The Frenet Serret equations | Differential Geometry 18 | NJ Wildberger

The Frenet Serret equations describe what is happening to a unit speed space curve, twisting and rotating around in three dimensional space. This is done with the language of vector valued derivatives.

The idea is to attach to each point of the curve, a triple of unit vectors, called traditionally T, N and B, the tangent, normal and bi-normal unit vectors. These form at every point a mutually perpendicular frame of basis vectors, much like the i,j and k standard unit basis vectors along the x,y and z axes.

The vector T is in the direction of the curve (T standards for tangent), while N is in the direction of the acceleration, which for a unit speed curve must be perpendicular to the tangent. The third vector B can be defined as the cross product of T and N.

The Frenet Serret equations describe what happens as we move along the curve with unit speed s, namely what are the derivatives of T,N and B with respect to s. The curvature k(s) comes into play, as does a new quantity called the torsion, usually denoted tau(s).

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