1.4: Arc Length and Curvature

In this section, we study formulas related to curves in both two and three dimensions, and see how they are related to various properties of the same curve. For example, suppose a vector-valued function describes the motion of a particle in space. We would like to determine how far the particle has traveled over a given time interval, which can be described by the arc length of the path it follows. Or, suppose that the vector-valued function describes a road we are building and we want to determine how sharply the road curves at a given point. This is described by the curvature of the function at that point. We explore each of these concepts in this section.

Arc Length for Vector Functions

We have seen how a vector-valued function describes a curve in either two or three dimensions. Recall that the formula for the arc length of a curve defined by the parametric functions and , for is given by

In a similar fashion, if we define a smooth curve using a vector-valued function , where , the arc length is given by the formula

In three dimensions, if the vector-valued function is described by over the same interval , the arc length is given by

Theorem: Arc-Length Formulas for Plane and Space curves

Plane curve: Given a smooth curve defined by the function , where lies within the interval , the arc length of over the interval is

Space curve: Given a smooth curve defined by the function , where lies within the interval , the arc length of over the interval is

The two formulas are very similar; they differ only in the fact that a space curve has three component functions instead of two. Note that the formulas are defined for smooth curves: curves where the vector-valued function is differentiable with a non-zero derivative. The smoothness condition guarantees that the curve has no cusps (or corners) that could make the formula problematic.

Example : Finding the Arc Length

Calculate the arc length for each of the following vector-valued functions:

Solution

1. Using Equation , , so

2. Using Equation , , so

   Here we can use a table integration formula

   so we obtain

Exercise

Calculate the arc length of the parameterized curve

Hint

Use Equation .

Answer

so units

We now return to the helix introduced earlier in this chapter. A vector-valued function that describes a helix can be written in the form

where represents the radius of the helix, represents the height (distance between two consecutive turns), and the helix completes turns. Let’s derive a formula for the arc length of this helix using Equation . First of all,

Therefore,

This gives a formula for the length of a wire needed to form a helix with turns that has radius and height .

Arc-Length Parameterization

We now have a formula for the arc length of a curve defined by a vector-valued function. Let’s take this one step further and examine what an arc-length function is.

If a vector-valued function represents the position of a particle in space as a function of time, then the arc-length function measures how far that particle travels as a function of time. The formula for the arc-length function follows directly from the formula for arc length:

If the curve is in two dimensions, then only two terms appear under the square root inside the integral. The reason for using the independent variable u is to distinguish between time and the variable of integration. Since measures distance traveled as a function of time, measures the speed of the particle at any given time. Since we have a formula for in Equation , we can differentiate both sides of the equation:

If we assume that defines a smooth curve, then the arc length is always increasing, so for . Last, if is a curve on which for all , then

which means that represents the arc length as long as .

Theorem: Arc-Length Function

Let describe a smooth curve for . Then the arc-length function is given by

Furthermore, If for all , then the parameter represents the arc length from the starting point at .

A useful application of this theorem is to find an alternative parameterization of a given curve, called an arc-length parameterization. Recall that any vector-valued function can be reparameterized via a change of variables. For example, if we have a function that parameterizes a circle of radius 3, we can change the parameter from to , obtaining a new parameterization . The new parameterization still defines a circle of radius 3, but now we need only use the values to traverse the circle once.

Suppose that we find the arc-length function and are able to solve this function for as a function of . We can then reparameterize the original function by substituting the expression for back into . The vector-valued function is now written in terms of the parameter . Since the variable represents the arc length, we call this an arc-length parameterization of the original function . One advantage of finding the arc-length parameterization is that the distance traveled along the curve starting from is now equal to the parameter . The arc-length parameterization also appears in the context of curvature (which we examine later in this section) and line integrals.

Example : Finding an Arc-Length Parameterization

Find the arc-length parameterization for each of the following curves:

Solution

1. First we find the arc-length function using Equation :

   which gives the relationship between the arc length and the parameter as so, . Next we replace the variable in the original function with the expression to obtain

   This is the arc-length parameterization of . Since the original restriction on was given by , the restriction on s becomes , or .

    

2. The arc-length function is given by Equation :
   Therefore, the relationship between the arc length and the parameter is , so . Substituting this into the original function yields
   This is an arc-length parameterization of . The original restriction on the parameter was , so the restriction on s is , or .

Exercise

Find the arc-length function for the helix

Then, use the relationship between the arc length and the parameter to find an arc-length parameterization of .

Hint

Start by finding the arc-length function.

Answer

, or . Substituting this into gives

.

Curvature

An important topic related to arc length is curvature. The concept of curvature provides a way to measure how sharply a smooth curve turns. A circle has constant curvature. The smaller the radius of the circle, the greater the curvature.

Think of driving down a road. Suppose the road lies on an arc of a large circle. In this case you would barely have to turn the wheel to stay on the road. Now suppose the radius is smaller. In this case you would need to turn more sharply to stay on the road. In the case of a curve other than a circle, it is often useful first to inscribe a circle to the curve at a given point so that it is tangent to the curve at that point and “hugs” the curve as closely as possible in a neighborhood of the point (Figure ). The curvature of the graph at that point is then defined to be the same as the curvature of the inscribed circle.

Figure : The graph represents the curvature of a function The sharper the turn in the graph, the greater the curvature, and the smaller the radius of the inscribed circle.

Definition: curvature

Let be a smooth curve in the plane or in space given by , where is the arc-length parameter. The curvature at is

As the unit vector is not changing in length, the curvature could be described as a measure of the speed at which the direction of the motion is changing. Since the parameter here is , we have forced the varying speed of travel along the path out of this measurement.

The formula in the definition of curvature is not very useful in terms of calculation. In particular, recall that represents the unit tangent vector to a given vector-valued function , and the formula for is

To use the formula for curvature, it is first necessary to express in terms of the arc-length parameter , then find the unit tangent vector for the function , then take the derivative of with respect to . This is a tedious process. Fortunately, there are equivalent formulas for curvature.

Theorem: Alternate Formulas of Curvature

If is a smooth curve given by , then the curvature of at is given by

If is a three-dimensional curve, then the curvature can be given by the formula

Note that in the context of motion, this formula could also be written

If is the graph of a function and both and exist, then the curvature at point is given by

Proof

The first formula follows directly from the chain rule:

where is the arc length along the curve . Dividing both sides by , and taking the magnitude of both sides gives

Since , this gives the formula for the curvature of a curve in terms of any parameterization of :

In the case of a three-dimensional curve, we start with the formulas and . Therefore, . We can take the derivative of this function using the scalar product formula:

Using these last two equations we get

Since , this reduces to

Since is parallel to , and is orthogonal to , it follows that and are orthogonal. This means that , so

Now we solve this equation for and use the fact that :

Then, we divide both sides by . This gives

This proves . To prove , we start with the assumption that curve is defined by the function . Then, we can define . Using the previous formula for curvature:

Therefore,

Note that in the context of motion, the first alternate formula can be rewritten

Considering what was stated above about the curvature being a measure of the speed at which the direction of motion is changing at a point, notice that we have normalized this measurement this time by dividing out the actual speed of travel along the curve at the given instant.

As indicated above, the second alternate formula can be rewritten as

Remember that the cross product measures the extent to which the two vectors are aligned, returning its largest value when they are orthogonal. Note also that the component of acceleration that is orthogonal to the velocity is the only part we care about here, as it is the component that will affect the change of direction of the motion. The component of acceleration that is aligned with the velocity (and tangent to the curve) will only affect the speed of the motion, which is being ignored in our measurement of curvature. Note that one factor of will multiply into the cross product to make the first vector be .

Example : Finding Curvature

Find the curvature for each of the following curves at the given point:

Solution

1. This function describes a helix.

The curvature of the helix at can be found by using . First, calculate :

Next, calculate

Last, apply :

The curvature of this helix is constant at all points on the helix.

2. This function describes a semicircle.

To find the curvature of this graph, we must use . First, we calculate and

Then, we apply :

The curvature of this circle is equal to the reciprocal of its radius. There is a minor issue with the absolute value in ; however, a closer look at the calculation reveals that the denominator is positive for any value of .

Exercise

Find the curvature of the curve defined by the function

at the point .

Hint

Use .

Answer

The Normal and Binormal Vectors

We have seen that the derivative of a vector-valued function is a tangent vector to the curve defined by , and the unit tangent vector can be calculated by dividing by its magnitude. When studying motion in three dimensions, two other vectors are useful in describing the motion of a particle along a path in space: the principal unit normal vector and the binormal vector.

Definition: binormal vectors

Let be a three-dimensional smooth curve represented by over an open interval . If , then the principal unit normal vector at is defined to be

The binormal vector at is defined as

where is the unit tangent vector.

Note that, by definition, the binormal vector is orthogonal to both the unit tangent vector and the normal vector. Furthermore, is always a unit vector. This can be shown using the formula for the magnitude of a cross product:

where is the angle between and . Since is the derivative of a unit vector, property (vii) of the derivative of a vector-valued function tells us that and are orthogonal to each other, so . Furthermore, they are both unit vectors, so their magnitude is 1. Therefore, and is a unit vector.

The principal unit normal vector can be challenging to calculate because the unit tangent vector involves a quotient, and this quotient often has a square root in the denominator. In the three-dimensional case, finding the cross product of the unit tangent vector and the unit normal vector can be even more cumbersome. Fortunately, we have alternative formulas for finding these two vectors, and they are presented in Motion in Space.

Example : Finding the Principal Unit Normal Vector and Binormal Vector

For each of the following vector-valued functions, find the principal unit normal vector. Then, if possible, find the binormal vector.

Solution

1. This function describes a circle.

To find the principal unit normal vector, we first must find the unit tangent vector

Next, we use :

Notice that the unit tangent vector and the principal unit normal vector are orthogonal to each other for all values of :

Furthermore, the principal unit normal vector points toward the center of the circle from every point on the circle. Since defines a curve in two dimensions, we cannot calculate the binormal vector.

2. This function looks like this:

To find the principal unit normal vector, we first find the unit tangent vector

Next, we calculate and :

Therefore, according to :

Once again, the unit tangent vector and the principal unit normal vector are orthogonal to each other for all values of :

Last, since represents a three-dimensional curve, we can calculate the binormal vector using Equation :

Exercise

Find the unit normal vector for the vector-valued function and evaluate it at .

Hint

First, find , then use .

Answer

For any smooth curve in three dimensions that is defined by a vector-valued function, we now have formulas for the unit tangent vector , the unit normal vector , and the binormal vector . The unit normal vector and the binormal vector form a plane that is perpendicular to the curve at any point on the curve, called the normal plane. In addition, these three vectors form a frame of reference in three-dimensional space called the Frenet frame of reference (also called the TNB frame) (Figure ). Last, the plane determined by the vectors and forms the osculating plane of at any point on the curve.

Figure : This figure depicts a Frenet frame of reference. At every point on a three-dimensional curve, the unit tangent, unit normal, and binormal vectors form a three-dimensional frame of reference.

Suppose we form a circle in the osculating plane of at point on the curve. Assume that the circle has the same curvature as the curve does at point and let the circle have radius . Then, the curvature of the circle is given by . We call the radius of curvature of the curve, and it is equal to the reciprocal of the curvature. If this circle lies on the concave side of the curve and is tangent to the curve at point , then this circle is called the osculating circle of at , as shown in Figure .

Figure : In this osculating circle, the circle is tangent to curve at point and shares the same curvature.

For more information on osculating circles, see this demonstration on curvature and torsion, this article on osculating circles, and this discussion of Serret formulas.

To find the equation of an osculating circle in two dimensions, we need find only the center and radius of the circle.

Example : Finding the Equation of an Osculating Circle

Find the equation of the osculating circle of the curve defined by the function at .

Solution

Figure shows the graph of .

Figure : We want to find the osculating circle of this graph at the point where .

First, let’s calculate the curvature at :

This gives . Therefore, the radius of the osculating circle is given by . Next, we then calculate the coordinates of the center of the circle. When , the slope of the tangent line is zero. Therefore, the center of the osculating circle is directly above the point on the graph with coordinates . The center is located at . The formula for a circle with radius and center is given by . Therefore, the equation of the osculating circle is . The graph and its osculating circle appears in the following graph.

Figure : The osculating circle has radius .

Exercise

Find the equation of the osculating circle of the curve defined by the vector-valued function at .

Hint

Use to find the curvature of the graph, then draw a graph of the function around to help visualize the circle in relation to the graph.

Answer

At the point , the curvature is equal to . Therefore, the radius of the osculating circle is .

A graph of this function appears next:

The vertex of this parabola is located at the point . Furthermore, the center of the osculating circle is directly above the vertex. Therefore, the coordinates of the center are . The equation of the osculating circle is

.

Key Concepts

• The arc-length function for a vector-valued function is calculated using the integral formula . This formula is valid in both two and three dimensions.
• The curvature of a curve at a point in either two or three dimensions is defined to be the curvature of the inscribed circle at that point. The arc-length parameterization is used in the definition of curvature.
• There are several different formulas for curvature. The curvature of a circle is equal to the reciprocal of its radius.
• The principal unit normal vector at is defined to be

• The binormal vector at is defined as , where is the unit tangent vector.
• The Frenet frame of reference is formed by the unit tangent vector, the principal unit normal vector, and the binormal vector.
• The osculating circle is tangent to a curve at a point and has the same curvature as the tangent curve at that point.

Key Equations

• Arc length of space curve
  
• Arc-length function
  
• Curvature

• Principal unit normal vector
  
• Binormal vector
  

Glossary

arc-length function

a function that describes the arc length of curve as a function of

arc-length parameterization

a reparameterization of a vector-valued function in which the parameter is equal to the arc length

binormal vector

a unit vector orthogonal to the unit tangent vector and the unit normal vector

curvature

the derivative of the unit tangent vector with respect to the arc-length parameter

Frenet frame of reference

(TNB frame) a frame of reference in three-dimensional space formed by the unit tangent vector, the unit normal vector, and the binormal vector

normal plane

a plane that is perpendicular to a curve at any point on the curve

osculating circle

a circle that is tangent to a curve at a point and that shares the same curvature

osculating plane

the plane determined by the unit tangent and the unit normal vector

principal unit normal vector

a vector orthogonal to the unit tangent vector, given by the formula

radius of curvature

the reciprocal of the curvature

smooth

curves where the vector-valued function is differentiable with a non-zero derivative

Contributors

• Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.