Arc Length and Curvature: Computations for Vector Calculus

The Arc Length and Curvature Question

Computing the arc length and curvature of various functions is one of the principal applications of the vectorial calculus we have just learned. Our first article provided a broad introduction into the world of calculus. The subsequent article takes time investigating the dot product and cross product in depth. The subject of vector functions and their space curves logically follows this topic. Vector differentiation and integration furthermore trail the subject of vector functions. Now, we have now arrived at our final subject of explicit vector calculus: computing the arc length and curvature. This is an essential function which relies heavily on both basic vector computations as well as differentiation and integration to accomplish the task. Through learning this topic, you will see the importance of several different vectors, which include the tangent, the normal, and binomial vector.

Introduction to the Length Function

We previously encountered how a vector function comprised of parametric equations defines a space curve. Such a vector function may appear as follows:

r(t) = < f(t), g(t), h(t) >

With this in mind for vectors of two-dimensions, the following function defines the length of the space curve.

L = \int_a^b \sqrt{[f'(t)]^2+[g'(t)]^2}dt

In order to compute the three-dimensional space curve, all we must do is incorporate our additional term h(t).

L = \int_a^b \sqrt{[f'(t)]^2+[g'(t)]^2+[h(t)]^2}dt

After this, we can rearrange this function taking into account the x, y, and z components of the field:

L=\int_a^b \sqrt{(\frac {dx}{dt})^2 + (\frac {dy}{dt})^2 + (\frac {dz}{dt})^2}

As a result, we can put the length function into its simplest form wherein we associate it with the absolute value of the derivative of the vector function:

L=\int_a^b|r'(t)|dt

Proof of Length Function

In this case, the length function as it relates to the previous function is proven by expanding the derivative of the vector function.

|r'(t)| = |f'(t) + g'(t) + h'(t)|\newline \newline |r'(t)|=\sqrt{[f'(t)|^2+[g'(t)]^2+[h'(t)]^2}

Example Computing Length

For example, suppose we have been given a vector function with its parameters and are required to compute the length of the space curve from the point (1,0,0) to (1, 0, 2π). We will do the following:

r(t) = < cos(t) + sin(t) + t >\newline r'(t) = < -sin(t) + cos(t) + 1 > \newline |r'(t)|=\sqrt{(-sin(t))^2 + cos^2(t) + 1}\newline L=\int_0^{2π} \sqrt{(-sin(t))^2+cos^2(t)+1}\newline L=\int_0^{2π}\sqrt{2}dt\newline L=2\sqrt{2}π

Arc Length Function

A given space curve may be modeled by several different vector functions with different component parameters. These different vector functions are called parameterizations of the space curve. By using a given parameterization of a space curve, we are enabled to create an arc length function that computes the length through any value of ‘t’. Consider a new arc length function defined by ‘s’:

s(t) = \int_a^t|r'(u)|du \newline s(t) = \int_a^t \sqrt{(\frac {dx}{du})^2+(\frac {dy}{du})^2+(\frac {dz}{du})^2}

For this function, s(t) represent the length of the arc on the space curve that occurs between the values r(a) and r(t). Therefore, we have a continuous function that can compute the length of the curve between any of the values for the domain.

In consideration of the arc length function, we can differentiate this entity, which proves to be useful for space curve parameterization.Thus, a curve becomes parameterized with respect to the length of the arc. To illustrate this, the operation results in a function that relates the derivative of the arc length to the derivative of the vector function:

\frac{ds}{dt}=|r'(t)|

In this case, suppose we have been given a vector function, and we desire to reparameterize it with respect to the length of the arc, beginning at a value of (1, 0, 0). As a result, we may execute this as follows:

r(t) = cos(t) + sin(t) + t \newline s(t)=\int_0^t|r'(u)|du \newline s(t)=\int_0^t\sqrt{2}du \newline s(t)=\sqrt{2}t

Firstly, observe that the original value of ‘t’ is set to a new value:

t=\frac{s}{\sqrt{2}}

Next, we substitute the value of ‘t’ into the original parameterization of the space curve but now it relates to the arc length. This reparameterization appears as:

r(t(s))=< cos(\frac{s}{\sqrt{2}}), sin(\frac{s}{\sqrt{2}}), (\frac{s}{\sqrt{2}}) >

Conceptualizing Curvature

A space curve is often mathematically described according to its curvature. In particular, with respect to its smoothness. Parameterizations of a space curve are smooth if the derivative of the parameterization is continuous on the interval I. If this is the case, then the curve itself is a smooth curve. One topological/graphical note is that smooth curves follow gradients of movement; there are no sharp changes in its course.

Curvature inherently is a measurement of the frequency of change in the direction of the curve. Recall that with general functions, change in direction can be computed in accordance with the tangent of the curve at a specific point. Additionally, tangents exist for the space curve defined by vector functions.The derivative of the vector function relates to the tangent vector as follows:

T(t)=\frac{r'(t)}{|r'(t)|}

Here, the tangent vector stipulates the direction of the space curve at a particular location. As a consequence of the tangent vector, the derivative of the tangent vector readily defines the curvature of the space curve. As a result, the function is defined as follows:

\kappa=|\frac{dT}{ds}|

We see by this function that the derivative of the tangent curve with respect to arc length provides the curvature of the curve at a particular point.

Defining the Curvature Function

Previously, we demonstrated that the arc length could be formulated as a function by relating arc length to the integral of the derivative of the vector function. Similarly, we may establish another function that executes a similar role, but actually defines a function for curvature along the space curve. As a result, we define the function of the arc curvature as follows:

\kappa(t)=\frac{|T'(t)|}{|r'(t)|}

More specifially, the curvature function is broken down further to procure:

\kappa(t)=\frac{|r'(t) \times r”(t)|}{|r'(t)|^3}

Because of this, we may observe that the quotient between the absolute value of the cross product for the first order and second order derivative of the vector function with the cubic of the first order derivative produces the curvature function.

Proof of the Curvature Function

For example, our original function is as follows:

\kappa(t)=\frac{|T'(t)|}{|r'(t)|}

We manipulate this overall function to demonstrate the proof of its validity:

r’=|r’|T=\frac{ds}{dt}T \newline r”=\frac{d^2s}{dt^2}T+\frac{ds}{dt}T’\newline r’\times r”=(\frac{ds}{dt})^2(T\times T)\newline |r’ \times r”|=(\frac{ds}{dt})^2|T\times T’|=(\frac{ds}{dt})^2|T’|\newline |T’|=\frac{|r’\times r”|}{(\frac{ds}{dt})^2}\newline\cdot\newline \kappa=\frac{|T’|}{|r’|}\newline \kappa=\frac{|r’\times r”|}{|r’|^3}

Binomial and Normal Vectors

The usefulness of arc length and curvature is evident in the determination of binomial and normal vectors. While an in depth discussion of these features is well beyond the scope of this article, I will briefly address these entities here. If you desire to investigate this topic in greater depth, check out this resource.

Now, the normal vector N(t) is a vector which is orthogonal to the tangent vector T(t). Therefore, the following function computes the normal vector:

N(t)=\frac{T'(t)}{|T'(t)|}

The binomial vector is a vector that is orthogonal to both the normal vector and the tangent vector. The following function constitutes the binomial vector:

B(t)=T(t)\times N(t)

A later article will address the use of these vectors in extremely thorough examples.

Arc Length and Curvature in Vector Calculus

Finally, our tutorial on vector calculus has come to an end. However, later tutorials will certainly build upon the content of these articles. Arc Length and Curvature are the primary applications for the vector operations discussed previously. However, these have more practical applications in the real world. Subsequently, a link to the previous articles on vector calculus may be found below. If you have completed this tutorial, keep on the look out for our next tutorial on partial derivatives. Happy Learning.

  1. General Vector Calculus
  2. Dot Product and Cross Product
  3. Vector Function
  4. Vector Differentiation and Integration
  5. Arc Length and Arc Curvature (current)

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