Curvature
If we talk about 1D manifolds, curvature measures how strongly a curve deviates from the straight line and for 2D manifolds, it measures how strongly a surface deviates from plane.
Curvature also have relations with derivative of tangent vector of a curve and curve normal. Hence, $$x’’(s) = k(s).n(s)$$ Here, $k(s)$ is curvature of a curve $s$. (see Arc Length) If the sign of normal vector flips, then curvature can be negative as well. We can also see that for a straight line the curvature value becomes 0.
There is also a relation of curvature with osculating circles.
Basically if I try to fit a circle in a curve (which is also called osculaing circle), I can find the curvature of that curve. So if my fitted circle is small, it means the curvature is more, and if fitted circle is large, curvature is less. Hence curvature is inversely proportional to the radius of osculating circle. Formally, $$ R(s) = \frac{1}{K(s)}$$
In geometry processing, we discuss three types of curvatures.
- Principal Curvature
- Mean Curvature
- Gaussian Curvature