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Differential Geometry

Space curves, surfaces, curvature, and geodesics.

Differential Geometry & Curves Handbook

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 \(\tau(t)\).

Space Curves

Frenet-Serret frame gives tangent T, normal N, and binormal B = T×N.

Frenet-Serret Formulas
T' = κN
N' = -κT + τB
B' = -τN

κ = |r'×r''|/|r'|³
τ = (r'×r'')·r'''/|r'×r''|²
Curvature κ—
Torsion τ—
Radius of curv ρ—
Arc length—