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When $ = oo, r = a; therefore there is an asymptotic circle, whose radius is a. Hence we tabulate as follows:
It appears then that the curve starts from infinity, as delineated in fig. 100, and periodically, when 0 = 77, = 277, = passes through the points A and B, which are the extremities of the diameter of the circle whose centre is the pole and whose radius is a; to which circle the curve continually approaches, being outside in the first and second quadrants, and inside in the third and fourth. There is then this peculiarity, that the curve on the outside is gradually becoming nearer and nearer to the circle, and the curve on the inside is receding from the diameter as 9 increases and approaching to coincidence with the circle.
ON THE CURVATURE OF PLANE CURVES.
280.] Imagine a tangent to be drawn at a point in a plane curve, which is such that the curve lies entirely on one side of the tangent; then the curve is said to be convex towards that side of space on which the tangent lies, and concave towards the other side; such is our definition of concavity and convexity; and on such a conception were investigated in Chapter X the analytical criteria for determining the direction of curvature. Let us moreover suppose that at the point of the curve under consideration there is no discontinuity, or indeterminateness of derived-functions; then, as the curve deviates from the tangent line, such a deviation may be greater or less, or, in other words, the curve may be more or less bent; herein then we have a new affection, viz. the amount of bending or of curvature, as it is called: the nature of which we propose to examine in the present Chapter.
And to consider it from another point of view; an infinitesimal element of the curve commencing from a given point being straight, it is in its length coincident with the tangent line at that point; and the next element being inclined at an angle to the former one deviates from the tangent. Now let the two consecutive elements be of equal lengths, and from the extremity of the second let a perpendicular be drawn to the tangent: as this perpendicular is longer or shorter, so will the deviation be greater or less, and the curve will be more or less bent.
These terms however are but relative; and accordingly it is necessary to fix on some standard with which to compare such amount of bending, and to investigate some means by which the comparison may be made.
The circle naturally suggests itself for a standard; whatever its curvature or bending is, it is the same at all points of the same circle: and in different circles, as the radius changes, so does the curvature. As the radius increases, the deviation from a straight line becomes less and less; and in the limit vanishes when the radius becomes infinite; see Art. 186; and as the radius decreases, the curvature increases, and in the limit when the radius becomes zero, the curvature becomes infinite; for the circle becomes a point, and the curve at once returns into itself; the radius of the circle therefore, and its curvature or deviation from a straight line, are so related that one varies inversely as the other. If then r is the radius of a circle, the amount of deviation of that circle from a straight line is a function of the reciprocal of r; let us give some definite name to this deviation, and in order that we may have a measure of it, let us define it. Curvature is the name which we shall adopt; and our mathematical definition of it is the simplest function of the reciprocal of r, viz.,
The curvature of a circle = -;
so that the curvature = 1, when r = 1; and therefore the curvature of a circle whose radius is unity is the unit-curvature.
Now when a finite arc of a circle is given, we can in many ways determine the radius of the circle; but when the arc is infinitesimal, the following is best adapted to our present conceptions.
The relation between an arc, the radius, and the angle subtended at the centre is, see Art. 24, (15),
the arc = the radius x the angle.
Let (x, y), (x + dx, y + dy) be any two points on a circle infinitesimally near to each other; and let ds be the arc between the two points; let normals be drawn to the circle at the points (x>y)> (x+dx, y + dy), and let d\jr be the angle at the centre contained between these normals, and let r be the radius: then
ds = rd\fr; (1)
- = <*>
and the curvature of the circle = * = ■
281.] Let us now extend these principles to any plane curve. Suppose the two points (x, y), (x + dx, y + dy) to be on a plane curve; and at neither of them let there be a point of disPbice, Vol. i. 3 K
continuity or of inflexion; then, although the ratio given in (2) may be no longer constant at all points of a curve, but may vary as we pass from one point to another; yet when the distance ds is infinitesimal, ^j- will assume some determinate
value, which we may call the curvature of the curve at the point; perhaps it may be said that it is a measure of the mean curvature of the arc, but the difference between that and the actual curvature at the point (x, y) is infinitesimal, and therefore must be neglected, so that the two become identical.
Imagine then two normals to be drawn at two consecutive points of the curve; these will generally meet at a finite distance; let d\f/ be the small angle included between them, and let p be the distance from the curve of the point at which they intersect; then, by reason of (2), and introducing + so as to include all the possible simultaneous changes of s and \fr, we have ,
From the analogy of the circle, p is called the radius of curvature, and the point at which the two consecutive normals intersect is called the centre of curvature; and therefore we have the following definition:
The distance from the curve at which two consecutive normals of a plane curve intersect is the radius of curvature of the curve at that point.
282.] To determine the analytical values of the radius of curvature.
Let the equation to the curve be y = f(x); see fig. 101; and let p, Q be the two points on it, infinitesimally near to one another, at which the normals are pn, Qn; n their point of intersection, which is therefore the centre of curvature; through G draw a line Kg parallel toQn; then Pq = ds, pnQ = dty = Pgk
It is to be observed that p is an absolute length; some principle therefore must be adopted by which the ambiguity of sign in the preceding expressions may be removed. Let us suppose the type-curve to be that represented in figs. 47,101, and 102; and let us assume the radius of curvature to be positive, when it is drawn downwards and towards the axis of x; and to be negative when it is drawn upwards from the curve and away from the axis of x, as in fig. 106. Now let us take the value of p
given in (7): in the former case is negative, because the
curve is concave downwards; and therefore, as p is to be positive, ,
and if in the result p is affected with a negative sign, it is evident that p is drawn upwards from the curve, and that the curve is convex downwards, as in fig. 106. Similarly in (8) the right-hand member is to have a positive sign, and we have
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