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

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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.

Wag 116-1 plane curve.jpg

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in which y has been replaced by f(x) from (1).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.
  1. Sometimes called the osculating circle. The circle of curvature was defined from another point of view on §104