The total curvature of a smooth closed curve in Eⁿ is an upper bound to the total swing of its position vector

Extract

Let γ:x = x(u) for a≤ u ≤ b be a closed curve in n dimensional euclidean space Eⁿ (for n ≥ 2) referred to some point P⁰, which does not lie on γ, as origin. We suppose that γ is smooth in the sense that the cartesian coordinates are class C² functions of the parameter u, and that dx/du is non-vanishing so that a tangent vector is everywhere well defined. These properties are also assumed to hold in an obvious way at the join of the end points u = a and u = b. As P moves on γ its position vector x(u) intersects the surface of the unit sphere centred at P₀ in a closed curve γ₀. Note that γ₀: x = x₀(u) may not be smooth everywhere. If there are points P on γ where P₀P is tangential to γ at P then dx₀/du =0 at the corresponding point on γ₀, and γ₀ may have a cusp there. We assume that γ₀ is smooth except at a finite number of points. We define the total swing of the position vector to γ to be the arc length L₀ of γ₀. Clearly L₀ is not an invariant but depends on the choice of the origin P₀ in relation to γ. The total (first) curvature of γ is an invariant and is defined by

where s is the arc length on γ, K(S) is the curvature and ‖v‖ is the euclidean norm of the vector v. Note that L_T is also the arc length of the closed curve γ₁, described on a unit sphere by the unit tangent t = dx/ds to γ as position vector, with the centre of the sphere as origin, γ₁ is the spherical indicatrix of t. Our purpose is to establish the result

for all choices of P₀.