GEOMETRY 39 



2.223 The center of curvature is a point C (fig. 2) on the normal at P such 

 that PC = p. If p is positive C lies on the positive normal (2.213) ; if negative, 

 on the negative normal. 



2.224 The circle of curvature is a circle with C as center and radius = p. 



2.225 The chord of curvature is the chord of the circle of curvature passing 

 through the origin and the point P. 



2.226 The coordinates of the center of curvature at the point x, y are , rj: 



% = x p sin T 



dy 



tan T = -j- 

 dx 



rj = y + p cos r 

 If /', m' are the direction cosines of the positive normal, 



77 = y -f m'p. 



2.227 If /, m are the direction cosines of the positive tangent and /', m' those 

 of the positive normal, 



dl I' dm m' 

 ds~ p ds ~ p 



I' = m , m' = -I, 



dV__ I dm' _ m 

 ds p ds p 



2.228 If the tangent and normal at P are taken as the x- and y- axes, then 



limit *_ 2 



x K> 2y 



2.229 Points of Inflexion. For a curve given in the form (a) of 2.200 a point 



of inflexion is a point at which one at least of j^ and j-r exists and is con- 



dx dy 



//2/tt H^if 



tinuous and at which one at least of "7-3 and j^ vanishes and changes sign. 



dx dy 



If the curve is given in the form (b) a point of inflexion, h, is a point at which 

 the determinant: 



/!' (4) I' ( 

 vanishes and changes sign. 



2.230 Eliminating x and y between the coordinates of the center of curvature 

 (2.226) and the corresponding equations of the curve (2.200) gives the equation 

 of the evolute of the curve - the locus of the center of curvature. A curve 

 which has a given curve for evolute is called an involute of the given curve. 



