Arc Length and Curvature

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Arc Length and Curvature
By Dr. Julia Arnold

Objectives: Find the arc length of a space curve. Use the arc length parameter to describe a plane curve or space curve. Find the curvature of a curve at a point on the curve. Use a vector-valued function to find frictional force.

Find the arc length of a space curve.
Objective 1 Find the arc length of a space curve.

Given a smooth plane curve C that has parametric equations
x = x(t) and y = y(t) where , the arc length s is given by (See Section 10.3) In vector form, where C is given by r(t)=x(t)i + y(t)j, the above equation can be written as We can extend this formula to space quite naturally as follows: If C is a smooth curve given by r(t)= x(t)i + y(t)j +z(t)k on an interval [a,b], then the arc length C on the interval is

To help you visualize what is taking place look at the curve and imagine taking steps from point to point.

To help you visualize what is taking place look at the curve and imagine taking steps from point to point.

To help you visualize what is taking place look at the curve and imagine taking steps from point to point.

To help you visualize what is taking place look at the curve and imagine taking steps from point to point.

To help you visualize what is taking place look at the curve and imagine taking steps from point to point.

To help you visualize what is taking place look at the curve and imagine taking steps from point to point.

To help you visualize what is taking place look at the curve and imagine taking steps from point to point.

To help you visualize what is taking place look at the curve and imagine taking steps from point to point.

To help you visualize what is taking place look at the curve and imagine taking steps from point to point.

To help you visualize what is taking place look at the curve and imagine taking steps from point to point.

To help you visualize what is taking place look at the curve and imagine taking steps from point to point.

To help you visualize what is taking place look at the curve and imagine taking steps from point to point.

To help you visualize what is taking place look at the curve and imagine taking steps from point to point.

To help you visualize what is taking place look at the curve and imagine taking steps from point to point.

To help you visualize what is taking place look at the curve and imagine taking steps from point to point.

To help you visualize what is taking place look at the curve and imagine taking steps from point to point.

Now let the points get closer and closer together and sum them up.
Let’s look a a few examples:

Example 1: Find the length of the space curve over the given interval.
Set up the integral Find the derivative of the vector. Substitute into the formula. Simplify Integrate and evaluate.

Objective 2 2. Use the arc length parameter to describe a plane curve or space curve.

Curves can be represented by vector-valued functions in different ways depending on the choice of parameter. For example the following two representations are equivalent. For motion along a curve the most convenient parameter is time t. However, for studying the geometric properties of a curve, the convenient parameter is often arc length s.

The arc length s is called the arc length parameter.
If C is a smooth curve given by r(t)= x(t)i + y(t)j +z(t)k on an interval [a,b], then the arc length of C on the interval [a,b] , with a<t<b is C = s(t) which is The arc length s is called the arc length parameter.

Example 2: Consider the curve represented by the vector-valued function
A. Write the length of the arc s as a function of t by evaluating the integral: Solution:

Example 2: Consider the curve represented by the vector-valued function
B. Solve for t in part A and substitute the result into the original set of parametric equations. This yields a parameterization of the curve in terms of the arc length parameter s. Solution:

Example 2: Consider the curve represented by the vector-valued function
C. Find the coordinates of the point on the curve for arc lengths Solution:

Example 2: Consider the curve represented by the vector-valued function
C. Find the coordinates of the point on the curve for arc lengths Solution:

Example 2: Consider the curve represented by the vector-valued function
D. Verify that Solution:

This brings us to a Theorem about the arc length parameter, namely
If C is a smooth curve given by Where s is the arc length parameter, then Moreover, if t is any parameter for the vector-valued function r such that Then t must be the arc length parameter. This theorem is stated without proof.

3. Find the curvature of a curve at a point on the curve.
Objective 3 3. Find the curvature of a curve at a point on the curve.

Curvature An important use of the arc length parameter is to find curvature. Curvature is the measure of how sharply the curve bends. For example, in this helix we get more bend here Than here.

We can calculate curvature by calculating the magnitude of the rate of change of the unit tangent vector T with respect to the arc length s. T2 T3 T1 Definition of Curvature Let C be a smooth curve ( in the plane or in space) given by r(s), where s is the arc length parameter. The curvature K at s is given by

Example 3: Find the curvature using s is the arc length parameter, for
Solution: This was the problem we did earlier and found the arc length parameter to be: and the function to be Using the formula for curvature K in terms of arc length s, namely we get: and knowing that Since curvature K is

Example 3: Find the curvature using s is the arc length parameter, for
Solution:

c Using winplot, this is the curve in question. Since s =
In terms of t the curvature would be

We can see that the curvature of a circle is the same everywhere and reason it to be a constant which turns out to be 1/r where r is the radius of the circle. See example 4 in your text. Other formulas for curvature. Since the previous definition depends on the arc length parameter, it might be good to have some alternative definitions which depend on an arbitrary parameter t. Two formulas for curvature Theorem 12.8 If C is a smooth curve given by r(t), then the curvature K of C at t is given by

Example 4: using the alternative curvature formula on the same vector-valued function
We can compare our answers. From Example 2A we already know that Next we need to find T(t). Now we need T’(t)

Using the formula Now we find Which is what we got back on slide 37 Click on the purple crayon to get back to this slide.

Using the other formula
We have

Example 5

Solution to question on previous slide.
At (4,0) the curvature would be: Thus 25/2 would be the radius of the circle which would be approximately 5.66 Using winplot I found the normal to the tangent line at x = 4 (blue line) and then found the center to be approximately at (0,-4). Figure 12.37

Use a vector-valued function to find frictional force.
Objective 4 Use a vector-valued function to find frictional force.

Example 6 Solution: Continued

Example 6 Solution continued

Figure 12.38

Hint for HW Confusion of Mass and Weight
A few further comments should be added about the single force which is a source of much confusion to many students of physics - the force of gravity. The force of gravity acting upon an object is sometimes referred to as the weight of the object. Many students of physics confuse weight with mass. The mass of an object refers to the amount of matter that is contained by the object; the weight of an object is the force of gravity acting upon that object. Mass is related to how much stuff is there and weight is related to the pull of the Earth (or any other planet) upon that stuff. The mass of an object (measured in kg) will be the same no matter where in the universe that object is located. Mass is never altered by location, the pull of gravity, speed or even the existence of other forces. For example, a 2-kg object will have a mass of 2 kg whether it is located on Earth, the moon, or Jupiter; its mass will be 2 kg whether it is moving or not (at least for purposes of our study); and its mass will be 2 kg whether it is being pushed upon or not. On the other hand, the weight of an object (measured in Newtons) will vary according to where in the universe the object is. Weight depends upon which planet is exerting the force and the distance the object is from the planet. Weight, being equivalent to the force of gravity, is dependent upon the value of g. On earth's surface g is 9.8 m/s2 (often approximated as 10 m/s2) or 32 feet/s2. On the moon's surface, g is 1.7 m/s2. Go to another planet, and there will be another g value. Furthermore, the g value is inversely proportional to the distance from the center of the planet. So if we were to measure g at a distance of 400 km above the earth's surface, then we would find the g value to be less than 9.8 m/s2.

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