Differential Geometry

Mathematical Principles & Theorems

Investigates local geometry of parametric curves \(\mathbf{r}(t) = (x(t), y(t), z(t))\). The moving Frenet-Serret orthonormal trihedron \((\mathbf{T}, \mathbf{N}, \mathbf{B})\) satisfies: \(\frac{d\mathbf{T}}{ds} = \kappa \mathbf{N}\), \(\frac{d\mathbf{N}}{ds} = -\kappa \mathbf{T} + \tau \mathbf{B}\), \(\frac{d\mathbf{B}}{ds} = -\tau \mathbf{N}\). Curvature \(\kappa = \frac{|\mathbf{r}' \times \mathbf{r}''|}{|\mathbf{r}'|^3}\) measures deviation from a straight line; Torsion \(\tau = \frac{(\mathbf{r}' \times \mathbf{r}'') \cdot \mathbf{r}'''}{|\mathbf{r}' \times \mathbf{r}''|^2}\) measures departure from the osculating plane.

Operating Instructions

• Select a 3D curve (Helix, Torus Knot, Trefoil, Viviani Curve) or enter custom parametric functions.
• Use the numeric value box to sweep parameter \(t \in [0, 1]\).
• Rotate the 3D visual stage to observe the tangent, normal, and binormal frame vectors.
• Review real-time numerical readouts for arc length, curvature \(\kappa(t)\), and torsion \(\tau(t)\).