Mathematical Methods for Physics and Engineering - Matematica.NET

2.1 DIFFERENTIATIONf(x)C∆θρPQθθ +∆θxFigure 2.4 Two neighbouring tangents to the curve f(x) whose slopes differby ∆θ. The angular separation of the corresponding radii of the circle ofcurvature is also ∆θ.point P on the curve f = f(x), with tan θ = df/dx evaluated at P . Now consideralso the tangent at a neighbouring point Q on the curve, and suppose that itmakes an angle θ +∆θ with the x-axis, as illustrated in figure 2.4.It follows that the corresponding normals at P and Q, which are perpendicularto the respective tangents, also intersect at an angle ∆θ. Furthermore, their pointof intersection, C in the figure, will be the position of the centre of a circle thatapproximates the arc PQ, at least to the extent of having the same tangents atthe extremities of the arc. This circle is called the circle of curvature.For a finite arc PQ, the lengths of CP and CQ will not, in general, be equal,as they would be if f = f(x) were in fact the equation of a circle. But, as Qis allowed to tend to P ,i.e.as∆θ → 0, they do become equal, their commonvalue being ρ, the radius of the circle, known as the radius of curvature. It followsimmediately that the curve and the circle of curvature have a common tangentat P and lie on the same side of it. The reciprocal of the radius of curvature, ρ −1 ,defines the curvature of the function f(x) at the point P .The radius of curvature can be defined more mathematically as follows. Thelength ∆s of arc PQ is approximately equal to ρ∆θ and, in the limit ∆θ → 0, thisrelationship defines ρ as∆sρ = lim∆θ→0 ∆θ = dsdθ . (2.15)It should be noted that, as s increases, θ may increase or decrease according towhether the curve is locally concave upwards (i.e. shaped as if it were near aminimum in f(x)) or concave downwards. This is reflected in the sign of ρ, whichtherefore also indicates the position of the curve (and of the circle of curvature)53