Kigazil Thus, the three unit vectors TNand B are all perpendicular to each other. The curve is thus parametrized in a preferred manner by its arc length. With a non-degenerate curve r sparameterized by its arc length, it is now possible to define the Frenet—Serret frame or TNB frame:. The normal vectorsometimes called the curvature vectorindicates the deviance of the curve from being a straight line. A rigid motion consists of a combination of a translation and a rotation.
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At each point of the curve, this attaches a frame of reference or rectilinear coordinate system see image. The Frenet—Serret formulas admit a kinematic interpretation. Imagine that an observer moves along the curve in time, using the attached frame at each point as their coordinate system. The Frenet—Serret formulas mean that this coordinate system is constantly rotating as an observer moves along the curve.
Hence, this coordinate system is always non-inertial. Concretely, suppose that the observer carries an inertial top or gyroscope with them along the curve. This is easily visualized in the case when the curvature is a positive constant and the torsion vanishes. The observer is then in uniform circular motion. If the top points in the direction of the binormal, then by conservation of angular momentum it must rotate in the opposite direction of the circular motion.
The general case is illustrated below. There are further illustrations on Wikimedia. The kinematics of the frame have many applications in the sciences.
In the life sciences , particularly in models of microbial motion, considerations of the Frenet-Serret frame have been used to explain the mechanism by which a moving organism in a viscous medium changes its direction.
Such is often the case, for instance, in relativity theory. Within this setting, Frenet-Serret frames have been used to model the precession of a gyroscope in a gravitational well. At the peaks of the torsion function the rotation of the Frenet-Serret frame T,N,B around the tangent vector is clearly visible. The kinematic significance of the curvature is best illustrated with plane curves having constant torsion equal to zero.
See the page on curvature of plane curves. Frenet—Serret formulas in calculus[ edit ] The Frenet—Serret formulas are frequently introduced in courses on multivariable calculus as a companion to the study of space curves such as the helix.
FRENET-SERRET FORMULA PDF