**CIRCLE OF CURVATURE. CENTER OF CURVATURE**

**116. Circle of curvature.**^{[1]} **Center of curvature.** If a circle be drawn through three points *P*_{0}, *P*_{1}, *P*_{2} on a plane curve, and if *P*_{1} and *P*_{2} be made to approach *P*_{0} along the curve as a limiting position, then the circle will in general approach in magnitude and position a limiting circle called the *circle of curvature of the curve at the point P*_{0}. The center of this circle is called the *center of curvature*.

Let the equation of the curve be

(1) | ; |

and let be the abscissas of the points respectively, the coördinates of the center, and the radius of the circle passing through the three points. Then the equation of the circle is

and since the coordinates of the points must satisfy this equation, we have

(2) |

Now consider the *function of * defined by

in which has been replaced by from (1).

Then from equations (2) we get