What specific quantity does curvature measure?
This worksheet only considers circles, but it illustrates the connection between curvature and the rate at which the unit tangent vector changes direction.
I don't get nearly as dizzy chasing my dog around my house as I do chasing him around my coffee table. My body rotates considerably faster going around the coffee table because the curvature of the path that I take is so much greater than that going around my house. This is in spite of the fact that the paths are essentially the same shape.
