What is the representation of Serret Frenet apparatus?

What is the representation of Serret Frenet apparatus?

The Frenet-Serret apparatus presents the curvature and torsion as numerical invariants of a space curve. Roughly speaking, two curves C and C′ in space are congruent if one can be rigidly moved to the other. A rigid motion consists of a combination of a translation and a rotation.

What is frenet Trihedron?

The Frenet Trihedron is the vectors consisting of the unit tangent vector, unit principal normal vector, and unit binormal vector.

What is torsion and curvature?

Curvature: Motion in several dimension has two aspects: one is its speed of motion; the other the shape of the curve it follows. The torsion of the curve is the magnitude of the rate of change of a unit vector in the direction of a v with distance along the curve. …

What is meant by Osculating plane?

In mathematics, particularly in differential geometry, an osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second order of contact at the point. An osculating plane is thus a plane which “kisses” a submanifold.

What is tangent and binormal?

According to mathworld, the binormal vector is defined as cross(tangent,normal) where tangent and normal are unit normal vectors. Note that, strictly speaking, order matters when you take cross products. cross(tangent,normal) points in the opposite direction from cross(normal,tangent) .

What are the Frenet-Serret formulae?

The formulae ( 2.56) are known as the Frenet-Serret formulae and describe the motion of a moving trihedron ( ) along the curve. From these , , the shape of the curve can be determined apart from a translation and rotation. For arbitrary speed curve the Frenet-Serret formulae are given by

What is the Kinematic interpretation of Frenet Serret?

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.

What are the unit vectors of the Frenet-Serret frame?

The tangent, normal, and binormal unit vectors, often called T, N, and B, or collectively the Frenet–Serret frame or TNB frame, together form an orthonormal basis spanning R3 and are defined as follows: T is the unit vector tangent to the curve, pointing in the direction of motion.

What does the Frenet Serret apparatus measure?

The associated collection, T, N, B, κ, and τ, is called the Frenet–Serret apparatus. Intuitively, curvature measures the failure of a curve to be a straight line, while torsion measures the failure of a curve to be planar.