Curvature and the Shape Operator

Newton and the Radius of Curvature

Sir Isaac Newton

Sir Isaac Newton is a man whose accomplishments reach deep into the heart of all sciences. He is most well known for his three basic laws of motion (now known simply as Newton's laws), and his work in the field of optics.

What does a circle that is tangential to a curve, however, tell us about the curvature of a line?

The answer to this question actually came long before the importance of this unique type of circle was known. In 1736, a book written by British mathematician and physicist Sir Isaac Newton (1642-1727) in 1671 was published called A Treatise on the Methods of Series and Fluxions.

Here, Newton lists the most basic properties of curves as he solves a problem he had himself written. These properties were:

  1. A circle has a constant curvature that is inversely proportional to its radius.
  2. The largest circle that is tangent to a curve (on its concave side) at a point has the same curvature as the curve at that point.
  3. The center of this circle is the "center of curvature" of the curve at that point.

The second of these properties describes the circle we now call the osculating circle.

Instead of measuring the curvature from a reference line (which can lead to problems we saw earlier), the curvature of a curve at a given point is measured in terms of the radius of this circle; thus, this radius is referred to as the radius of curvature.

radius of curvature

As we can see here, the normal line at the point we are examining points directly into the center of the osculating circle.