Let P move in space with speed 1 as in Problem. The cun’ature of the path is defined as κ,...

Let P move in space with speed 1 as in Problem. The cun’ature of the path is defined as κ, where

Thus if κ ≠ 0, T and (1/κ)dT/ds = N are perpendicular unit vectors; T is tangent to the curve at P, N is normal to the curve at P and is termed the principal normal. The radius of curvature is ρ = 1/κ. The vector B = T × N is called the binormal. Establish the following relations:

a) B is a unit vector and (T, N, B) is a positive triple of unit vectors

c) there is a scalar – τ such that = −τN; τ is known as the torsion

are known as the Frenetformulas. For further properties of curves, see the book by Struik listed at the end of the chapter.

Problem

Let a point P move in space, so that = r = r(t) and the velocity vector is v = dr/dt. The acceleration vector of P is

Assume further that the speed is 1, so that t can be identified with are length s and

Show that v is perpendicular to the acceleration vector a = dv/ds (cf. Problem 1). The plane through P determined by v and a is known as the osculating plane of the curve at P. It is given by the equation

where Q is an arbitrary point of the plane, provided that v× a≠ 0. Show that the osculating plane is also given by

where t is an arbitrary parameter along the curve, provided that (dr/dt) × (d2r/dt2) ± 0.

b) Find the osculating plane of the curve of Problem 2 at the point t = π/3 and graph. [It can be shown that this plane is the limiting position of a plane through three points P1, P2, P3 on the curve, corresponding to parameter values t1, t2, t3, as the values t1, t2, t3 approach the value t at P.]

Problem 1

Let u = u(t) have constant magnitude, |u(t)| ≡ a = const. Show, assuming appropriate differentiability, that u is perpendicular to du/dt. What can be said of the locus of P such that = u?

Problem 2

Given the equations x = sin t, y = cos t, z = sin2t of a space curve,

a) sketch the curve,

b) find the equations of the tangent line at the point P for which t =

c) find the equation of a plane cutting the curve at right angles at P.