Classical Mechanics (Kibble & Berkshire) · Problem 1.17
Velocity, acceleration and scalar angular velocity — T.W.B. Kibble, F.H. Berkshire
Physics — Mechanics · tangent-normal-vectors-curvature
Problem
Find the unit tangent vector, the unit normal vector and the curvature of the helix , , at the point with parameter .Intuitive explanation
A helix is a 3D curve that spirals upward while circling—think of a spring or DNA strand. To understand its geometry, you need to find how sharply it turns (curvature) and the directions it points in (tangent and normal). The key is to differentiate the position vector with respect to , normalize to get the unit tangent, then differentiate again and normalize to find how the curve bends away from its tangent direction.
Intuitive explanation
A helix is a 3D curve that spirals upward while circling—think of a spring or DNA strand. To understand its geometry, you need to find how sharply it turns (curvature) and the directions it points in (tangent and normal). The key is to differentiate the position vector with respect to , normalize to get the unit tangent, then differentiate again and normalize to find how the curve bends away from its tangent direction.