Curvature is a mathematical concept that measures the bending properties of a curve. The curvature gives a numerical value at each point on a parametric curve. It varies from large, at places where the curve bends sharply, to small, where the curve is "flat." For straight lines and circles, the curvature is constant, giving the same value for all points along the curve. For a straight line, the curvature is the constant 0, and for a circle of radius R, the curvature is the constant 1/R.
Definition: The curvature plot is a graph of s vs. k.
Go to the discover page and get some more intuition about the values of curvature.
Frequently, we can see the curvature values just by looking at the shape of the parametric curve. For example, in the figures below we have marked locations where the curvature is a local maximum with green dots and a local minimum with blue dots. You should be able to see that the curve bends sharpest at the local maxima of curvature, and is flattest at the local minima in curvature.
Definition: A vertex on a parametric curve is a point on the curve where the curvature is a local maximum or a local minimum.
Try to discern the vertices on some curves in the practice page.
When the curvature values vary only slightly over an interval, the corresponding portion of the curve looks pretty much like a circle and we cannot see the curvature variation. When the variation in curvature is large enough, the shape can be seen as non-circular. Also, sharp bends and flat spots at the vertices will appear when the variation in curvature is large. Thus the curvature can both reflect our visual intuition about the shape and provide precise numerical values for each point on the curve. Therefore we can use curvature to give a precise mathematical description of the shape of a curve.