The Frenet Frame

Differential Geometry · Axiom Academy

An orthonormal moving frame that captures the local geometry of curves in space The tangent vector points in the direction of motion along the curve. For a parametrized curve r(t) , the unit tangent vector is obtained by normalizing the velocity vector. The principal normal measures how the tangent vector is changing—it points in the direction the curve is turning. Since |T| = 1 always, the derivative T' is perpendicular to T. The binormal vector completes the orthonormal frame. It is perpendicular to both T and N, defined by their cross product. The Frenet frame T, N, B moves along the curve, rotating and changing orientation as the curve bends and twists through space. This moving frame is fundamental to understanding curve geometry. 5. Osculating, Normal, and Rectifying Planes The Frenet frame defines three important planes at each point of the curve: Osculating plane: spanned by T and N (contains the curve's local curvature) Normal plane: spanned by N and B (perpendicular to the curve) Rectifying plane: spanned by T and B (contains T and B)

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