Final answer:
The Frenet-Serret theorem describes the kinematic properties of a particle on a differentiable curve in three dimensions with a set of equations involving the TNB frame vectors and their derivatives with respect to arc length.
Step-by-step explanation:
The Frenet-Serret theorem is a set of differential equations describing the kinematic properties of a particle moving along a continuous, differentiable curve in three-dimensional Euclidean space ℝ³. These properties are encapsulated in the Frenet-Serret frame or TNB frame, consisting of the tangent vector (τ), the normal vector (η), and the binormal vector (β). The equations describe how this frame changes along the curve and are given by:
where κ is the curve's curvature, τ is the torsion, and the prime denotes differentiation with respect to the arc length parameter of the curve. This theorem thus offers a complete description of the curve's geometry and is vital for understanding the behavior of particles in space, as well as in the fields of mechanics and graphics.