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

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CIRCLE OF CURVATURE. CENTER OF CURVATURE

116. Circle of curvature. 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)	$y\ =\ f(x)$;

and let $x_0, x_1, x_2$ be the abscissas of the points $P_0, P_1, P_2$ respectively, $(\alpha', \beta')$ the coördinates of the center, and $R'$ the radius of the circle passing through the three points. Then the equation of the circle is

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

and since the coordinates of the points $P_0, P_1, P_2$ must satisfy this equation, we have

(2)	$\begin{cases} (x_0 - \alpha')^2 + (y_0 - \beta')^2 - R'^2 = 0, \\ (x_1 - \alpha')^2 + (y_1 - \beta')^2 - R'^2 = 0, \\ (x_2 - \alpha')^2 + (y_2 - \beta')^2 - R'^2 = 0.\end{cases}$

Now consider the function of $X$ defined by

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

in which $y$ has been replaced by $f(x)$ from (1).

Then from equations (2) we get

$F(x_0)\ =\ 0, F(x_1)\ =\ 0, F(x_2)\ =\ 0.$