# Normal plane and osculating plane

More specifically, the formulas describe the derivatives of the so-called tangent, normal, and binormal unit vectors in terms of each other. Vector notation and linear algebra currently used to write these formulas were not yet in use at the time of their discovery.

Intuitively, curvature measures the failure of a curve to be a straight line, while torsion measures the failure of a curve to be planar. Let r t be a curve in Euclidean spacerepresenting the position vector of the particle as a function of time. The Frenet—Serret formulas apply to curves which are non-degeneratewhich roughly means that they have nonzero curvature. Let s t represent the arc length which the particle has moved along the curve in time t. The quantity s is used to give the curve traced out by the trajectory of the particle a natural parametrization by arc length, since many different particle paths may trace out the same geometrical curve by traversing it at different rates.

In detail, s is given by. 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 :. From equation 2 it follows, since T always has unit magnitudethat N the change of T is always perpendicular to Tsince there is no change in length of T. From equation 3 it follows that B is always perpendicular to both T and N. Thus, the three unit vectors TNand B are all perpendicular to each other.

The Frenet—Serret formulas are also known as Frenet—Serret theoremand can be stated more concisely using matrix notation: [1]. This matrix is skew-symmetric. The Frenet—Serret formulas were generalized to higher-dimensional Euclidean spaces by Camille Jordan in Suppose that r s is a smooth curve in R nand that the first n derivatives of r are linearly independent. In detail, the unit tangent vector is the first Frenet vector e 1 s and is defined as.

The normal vectorsometimes called the curvature vectorindicates the deviance of the curve from being a straight line. It is defined as. Its normalized form, the unit normal vectoris the second Frenet vector e 2 s and defined as. The tangent and the normal vector at point s define the osculating plane at point r s. Notice that as defined here, the generalized curvatures and the frame may differ slightly from the convention found in other sources.

As a result, the transpose of Q is equal to the inverse of Q : Q is an orthogonal matrix.

## Osculating Plane

It suffices to show that. The Frenet—Serret frame consisting of the tangent Tnormal Nand binormal B collectively forms an orthonormal basis of 3-space. 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. The angular momentum of the observer's coordinate system is proportional to the Darboux vector of the frame. 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. In the limiting case when the curvature vanishes, the observer's normal precesses about the tangent vector, and similarly the top will rotate in the opposite direction of this precession. The general case is illustrated below. There are further illustrations on Wikimedia.In differential geometry of curvesthe osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p.

Its center lies on the inner normal lineand its curvature defines the curvature of the given curve at that point.

This circle, which is the one among all tangent circles at the given point that approaches the curve most tightly, was named circulus osculans Latin for "kissing circle" by Leibniz.

The center and radius of the osculating circle at a given point are called center of curvature and radius of curvature of the curve at that point. A geometric construction was described by Isaac Newton in his Principia :. There being given, in any places, the velocity with which a body describes a given figure, by means of forces directed to some common centre: to find that centre. Imagine a car moving along a curved road on a vast flat plane.

Suddenly, at one point along the road, the steering wheel locks in its present position.

Thereafter, the car moves in a circle that "kisses" the road at the point of locking. The curvature of the circle is equal to that of the road at that point. That circle is the osculating circle of the road curve at that point. This determines the unit tangent vector T sthe unit normal vector N sthe signed curvature k s and the radius of curvature R s at each point for which s is composed:.

The corresponding center of curvature is the point Q at distance R along Nin the same direction if k is positive and in the opposite direction if k is negative.

If C is a regular space curve then the osculating circle is defined in a similar way, using the principal normal vector N.

It lies in the osculating planethe plane spanned by the tangent and principal normal vectors T and N at the point P. Then the formulas for the signed curvature k tthe normal unit vector N tthe radius of curvature R tand the center Q t of the osculating circle are. If we do the calculations the results for the X and Y coordinates of the center of the osculating circle are:.

For a curve C given by a sufficiently smooth parametric equations twice continuously differentiablethe osculating circle may be obtained by a limiting procedure: it is the limit of the circles passing through three distinct points on C as these points approach P. The osculating circle S to a plane curve C at a regular point P can be characterized by the following properties:.

This is usually expressed as "the curve and its osculating circle have the second or higher order contact " at P.

Loosely speaking, the vector functions representing C and S agree together with their first and second derivatives at P. If the derivative of the curvature with respect to s is nonzero at P then the osculating circle crosses the curve C at P. Points P at which the derivative of the curvature is zero are called vertices. If P is a vertex then C and its osculating circle have contact of order at least three. If, moreover, the curvature has a non-zero local maximum or minimum at P then the osculating circle touches the curve C at P but does not cross it.

The curve C may be obtained as the envelope of the one-parameter family of its osculating circles. Their centers, i. Vertices of C correspond to singular points on its evolute. Within any arc of a curve C within which the curvature is monotonic that is, away from any vertex of the curvethe osculating circles are all disjoint and nested within each other.

This result is known as the Tait-Kneser theorem. The parabola has fourth order contact with its osculating circle there. A Lissajous curve with ratio of frequencies can be parametrized as follows.The oscillating plane can also be defined as the plane in the limiting.

Figure 1. From the standpoint of mechanics, the osculating plane can be characterized as the acceleration plane. As a mass point moves arbitrarily along lthe acceleration vector lies in the osculating plane. Except for special cases, l usually penetrates the osculating plane at M see Figure 1. If all three coefficients of XYand Z in the equation vanish, the osculating plane is undefined—it can coincide with any plane through the tangent. The following article is from The Great Soviet Encyclopedia It might be outdated or ideologically biased.

The oscillating plane can also be defined as the plane in the limiting Figure 1. Moscow, The Great Soviet Encyclopedia, 3rd Edition All rights reserved. Mentioned in? References in periodicals archive? On the elliptic cylindrical Tzitzeica curves in Minkowski 3-space. Further, we define the so called osculating plane of r spanned by the vectors r' x and r" x in the same point.

On the explicit characterization of admissible curve in 3-dimensional pseudo-Galilean space. Contributions to classical differential geometry of the curves in [E.

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Namely, rectifying, normal, and osculating planes of such curves always contain a particular point. Encyclopedia browser? Full browser?Differentiate r t with respect to t. Differentiate T t with respect to t. By Formula 3 of Theorem Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

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### Frenet–Serret formulas

Problem 8E. Problem 9E. Problem 10E. Problem 11E. Problem 12E. Problem 13E.Write the expression for tangent vector of a vector function r t T t. Write the equation for normal vector of vector function r t N t. Write the expression for binormal vector of vector function r t B t.

Write the expression for magnitude of vector a a. Equate the components of r t with point 021. Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees! Operations Management.

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Problem 1E. Problem 2E. Problem 3E. Problem 4E.In mathematicsparticularly in differential geometryan 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. The word osculate is from the Latin osculatus which is a past participle of oscularimeaning to kiss.

An osculating plane is thus a plane which "kisses" a submanifold.

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The osculating plane in the geometry of Euclidean space curves can be described in terms of the Frenet-Serret formulas as the linear span of the tangent and normal vectors. From Wikipedia, the free encyclopedia. This article does not cite any sources.

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It only takes a minute to sign up. You know that for describing a plane we need a point and a normal vector. This normal vector is perpendicular to the plane.