# Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/202

CIRCLE OF CURVATURE. CENTER OF CURVATURE

116. Circle of curvature.[1] Center of curvature. If a circle be drawn through three points P0, P1, P2 on a plane curve, and if P1 and P2 be made to approach P0 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 P0. The center of this circle is called the center of curvature.

Let the equation of the curve be

 (1) ${\displaystyle y\ =\ f(x)}$;

and let ${\displaystyle x_{0},x_{1},x_{2}}$ be the abscissas of the points ${\displaystyle P_{0},P_{1},P_{2}}$ respectively, ${\displaystyle (\alpha ',\beta ')}$ the coördinates of the center, and ${\displaystyle R'}$ the radius of the circle passing through the three points. Then the equation of the circle is

${\displaystyle (x-\alpha ')^{2}+(y-\beta ')^{2}\ =\ R'{^{2}};}$

and since the coordinates of the points ${\displaystyle P_{0},P_{1},P_{2}}$ must satisfy this equation, we have

 (2) ${\displaystyle {\begin{cases}(x_{0}-\alpha ')^{2}+(y_{0}-\beta ')^{2}-R'^{2}=0,\\(x_{1}-\alpha ')^{2}+(y_{1}-\beta ')^{2}-R'^{2}=0,\\(x_{2}-\alpha ')^{2}+(y_{2}-\beta ')^{2}-R'^{2}=0.\end{cases}}}$

Now consider the function of ${\displaystyle X}$ defined by

${\displaystyle F(x)\ =\ (x-\alpha ')^{2}+(y-\beta ')^{2}=R'2,}$

in which ${\displaystyle y}$ has been replaced by ${\displaystyle f(x)}$ from (1).

Then from equations (2) we get

${\displaystyle F(x_{0})\ =\ 0,F(x_{1})\ =\ 0,F(x_{2})\ =\ 0.}$
1. Sometimes called the osculating circle. The circle of curvature was defined from another point of view on §104