bird's head. 
I want to know: 

the radius of the circle that correspond to the beak curvature
how "steep" is that curvature (probably a lack of vocabulary on my side but see the image down to see what I mean by "steep")

You can see this image as the different birds in the image above. Bird #4 would have a flatter beak than bird #1.
I need to be able to calculate the "steepness" even if the x, y or z are not of the same length. 
The idea of bringing a mechanical tool is to calculate rapidly and conveniently a number and add it directly to a dataset. It's much faster than to look at photos and analyse with a software. So if there is a mechanical tool designed for this, it would be super useful. 
Again, the image is just a visualization of the problem. I would like to measure the beak of different birds on the field.

A:

For a circle, yes. Such a tool exists:
Digital Radius Caliper
There are also Digital Radius Gauges, which may be useful for smaller curves.
For a general curve, you can't measure the curvature exactly since it varies continuously along the curve. You could perhaps assume a region is approximately circular and measure that. But depending on the size of the tool and the rate of curvature change this could be wildly inaccurate.

A:

How about a Contour Gauge:

And here:
Wikipedia Article

Make your measurement
Trace it
Scan/digititize it
Analyze it (spline/curve fit) with some type of CAD software. 

