Learning Outcomes
- Determine the length of a particle’s path in space by using the arc-length function.
Arc Length for Vector Functions
We have seen how a vector-valued function describes a curve in either two or three dimensions. Recall Alternative Formulas for Curvature, which states that the formula for the arc length of a curve defined by the parametric functions x=x(t), y=t(t), t1≤t≤t2x=x(t), y=t(t), t1≤t≤t2 is given by
s=∫t2t1 √(x′(t))2+(y′(t))2dts=∫t2t1 √(x′(t))2+(y′(t))2dt
In a similar fashion, if we define a smooth curve using a vector-valued function r(t)=f(t)I+g(t)jr(t)=f(t)I+g(t)j, where a≤t≤ba≤t≤b, the arc length is given by the formula
s=∫ba √(f′(t))2+(g′(t))2dts=∫ba √(f′(t))2+(g′(t))2dt
In three dimensions, if the vector-valued function is described by r(t)=f(t)I+g(t)j+h(t)kr(t)=f(t)I+g(t)j+h(t)k over the same interval a≤t≤ba≤t≤b, the arc length is given by
s=∫ba √(f′(t))2+(g′(t))2+(h′(t))2dts=∫ba √(f′(t))2+(g′(t))2+(h′(t))2dt
Arc-Length formulas Theorem
- Plane curve: Given a smooth curve CC defined by the function r(t)=g(t)i+g(t)jr(t)=g(t)i+g(t)j, where tt lies within the interval [a, b][a, b], the arc length of CC over the interval is
- Space curve: Given a smooth curve CC defined by the function r(t)=f(t)i+g(t)j+h(t)kr(t)=f(t)i+g(t)j+h(t)k, where tt lies within the interval [a, b][a, b], the arc length of CC 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 r(t)r(t) 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:
try it
Calculate the arc length of the parameterized curve
r(t)=⟨2t2+1, 2t2−1, t3⟩, 0≤t≤3.
Watch the following video to see the worked solution to the above Try It
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
r(t)=Rcos(2πNth)i+Rsin(2πNth)j+tk, 0≤t≤h,
where R represents the radius of the helix, h represents the height (distance between two consecutive turns), and the helix completes N turns. Let’s derive a formula for the arc length of this helix using Arc-Length Formulas. First of all,
r′(t)=−2πNRhsin(2πNth)i+2πNRhcos(2πNth)j+k.
Therefore,
s=∫ba ‖r′(t)‖ dt=∫h0 √(−2πNRhsin(2πNth))2+(2πNRhcos(2πNth))2+12dt=∫h0 √4π2N2R2h2(sin2(2πNth)+cos2(2πNth))+1dt=∫h0 √4π2N2R2h2+1dt=[t√4π2N2R2h2+1]h0=h√4π2N2R2+h2h2=√4π2N2R2+h2.
This gives a formula for the length of a wire needed to form a helix with N turns that has radius R and height h.
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:
s(t)=∫ta √(f′(u))2+(g′(u))2+(h′(u))2du
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 s(t) measures distance traveled as a function of time, s′(t) measures the speed of the particle at any given time. Since we have a formula for s(t), we can differentiate both sides of the equation:
s′(t)=ddt[∫ta √(f′(u))2+(g′(u))2+(h′(u))2du]=ddt[∫ta ‖r′(u)‖ du]=‖r′(t)‖.
If we assume that r(t) defines a smooth curve, then the arc length is always increasing, so s′(t)>0 for t>a. Last, if r(t) is a curve on which ‖r′(t)‖=1 for all t, then
s(t)=∫ta ‖r′(u)‖ du=∫ta 1du=t−a,
which means that t represents the arc length as long as a=0.
Arc-Length function Theorem
Let r(t) describe a smooth curve for t≥a. Then the arc-length function is given by
Furthermore, dsdt=‖r′(t)‖>0. If ‖r′(t)‖=1 for all t≥a, then the parameter t represents the arc length from the starting point at t=a.
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 r(t)=⟨3cost, 3sint⟩, 0≤t≤2π that parameterizes a circle of radius 3, we can change the parameter from t to 4t, obtaining a new parameterization r(t)=⟨3cos4t, 3sin4t⟩. The new parameterization still defines a circle of radius 3, but now we need only use the values 0≤t≤π2 to traverse the circle once.
Suppose that we find the arc-length function s(t) and are able to solve this function for t as a function of s. We can then reparameterize the original function r(t) by substituting the expression for t back into r(t). The vector-valued function is now written in terms of the parameter s. Since the variable s represents the arc length, we call this an arc-length parameterization of the original function r(t). One advantage of finding the arc-length parameterization is that the distance traveled along the curve starting from s=0 is now equal to the parameter s. The arc-length parameterization also appears in the context of curvature (which we examine later in this section) and line integrals, which we study in the Introduction to Vector Calculus.
Example: finding an arc-length parameterization
Find the arc-length parameterization for each of the following curves:
try it
Find the arc-length function for the helix
r(t)=⟨3cost, 3sint, 4t⟩, t≥0.
Then, use the relationship between the arc length and the parameter t to find an arc-length parameterization of r(t).
Watch the following video to see the worked solution to the above Try It
Candela Citations
- CP 3.10. Authored by: Ryan Melton. License: CC BY: Attribution
- Calculus Volume 3. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/books/calculus-volume-3/pages/1-introduction. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-3/pages/1-introduction