File Name: tangents and normals calculus .zip
- 1.7: Tangent Planes and Normal Lines
- IB Math SL
- 12.7: Tangent Lines, Normal Lines, and Tangent Planes
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.
If the secant line PQ approaches the same limiting position as Q approaches P along the curve from either side then the limiting position is called the tangent line to the curve at the point P. The point P is called the point of contact of the tangent line to the curve. The tangent at a point to a curve, if it exists, is unique. Therefore, there exists at most one tangent at a point to a curve.
1.7: Tangent Planes and Normal Lines
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 space , representing the position vector of the particle as a function of time.
The Frenet—Serret formulas apply to curves which are non-degenerate , which 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 s , parameterized 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 magnitude , that N the change of T is always perpendicular to T , since 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 T , N , and B are all perpendicular to each other. The Frenet—Serret formulas are also known as Frenet—Serret theorem , and can be stated more concisely using matrix notation: .
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 n , and 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 vector , sometimes called the curvature vector , indicates the deviance of the curve from being a straight line. It is defined as. Its normalized form, the unit normal vector , is 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. It suffices to show that. The Frenet—Serret frame consisting of the tangent T , normal N , and 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. 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. The curvature and torsion of a helix with constant radius are given by the formulas.
The sign of the torsion is determined by the right-handed or left-handed sense in which the helix twists around its central axis. In his expository writings on the geometry of curves, Rudy Rucker  employs the model of a slinky to explain the meaning of the torsion and curvature.
The slinky, he says, is characterized by the property that the quantity. In particular, curvature and torsion are complementary in the sense that the torsion can be increased at the expense of curvature by stretching out the slinky.
The Frenet—Serret apparatus allows one to define certain optimal ribbons and tubes centered around a curve. These have diverse applications in materials science and elasticity theory ,  as well as to computer graphics. This surface is sometimes confused with the tangent developable , which is the envelope E of the osculating planes of C. This is perhaps because both the Frenet ribbon and E exhibit similar properties along C. Namely, the tangent planes of both sheets of E , near the singular locus C where these sheets intersect, approach the osculating planes of C ; the tangent planes of the Frenet ribbon along C are equal to these osculating planes.
The Frenet ribbon is in general not developable. In classical Euclidean geometry , one is interested in studying the properties of figures in the plane which are invariant under congruence, so that if two figures are congruent then they must have the same properties. The Frenet-Serret apparatus presents the curvature and torsion as numerical invariants of a space curve.
A rigid motion consists of a combination of a translation and a rotation. Such a combination of translation and rotation is called a Euclidean motion. In terms of the parametrization r t defining the first curve C , a general Euclidean motion of C is a composite of the following operations:. The Frenet—Serret frame is particularly well-behaved with regard to Euclidean motions. First, since T , N , and B can all be given as successive derivatives of the parametrization of the curve, each of them is insensitive to the addition of a constant vector to r t.
This leaves only the rotations to consider. Intuitively, if we apply a rotation M to the curve, then the TNB frame also rotates. More precisely, the matrix Q whose rows are the TNB vectors of the Frenet-Serret frame changes by the matrix of a rotation.
Moreover, using the Frenet—Serret frame, one can also prove the converse: any two curves having the same curvature and torsion functions must be congruent by a Euclidean motion.
If the Darboux derivatives of two frames are equal, then a version of the fundamental theorem of calculus asserts that the curves are congruent. In particular, the curvature and torsion are a complete set of invariants for a curve in three-dimensions. The formulas given above for T , N , and B depend on the curve being given in terms of the arclength parameter.
This is a natural assumption in Euclidean geometry, because the arclength is a Euclidean invariant of the curve. In the terminology of physics, the arclength parametrization is a natural choice of gauge. However, it may be awkward to work with in practice. A number of other equivalent expressions are available. Suppose that the curve is given by r t , where the parameter t need no longer be arclength. Then the unit tangent vector T may be written as. The resulting ordered orthonormal basis is precisely the TNB frame.
This procedure also generalizes to produce Frenet frames in higher dimensions. The torsion may be expressed using a scalar triple product as follows,.
If the curvature is always zero then the curve will be a straight line. Here the vectors N , B and the torsion are not well defined. A curve may have nonzero curvature and zero torsion. The converse, however, is false. That is, a regular curve with nonzero torsion must have nonzero curvature. This is just the contrapositive of the fact that zero curvature implies zero torsion. Given a curve contained on the x - y plane, its tangent vector T is also contained on that plane.
Its binormal vector B can be naturally postulated to coincide with the normal to the plane along the z axis. From Wikipedia, the free encyclopedia. For the category-theoretic meaning of this word, see normal morphism. See Griffiths where he gives the same proof, but using the Maurer-Cartan form. Our explicit description of the Maurer-Cartan form using matrices is standard. See, for instance, Spivak, Volume II, p. A generalization of this proof to n dimensions is not difficult, but was omitted for the sake of exposition.
Again, see Griffiths for details. San Jose State University. Archived from the original on 15 October Lectures on Differential Geometry. Englewood Cliffs, N. Various notions of curvature defined in differential geometry. Curvature Torsion of a curve Frenet—Serret formulas Radius of curvature applications Affine curvature Total curvature Total absolute curvature.
Principal curvatures Gaussian curvature Mean curvature Darboux frame Gauss—Codazzi equations First fundamental form Second fundamental form Third fundamental form. Curvature of Riemannian manifolds Riemann curvature tensor Ricci curvature Scalar curvature Sectional curvature. Curvature form Torsion tensor Cocurvature Holonomy. Categories : Differential geometry Multivariable calculus Curves Curvature mathematics.
IB Math SL
In this section we want to look at an application of derivatives for vector functions. Actually, there are a couple of applications, but they all come back to needing the first one. With vector functions we get exactly the same result, with one exception. While, the components of the unit tangent vector can be somewhat messy on occasion there are times when we will need to use the unit tangent vector instead of the tangent vector. First, we could have used the unit tangent vector had we wanted to for the parallel vector. However, that would have made for a more complicated equation for the tangent line. Do not get excited about that.
Tangents and normals mc-TY-tannorm This unit explains how differentiation can be used to calculate the equations of the tangent and normal to a curve.
12.7: Tangent Lines, Normal Lines, and Tangent Planes
Generating Bitcoin for You and Me. Tangent Line - A straight line that passes through a point on a function and has the same slope as that function at that point. Normal Line - A straight line that passes through a point on a function is perpendicular to the function at that point. A common question is to find an equation for the normal or tangent line at a given point on a function. Since the equation is linear the answer will take the form of:.
There are two kinds of tangent lines — oblique slant tangents and vertical tangents. As a result, the equations of the tangent and normal lines are written as follows:. The study of curves can be performed directly in polar coordinates without transition to the Cartesian system. This allows to find the tangency point:.