The curvature as determined by a bug on a sphere. Problem: a bug on a sphere wants to evaluate the curvature of his world. "The bug might make a circle like the one shown in" the following figure "and measure circumferences". https://www.feynmanlectures.caltech.edu/II_42.html
“He would discover that the circumference is less than 2π times the radius [here in this figure, the radius of the circumference is "r"]. (You can see that because from the wisdom of our three-dimensional view it is obvious that what he calls the “radius” [here in the figure, "radius" is "s"] is a curve which is longer than the true radius of the circle.) Suppose that the bug on the sphere had read Euclid, and decided to predict a radius by dividing the circumference C by 2π. Then he would find that the measured radius was larger than the predicted radius. Pursuing the subject, he might define the difference to be the “excess radius,” … and study how the excess radius effect depended on the size of the circle." (Feynman).
The curvature is: determined by:
This is also the Gauss curvature.
