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curvature, in the other a gauche polygon. The term 'curve of double curvature' is very bad, and, though in very general use, is, we hope, not ineradicable. The fact is, that there are not two curvatures, but only a curvature (as above defined), of which the plane is continuously changing, or twisting, round the tangent line; thus exhibiting a torsion. The course of such a curve is, in common language, well called 'tortuous;' and the measure of the corresponding property is conveniently called Tortuosity.
8. The nature of this will be best understood by considering the curve as a polygon whose sides are indefinitely small. Any two consecutive sides, of course, lie in a plane—and in that plane the curvature is measured as above, but in a curve which is not plane the third side of the polygon will not be in the same plane with the first two, and, therefore, the new plane in which the curvature is to be measured is different from the old one.
The plane of the curvature on each side of any point of a tortuous curve is sometimes called the Osculating Plane of the curve at that point. As two successive positions of it contain the second side of the polygon above mentioned, it is evident that the osculating plane passes from one position to the next by revolving about the tangent to the curve.
9. Thus, as we proceed along such a curve, the curvature in general varies; and, at the same time, the plane in which the curvature lies is turning about the tangent to the curve. The tortuosity is therefore to be measured by the rate at which the osculating plane turns about the tangent, per unit length of the curve.
To express the radius of curvature, the direction cosines of the osculating plane, and the tortuosity, of a curve not in one plane, in terms of Cartesian triple co-ordinates, let, as before, 89 be the angle between the tangents at two points at a distance ds from one another along the curve, and let od be the angle between the osculating planes at these points. Thus, denoting by p the radius of curvature, and the tortuosity, we have
Curvature and tortuosity.
80 Curvature according to the regular notation for the limiting values of and tortu.
δώ and when ds is diminished without limit. Let OL, OL'
as be lines drawn through any fixed point o parallel to any two successive positions of a moving line PT, each in the directions indicated by the order of the letters. Draw OS perpendicular to their plane in the direction from 0, such that OL, OL', OS lie in the same relative order in space as the positive axes of co-ordinates, OX, OY, 02. Let OQ bisect LOL', and let OR bisect the angle between OL' and LO produced through 0.
Let the direction cosines of
be a, b, c;
a, b, y;
OS λ, μ, ν: and let 88 denote the angle LOL. We have, by the elements of analytical geometry, cos 80 = aa' + bb' + cc' ........
(3); ! (a + a')
(b +6') } (c +c')
6' - 6
2 sin dou
ca – ca ab' - a'b
cos 1 80
Now let the two successive positions of PT be tangents to a
Curvature and a, B, y become the direction cosines of the normal, PC,
drawn towards the centre of curvature, C; and ), M, v those of
ds This, modified by differentiation, and application of the formula
de los = dcdc + ddy + d dog ............(13), becomes
ds Another formula for 1 is obtained immediately from equations (11); but these equations may be put into the following simpler form, by differentiation, &c.,
dy doz – dzd’y d"c – c”, dxdRy - dy dox λ.
from which we find
dф To find the tortuosity, a, we have only to apply the general
1 da equation above, with d, M, v substituted for l, m, n, and 1 du 1 dv
da edul dv
have r'= Tds fds
for a, b, y.
da dv du da v λ
and tortuds ds ds ds ds ds
osity. where , M, v, denote the direction cosines of the osculating
plane, given by the preceding formula. 10. The integral curvature, or whole change of direction of Integral an arc of a plane curve, is the angle through which the tangent of a curve has turned as we pass from one extremity to the other. The $ 136). average curvature of any portion is its whole curvature divided by its length. Suppose a line, drawn from a fixed point, to move so as always to be parallel to the direction of motion of à point describing the curve: the angle through which this turns during the motion of the point exhibits what we have thus defined as the integral curvature. In estimating this, we must of course take the enlarged modern meaning of an angle, including angles greater than two right angles, and also negative angles. Thus the integral curvature of any closed curve, whether everywhere concave to the interior or not, is four right angles, provided it does not cut itself. That of a Lemniscate, or figure of 8, is zero. That of the Epicycloid is eight right angles; and so on.
11. The definition in last section may evidently be extended to a plane polygon, and the integral change of direction, or the angle between the first and last sides, is then the sum of its exterior angles, all the sides being produced each in the direction in which the moving point describes it while passing round the figure. This is true whether the polygon be closed or not. If closed, then, as long as it is not crossed, this sum is four right angles,-an extension of the result in Euclid, where all re-entrant polygons are excluded. In the case of the star-shaped figure ☆, it is ten right angles, wanting the sum of the five acute angles of the figure; that is, eight right angles.
12. The integral curvature and the average curvature of a curve which is not plane, may be defined as follows:-Let successive lines be drawn from a fixed point, parallel to tangents at successive points of the curve.
These lines will form a conical surface. Suppose this to be cut by a sphere of unit radius baving its centre at the fixed point. The length of the
Integral curve of intersection measures the integral curvature of the of a curve given curve. The average curvature is, as in the case of a
plane curve, the integral curvature divided by the length of the 136
For a tortuous curve approximately plane, the integral curvature thus defined, approximates (not to the integral curvature according to the proper definition, $ 10, for a plane curve, but) to the sum of the integral curvatures of all the parts of an approximately coincident plane curve, each taken as positive. Consider, for examples, varieties of James Bernouilli's plane elastic curve, § 611, and approximately coincident tortuous curves of fine steel piano-forte wire. Take particularly the plane lemniscate and an approximately coincident tortuous closed curve.
13. Two consecutive tangents lie in the osculating plane. This plane is therefore parallel to the tangent plane to the cone described in the preceding section. Thus the tortuosity may, be measured by the help of the spherical curve which we have just used for defining integral curvature. We cannot as yet complete the explanation, as it depends on the theory of rolling, which will be treated afterwards (SS 110—137). But it is enough at present to remark, that if a plane roll on the sphere, along the spherical curve, turning always round an instantaneous axis tangential to the sphere, the integral curvature of the curve of contact or trace of the rolling on the plane, is a proper measure of the whole torsion, or integral of tortuosity. From this and § 12 it follows that the curvature of this plane curve at any point, or, which is the same, the projection of the curvature of the spherical curve on a tangent plane of the spherical surface, is equal to the tortuosity divided by the curvature of the given curve.
1 Let be the curvature and = the tortuosity of the given р
ds curve, and ds an element of its length. Then and Tds, each integral extended over any stated length, 1, of the curve, are respectively the integral curvature and the integral tortuosity. The mean curvature and the mean tortuosity are respectively
and Tds. P