For a unit-speed space curve with nonzero curvature, , , , and . With the torsion convention used here,
Smooth functions and determine a unit-speed space curve with curvature and torsion , uniquely up to a proper Euclidean motion of Euclidean three-space. Solve the Frenet-Serret formulas for an oriented orthonormal frame and integrate its tangent for existence; uniqueness for the frame ordinary differential equation gives rigidity.
This is the space-curve version of the fundamental theorem of curves.
For a sufficiently smooth unit-speed curve with , the Frenet frame is the positively oriented orthonormal frame
The binormal is , the cross product of the unit tangent vector and principal normal vector. It is a unit vector perpendicular to both factors and completes the oriented Frenet frame.
For a unit-speed curve with positive curvature of a space curve , the principal normal is , with unit tangent vector . It is perpendicular to and points in the direction in which the tangent is turning. At zero curvature this formula does not define .
For a regular curve with positive curvature of a space curve, the Frenet-Serret formulas give . If the unit tangent vector has constant component along a fixed vector , differentiation of gives . Hence every principal normal vector is perpendicular to that direction. Nonzero curvature is needed to define the principal normal.
A Euclidean motion has the form with . It preserves Euclidean distance, angles, and all extrinsic geometric quantities transformed with their orientations.
A proper Euclidean motion has the form with . It preserves lengths, dot products, cross products, orientation, curvature, and torsion.
This is the orientation-preserving subgroup of the three-dimensional Euclidean group.
A screw motion combines a rotation in three dimensions about a fixed line with a translation parallel to that line. With unit axis direction , axial speed , and rotation matrix fixing , each map is a proper Euclidean motion of Euclidean three-space, and together these maps form a one-parameter subgroup. Each off-axis group orbit is a circular helix when both rotational speed and axial speed are nonzero. A pure rotation or translation is a limiting case.
The derivative at the identity of a smooth curve in is a vector fieldAfter shifting the origin perpendicular to , it becomes rotation about the axis parallel to plus translation along that axis.
The integral curves of are lines when , and helices around an axis when . The helix is closed exactly when its axial translation component vanishes, in which case every nonfixed orbit is a circle.
A simple closed curve in preserved by a nonconstant smooth family of proper rigid motions is a circle. Differentiating the normalized family at the identity gives a nonzero infinitesimal proper Euclidean motion tangent to the curve. Compactness rules out its line and nonclosed helical orbits, leaving a circular rotation orbit.
A scalar assignment is a pointwise Euclidean invariant in the weak covariance sense when it is unchanged by proper Euclidean motions and satisfiesunder translation of the arc-length parameter.
For sufficiently smooth curves, obeys Euclidean and parameter-translation covariance but is not determined by the pair . Under the bare covariance definition, nonlocal examples such as work as well.
For a unit-speed curve on the unit sphere,under the compatible signed-torsion and geodesic-curvature conventions.
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The Frenet–Serret formulas are a set of differential equations that describe the intrinsic geometry of a space curve in three-dimensional space. They provide a way to relate the curvature and torsion of a curve to the behavior of its tangent vector, normal vector, and binormal vector. The formulas are fundamental in the study of curves in differential geometry and are named after the mathematicians Jean Frédéric Frenet and Joseph Alain Serret.