. Differential and integral calculus, an introductory course for colleges and engineering schools. re will be no occasion to distinguish between positive andnegative curvature, that sign is to be given to the radical in IIwhich will render a +. This formula enables us to compare thecurvatures of a curve at different points. For example, in theparabola y = x2 we have yf = 2x, y = 2, 1 + y> = 1 + 4*2, and .\ a = +\x2)f It is at once apparent that the parabola is curved most at theorigin [x = 0), and that, since a = 0 as x = oo, the curve ap-proaches more and more nearly the form of a straight
. Differential and integral calculus, an introductory course for colleges and engineering schools. re will be no occasion to distinguish between positive andnegative curvature, that sign is to be given to the radical in IIwhich will render a +. This formula enables us to compare thecurvatures of a curve at different points. For example, in theparabola y = x2 we have yf = 2x, y = 2, 1 + y> = 1 + 4*2, and .\ a = +\x2)f It is at once apparent that the parabola is curved most at theorigin [x = 0), and that, since a = 0 as x = oo, the curve ap-proaches more and more nearly the form of a straight line, as wego out along the curve. Obviously the curvature of a right line is 0. Consider now thecircle of radius r. The mean curvature of an arc PQ{ — ra) is a _ a _ 1PQ~n*~~r) and therefore the actual curvature,which is the limit of the mean curva-ture, is also -; that is, 1a = — ■ r Expressing this result in words, the curvature of a circle is the sameat every point, and is equal to the reciprocal of the radius. A circlemay therefore be drawn having any curvature from 0 to oo . Since. §109 CURVATURE. EVOLUTES AND INVOLUTES 155 r = -i a circle of 0 curvature is a circle of oo radius; that is, it isa a right line, while a circle of oo curvature is a circle of 0 radius, that is, a point. Hence the point and the right line are the extremes of curvature. When a = 1, r = 1, or the circle of unit curvature is the circle of unit radius. 109. The Circle of Curvature. From what has been said itis plain that we may calculate the curvature of a curve at anygiven point P, and then by constructing a circle whose radius isthe reciprocal of the curva-ture, shall have a circle whichhas the same curvature as thegiven curve at P. If now weso place this circle that it shallbe tangent to the curve at P(have with the curve a commontangent line at P) and shallhave its center on the concaveside of the arc, it is termedthe circle of curvature or theosculating circle of the curveat P. Its rad
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