Circle of Curvature

Definition

The circle of curvature (or osculating circle) at a point $P$ on a curve is the circle that best approximates the curve locally, sharing the same tangent, normal, and curvature at $P$.

Why It Matters

It provides a local circular model of a curve, helping visualize bending and analyze coordinate shifts in curved kinematics.

Core Concepts

• Center of Curvature: The center of the osculating circle, located at distance $\rho$ (radius of curvature) from the curve in the direction of the principal unit normal $\mathbf{N}$: $C = \mathbf{r} + \rho\mathbf{N}$
  • How to read: “The center of curvature C equals the position vector r plus rho times the unit normal vector N.”
  • Meaning: Step from the curve along the inward normal vector by one radius of curvature to find the center of the best-fit circle.
• Osculating plane: The plane containing the tangent and normal vectors where the circle lies.

Connected Concepts

• Radius of Curvature
• Curvature