Frenet–Serret Frame of a Helix

The Frenet–Serret frame (also known as the TNB frame) is a standard frame of reference for studying the geometry of space curves. For a helix described by the parametric equation r(θ) = (R cos θ, R sin θ, h θ), the frame provides an elegant way to understand its curvature and torsion. This module explores how the radius R and signed rise h determine the constant curvature and torsion of a helix, and visualizes the local frame.

Experiment

Follow these steps to experiment with the module and observe the Frenet frame properties.

1. Set Helix radius R to 1, Rise per radian h to 1, and Angle theta / pi to 0 to view a standard helix. Note the curvature and torsion are both 0.5.
2. Change Angle theta / pi to move the frame along the curve. Observe how the normal vector N always points directly toward the central axis, while the tangent and binormal vectors rotate.
3. Set Rise per radian h to 0. This creates a planar circle. Notice that torsion becomes exactly 0, meaning the curve never leaves the XY-plane. The binormal B becomes a constant vertical vector (0, 0, 1).
4. Set Rise per radian h to a negative value, such as -1. This produces a left-handed helix with negative signed torsion. Observe how the tangent vector slopes downward and the binormal vector tilts differently compared to the positive rise case.

You can try this interactive module yourself: Frenet–Serret Frame of a Helix