What Is the Contribution of Curvature in Different Curves?

Constant Curvature

• A circle represents the concept of constant curvature. Every point on a circle is the same distance from a point at the center of the circle. The circle bends at an even rate. If any part of the circle doesn't bend at that same rate, then the figure isn't a circle. A circle is the only curve that bends at a constant rate.

Point

• A straight line is a series of points that form a line. A curve is also a line with a series of points. To examine the concept of curvature requires looking at a curve from one particular point. Each point represents part of an imaginary circle that would represent the curvature at that particular location. This concept is known as an osculating circle.

Osculating Circle

• An osculating circle defines curvature at one particular point along a curve. The osculating circle "touches" the curve at that point and no other point. At each point the osculating circle would be different. It points upward or downward depending upon the particular curve and its shape. Using an osculating circle involves examining a curve as a person standing at one point on the curve.

Radius of Curvature

• A straight line doesn't curve. Make a circle bigger and each point looks more like a straight line. The radius of curvature is the distance from a point in the center of a circle to a point on the circle. The radius changes from point to point along any curve as the osculating circle moves.