Of importance in the theory is the circle of curvature, namely, the limit circle defined by the fixed point P and two variable points PI, Ps as the variable points move along the curve toward coincidence with P. It is referred to as the circle through three consecutive points P, P., and obviously lies in the plane of osculation at P. Its centre, radius and the reciprocal of the radius are called respectively the centre of curvature, the radius of curvature and the curvature of the curve at P. Relative to the name, curve of double curvature, the cur vature here described is properly first curva ture, but it is customary to omit the word "firse. The second curvature, to be noted shortly, is usually referred to as torsion.
Four points not in a plane determine one sphere. Through P and three variable points of the curve a sphere may be passed, which, when the variable points come into coincidence with P, becomes the sphere of osculation at P. It will be subsequently seen that its centre and the centre of first curvature are on a line paral lel to the binormal at P.
The further development of the theory will employ the methods of the differential geom etry in preference to limit processes. The ele ments already defined, when taken in pairs, give rise to new elements. The normal planes at the consecutive points P and Pi intersect in a line passing through the centre of curvature of P and parallel to the binormal of P. It is called the axis of curvature of the curve at P. The planes of rectification at P, Pi intersect in the line of rectification at P. This line passes through P, makes an angle with the binormal i of P and perpendicular to the principal nor mals of P and Pi. The planes of osculation at P, intersect in the tangent of the curve. The principal normals at P and P3 do not intersect, nor do the binormals of P and The con secutive tangents do intersect in a point of the curve.
The Two Curvatures, or Curvature and Torsion.— As a point moves on the curve, the tangent continually changes its direction, as does also the plane of osculation: the first marks the tendency of the curve to depart from a right line, and the second the tendency to depart from a plane. Measures of these tendencies at a point P are defined as follows: If PP, be an infinitely small arc of length ds, and de the in finitely small angle between the tangents at P and P3; and dr the infinitely small angle between the planes of osculation at P and Pi or, what is the same thing, the angle between the binormals of P and Pi; then the ratio de — is called the de curvature of the curve at P. and the ratio ds
is called the torsion of the curve at P. The angles dr and dr are called respectively angle of contingence, and angle of torsion at P. This point of view of curvature leads at once to the circle of curvature previously described. The ratio has the same value for the curve and d the circle at P and if R designates the radius of the circle, there results curvature at a point =-- — On the other hand, there is no circle R ds of torsion connected with the curve, and it is only by analogy that the term radius of torsion, T, is defined to be the reciprocal of the torsion. Expressed in equational form, torsion 1 dr at a point = T ds A third curvature is sometimes regarded, merely as a convenience. It is the linuting ratio of the infinitely small angle d K between the principal normals at P and P. to the arc ds. It is in a sense the resultant of the other two cur vatures, and has received the name of entire curvature. The angle du is called the angle of entire curvature. Entire curvature is not an independent curvature, for, as will shortly be seen, its angle dn is a function of the other two through the equation de2=del-Fdr2.
Before clothing the foregoing definitions in analytical garb, it is necessary to adopt conven tions as to the signs of directions and to com plete the notation. Of the two directions on a curve that is taken as positive which corre sponds to increasing values of u. The positive direction of the tangent is taken to coincide with the positive direction of the curve, and the di rection cosines of the angles the positive tangent makes with the positive directions of the co-ordi nate axes x, y is are designated by eg, y,, re spectively. The positive direction of ;I:ie princi pal normal is the direction from P toward the centre of curvature, and its direction cosines are designated by a,, Y2. The positive direction of the binormal is so taken that it is directed with respect to the positive tangent and positive principal normal as the positive z axis is di rected to the positive x axis and positive y axis. Its direction cosines are represented by as, y..