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Calculus II: Vector Calculus and Multivariable Analysis / Vectors and Curves in Space

Arc Length, Unit Tangent Vector, Curvature, and the Frenet Frame

A bond price path and a yield curve both trace curves in space. Two questions arise immediately: how long is the path (total variation), and how sharply does it bend (curvature)? Arc length answers the first; curvature κ\kappa answers the second. Curvature is the engine behind the convexity of a fixed-income position — the price–yield curve bends, and the degree of bending is κ\kappa. The Frenet–Serret frame {T,N,B}\{\mathbf{T}, \mathbf{N}, \mathbf{B}\} — unit tangent, principal unit normal, binormal — is the moving coordinate system attached to the curve at each point; it decomposes acceleration into the component changing speed (tangential) and the component changing direction (normal), which maps directly onto the delta/gamma decomposition in option risk.
Learning objectives. By the end of this note you should be able to:
  • compute the unit tangent vector T(t)=r′(t)/∥r′(t)∥\mathbf{T}(t) = \mathbf{r}'(t)/\|\mathbf{r}'(t)\|;
  • set up and evaluate the arc length integral L=∫ab∥r′(t)∥ dtL = \int_a^b \|\mathbf{r}'(t)\|\,dt;
  • use the arc length function σ(t)\sigma(t) to reparameterise a curve by arc length;
  • compute the principal unit normal N(t)\mathbf{N}(t) and binormal B(t)\mathbf{B}(t);
  • define curvature κ\kappa from the arc length parameterisation and compute it using the cross-product formula κ=∥r′×r′′∥/∥r′∥3\kappa = \|\mathbf{r}' \times \mathbf{r}''\|/\|\mathbf{r}'\|^3.

In this lesson

    1. The Unit Tangent Vector
    1. Arc Length
    1. The Arc Length Function and Reparameterisation
    1. The Frenet–Serret Frame
    1. Curvature
    1. Decomposition of Acceleration

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