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ON CURVATURE OF CURVES IN SPACE, AND ON
323.] * Certain principles names and modes of estimation which were discussed in Chapter XII, as to the curvature of curves, are stated with breadth sufficient to include kindred properties of curves in space; a difference however of great importance exists between the two classes, and which it is necessary at once to bring out into greater prominence. In the former case the whole of the curve lies in one plane, and the curve is therefore called a plane curve; in the present, although every two consecutive elements, or every three consecutive points, must be in one plane, viz. the osculating plane, yet the third element, or the fourth point, may be, and generally will be, in a different plane. For this reason such curves are called non-plane curves, and from this general property arise other affections of a more complex character, and which we proceed to enquire into.
Consider a portion of a curve in space, at no point of the part of which under investigation is there a point of abrupt termination, or of discontinuity, and at which the derived-functions of the equations to the curve are not indeterminate. Now as every three consecutive points must be in one plane, and as the mode of estimating curvature as explained in Art. 231 requires only three points in the curve's plane, the principles therein investigated are immediately applicable, and we propose to apply them by a similar process, viz. by drawing in the osculating plane, which is the plane containing three consecutive points, two consecutive normals, which will generally meet at a finite distance from the curve; the ratio of the infinitesimal angle contained between which to the element of the curve is what we have before called curvature (see Art. 231), but which, for the sake of greater distinctness, we shall now call absolute curvature; and in accordance with the expression we shall use the following terms: radius of absolute curvature is the distance from the curve of the intersection of two consecutive normals drawn in the osculating plane; centre of absolute curvature is the point of intersection of two such normals; angle of curvature is the angle contained between them; and since two such normals are perpendicular to two consecutive tangents, it is equal to the angle between them, and is accordingly called the angle of contingence (see Art. 235).
* For a most masterly exposition of the properties considered in this Chapter, and for geometrical proofs of them by the infinitesimal method, the reader is requested to consult a Memoir by M. de Saint Venant in "Trentieme Cahier du Journal de 1' Ecole Royal Polytechnique."—Bachelier, Para, 1845.
Suppose then (x, y, z), (a? + dx, y + dy, z + dz) to be two consecutive points on a curve, and da the distance between them; and suppose two consecutive normals to be drawn in the osculating plane, and to contain between them the angle dr; then, if p = radius of absolute curvature, and dr = angle of curvature, agreeably to Art. 235,
ds — ± pdr, (1)
The radius of absolute curvature, which is mathematically defined by equation (2), is therefore the distance from the curve at which two consecutive normals drawn in the osculating plane intersect, or is the ratio of the element of the curve to the angle of curvature.
324.] In a curve however of the most general nature, as we pass continuously along it, the third element will be in a plane different to that of the two preceding ones; that is, two consecutive osculating planes will be inclined to each other; or, what is the same thing, two consecutive binormals are not parallel. This then is an affection different to any of those of plane curves, and which has been called by various names*, "second curvature," "torsion," "flexure," "cambrure;" we shall call it torsion, and curves which are affected with it we shall call non-plane curves. In plane curves it vanishes, and is of greater or less amount according to the deviation of the curve from a plane curve; we propose to measure it according to the principles of Art. 231. If therefore ds be an element of the curve, and da> be the angle contained between two consecutive binormals, the torsion will vary directly as da>, and inversely as ds. Let us therefore define as follows:
* On the use of these terms, consult Note I. of M. de Saint Venant's Memoir in the Journal de 1' Ecole Polytechnique.
torsion = -=-: (3)
and suppose R to be the radius of a circle, an arc ds of which subtends an angle it a at the centre, then
and torsion = -; calling then R the radius of torsion, dto the
angle of torsion, we have the following definition of R:
The radius of torsion is the ratio of the element of a curve to the angle of inclination of two consecutive binormals.
It is of course manifest that two consecutive binormals do not of necessity intersect; but this will appear more distinctly hereafter.
It is also to be observed, that the torsion vanishes in the case of plane curves, but that the absolute curvature vanishes only in the case of straight lines; hence we shall derive analytical conditions of lines in space being plane and being straight.
And on account of these two affections, such curves have been called "curves of double curvature."
325.] On the Radius of Absolute Curvature.
Let x, y, z be the coordinates to the point on the curve at which the radius of absolute curvature is drawn: ds an element of the curve, £ , rj, ( the coordinates to the centre of curvature, and p its length; so that
P2 = (£-#)a + (I-?)2 + (f-*)2, (5)
and the point (f , ?;, f) lies in the osculating plane.