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 Ca 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 TNand 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. Home Contact Privacy. Congruence of Curves 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. In terms of the parametrization r t defining the first curve Ca general Euclidean motion of C is a composite of the following operations: Translation.
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In this section, we introduce the moving frame of a path inwhich is also called the TNB frame. This is a set of three mutually perpendicular unit vectors an orthonormal set which provide a consistent reference frame for a particle moving along a path.
Imagine yourself walking around, and think about the following three directions:. You can describe locations relative to your reference frame:. However, relative to the rest of the universe, these directions change as you walk around. If you turn around to face in the opposite direction, ahead and right would be pointing in the opposite directions that they were before.
If you fly across the world to Australia, the up direction is now in a different direction. So, this frame of reference is unique to your position, and how you are moving around. Our goal is to define a similar reference frame for a particle moving along a path inwhich will consist of three orthonormal vectors. Suppose we have a path inand we want to define a reasonable reference frame for a particle moving along this path, which will consist of three mutually orthogonal unit vectors, hence an orthonormal basis for.
Since the velocity vector points in the direction of instantaneous motion along the path, this gives us the correct direction for our first vector. In order to obtain a unit vector in the same direction as the velocity vector, we divide by its length, obtaining the unit tangent vector :. In the video below, you can see how the unit tangent vector changes as we traverse a curve. Notice that the unit tangent vector is always pointing in the direction of motion. Thinking back to our definition of curvature, we were able to see how a particle was turning by looking at the change in the unit tangent vector.
It only takes a minute to sign up. I found this to be a rather odd question, but one nevertheless oddly engaging. In an effort to understand curves satisfying the OPs hypothesis see 7 belowI tried to develop a few of its implications. It is my hope that others with curiosity similar to mine will find the following "analysis" helpful.
Said torsion may be defined via the third Frenet-Serret equation. We may re-express this result via 14 and 21 :. Indeed, 32 implies. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. How did we get from this to this? Asked 17 days ago. Active 6 days ago. Viewed times. Nono Nono 79 3 3 bronze badges.
Active Oldest Votes. Robert Lewis Robert Lewis Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Featured on Meta. Responding to the Lavender Letter and commitments moving forward.
The mathematical function that describes a spiral can be expressed using rectangular or Cartesian coordinates. However, if we change our coordinate system to something that works a bit better with circular patterns, the function becomes much simpler to describe. The polar coordinate system is well suited for describing curves of this type. How can we use this coordinate system to describe spirals and other radial figures?
See Example 1. In this chapter we also study parametric equations, which give us a convenient way to describe curves, or to study the position of a particle or object in two dimensions as a function of time.
We will use parametric equations and polar coordinates for describing many topics later in this text. Want to cite, share, or modify this book?
Skip to Content. Calculus Volume 3 Introduction. Table of contents.
Frenet–Serret Formulas - Applications and Interpretation - Congruence of Curves
My highlights. Answer Key. Chapter Outline 1. Figure 1. Scientists think they have existed mostly unchanged for about million years. Previous Next. Order a print copy. We recommend using a citation tool such as this one.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: . 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.Calculus 1 Lecture 1.1: An Introduction to Limits
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 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.
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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.To execute all commands select " Edit Execute Worksheet ". Be warned: This takes about minutes to execute on my office desktop.
Maple worksheet: curvature. As usual we begin by wiping memory and reloading the relevant packages. We also let Maple know that the variable "t" is a real number for the rest of this worksheet.
Let's define a vector valued function called " " so that the graph of is this ellipse. Next, we define the arc length function for this parametrization and then evalute the arc length function when to compute the arc length of the ellipse. Notice that the arc length of an ellipse involves elliptic functions -- surprise! Alternatively we could use the VectorCalculus command "ArcLength" this might be the smarter thing to do rather than reinventing the wheel.
Next, let's plot the ellipse together with its TNB-frame at the point. Now we define the curvature function and plot curvature. Remember that. Or using the built-in function The osculating circle at has radius one over curvature.
The osculating plane is the plane through the point r t with normal vector is given by B t or alternatively it's parallel to both T t and N t. Let's plot the curve, the osculating circle atand the osculating plane at together.
Torsion is a measure of how a curve "bends" out of its osculating planes. This curve's torsion is zero because the curve is planar it doesn't "bend" out of it's osculating plane at all. We can compute torsion using the built in function.
First, we'll plot this curve. Next, let's compute its arc length and plot its curvature. Since Maple cannot find an exact arc length, we'll use "evalf" to get a decimal approximation. Example: Consider the twisted cubic:. Let's find the TNB-frame of and then plot the curvature and torsion for this curve. Now let's plot the osculating circle and tangent line for the twisted cubic at the point along with a plot of the twisted cubic itself.
Now let's try that again using a more "general approach". Instead of punching the parameterization into "spacecure" let's first define it separately so we have it saved for later.
Note : Maple uses the notation " " for the unit vector " ", " " for the unit vector " ", and " " for the unit vector " ". It's easy to differentiate and integrate vector valued functions.
Let's plot the helix together with it's tangent line at. The derivative gives the direction of the line, and the parameterization will give a point on the line. The "subs" command allows us to substitute into. Next, let's compute the arc length function for the helix. Note : Sometimes we have to tell Maple some extra information -- in this problem Maple should know that our dummy variable " " is a real number.
It's interesting to note that Maple's built in ArcLength function does not yield a useful formula for arc length here. The above formula is far more complicated than the one we originally derived. Oh well, Maple's not perfect. Now we can make Maple solve for t and then reparametrize our helix in terms of arc length.
Warning, solve may be ignoring assumptions on the input variables. Of course, our graph looks no different, since we have the same curve with a different parametrization.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. Next, we need to talk about the unit normal and the binormal vectors. The unit normal is orthogonal or normal, or perpendicular to the unit tangent vector and hence to the curve as well. They will show up with some regularity in several Calculus III topics. The definition of the unit normal vector always seems a little mysterious when you first see it.
It follows directly from the following fact. To prove this fact is pretty simple. From the fact statement and the relationship between the magnitude of a vector and the dot product we have the following. Also, recalling the fact from the previous section about differentiating a dot product we see that. The definition of the unit normal then falls directly from this. Because the binormal vector is defined to be the cross product of the unit tangent and unit normal vector we then know that the binormal vector is orthogonal to both the tangent vector and the normal vector.
Notes Quick Nav Download. You appear to be on a device with a "narrow" screen width i. Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device should be able to scroll to see them and some of the menu items will be cut off due to the narrow screen width. Here is the tangent vector to the curve. Show Solution We first need the unit tangent vector so first get the tangent vector and its magnitude.