Question

Let α be a regular curve with κ(t)=0 for all t.

(i) State the Frenet-Serret Theorem for α.
(ii) Define what is meant by the osculating plane of α at t₀.
(iii) Prove that if α lies inside an osculating plane at any point, then it has zero torsion.

Answer 1

The Frenet-Serret Theorem dictates that a curve with zero curvature is a straight line. An osculating plane at a point is the plane that most precisely approximates the curve, involving the tangent and normal vectors at that point. If a curve lies within its osculating plane at any point, then it has zero torsion as there's no change in the binormal vector.

Step-by-step explanation:

The Frenet-Serret Theorem for a regular curve α with a curvature κ(t)=0 at all points t states that the curve is a straight line. This is because a curvature of zero indicates that there is no bending of the curve at any point, and hence, it does not change direction.

The osculating plane of α at a point t₀ is defined as the plane that most closely approximates the curve at that point. It is determined by the tangent and normal vectors of the curve at t₀. Since the curvature is zero for all points, the osculating plane is not well-defined as the normal vector does not exist.

To prove that a curve α that lies entirely in an osculating plane at any point has zero torsion, we consider the definition of torsion, which measures the rate of change of the curve's binormal vector. If a curve lies completely within an osculating plane, there is no change in the binormal vector, implying that the torsion is zero.