Differential Geometry

Fundamental Theorem of Space Curves Visualization

In differential geometry, the fundamental theorem of space curves states that every regular curve in three-dimensional space, with non-zero curvature, has its shape (and size) completely determined by its curvature and torsion.

Given curvature \kappa and torsion \tau, the Frenet equation relates the tangent, normal, and binormal vector of the curve at each position.

    \[ \begin{cases} T'_{\alpha} & =\kappa_{\alpha}N_{\alpha}\\ N'_{\alpha} & =-\tau T_{\alpha}-\kappa_{\alpha}B_{\alpha}\\ B'_{\alpha} & =\tau N_{\alpha} \end{cases} \]

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