The Arc length of a Parameterized Curve
Math Insight
A vector-valued function of a single variable, c:R→Rn (confused see note 1), can
be viewed as parameterizing a curve2. Such a function c(t)traces out a curve as
you vary t.
You could think of a curve c:R→R3 as being a wire. For
example, c(t)=(cost,sint,t), for 0≤t≤6π, is the parameterization of a helix. You can
view it as a slinky or a spring.
Parameterized helix. The vector-valued function c(t)=(cost,sint,t) parameterizes a helix, shown in
red. This helix is the image of the interval [0,6π] (shown in magenta) under the mapping of c. For
each value of t, the cyan point represents the vector c(t).
Imagine we wanted to estimate the length of the slinky, which we call the arc
length of the parameterized curve. Unfortunately, it's difficult to calculate the
length of a curved piece of wire. It's much easier to calculate the length of
straight pieces of wire. Probably the easiest way to calculate the length of the
slinky would be to stretch it out into one straight line. But, if you ever tried to do
that with a slinky (or a strong spring), you'd discover that stretching it into a
straight line is virtually impossible.
If you can't stretch the slinky into one straight line, what could you do to estimate
its length? One thing you could do is pretend that the slinky, rather than being a
curved wire, was really composed of a bunch of short straight wires. In other
words, you could approximate the curved slinky with line segments.
Source URL: http://mathinsight.org/parametrized_curve_arc_length
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
www.saylor.org
Page 1 of 5
Helix arc length. The vector-valued function c(t)=(cost,sint,t) parameterizes a helix, shown in red.
The green lines are line segments that approximate the helix.
The length of the line segments is easy to measure. If you add up the lengths of
all the line segments, you'll get an estimate of the length of the slinky.
Let Δt specify the discretization interval of the line segments, and denote the total
length of the line segments by L(Δt). (In the above applet, Δt is written as dt.) As
the line segments take shortcuts, the length of the line segments underestimate
the arc length of the slinky.
However, if you increase the number of line segments (decreasing the length of
each line segment), the total length of the line segments becomes a better
estimate of the slinky arc length. As Δt approaches zero, the length of each line
segment shrinks toward zero, the number of line segments increases, and the
line segments become closer and closer to the slinky. Consequently, the total
length L(Δt) of the line segments approaches the slinky arc length.
What's the length of each line segment? If there are n line segments, we could
define t0,t1,…,tn so that the first line segment goes from the point c(t0) to the
point c(t1), the second line segment goes from the point c(t1) to the point c(t2),
etc. The vector from c(t0) to c(t1) is simply c(t1)−c(t0), so the length of the line
segment must be ∥c(t1)−c(t0)∥. The length of the second line segment
is ∥c(t2)−c(t1)∥, etc.
Source URL: http://mathinsight.org/parametrized_curve_arc_length
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
www.saylor.org
Page 2 of 5
To find the total length of the line segments, we just add up those lengths from
all n line segments:
Now we do some tricks to put this into a different form. First, if Δti=ti−ti−1, then
we can rewrite ti as ti−1+Δti. Next, we can divide each term of the above
equation by Δti and multiply it by Δti so that our expression for the length
becomes
Maybe this new equation doesn't look like much of an improvement. But if you
were a real math nerd, you might have noticed that the quotient
involving c(ti−1) is exactly the expression used in the limit definition of the
derivative3 c′(t) of a parameterized curve (if we replace h with Δti). In fact,
equation (14) is a Riemann sum for an integral, analogous to the ones used to
define integrals such as bricK2927double integrals5. If we let the number of line
segments increase (as we take the limit Δti→0) the quotient becomes c′(t), and
equation (14) approaches the integral
which is the true arc length of the slinky. The numbers a and b are the values
of t at the ends of the slinky (i.e., the numbers so that the slinky is defined
by c(t) for a≤t≤b). In our example, the slinky was defined by c(t) for 0≤t≤6π, so
we would use a=0 and b=6π.
The magnitude of the derivative ∥c′(t)∥ is the speed of a particle6 that is at
position c(t) at time t. The above equation simply says that the total length of the
curve traced by the particle is the integral of its speed. (This length must, of
course, be independent of the particle's speed7.)
You can see some examples here8.
Source URL: http://mathinsight.org/parametrized_curve_arc_length
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
www.saylor.org
Page 3 of 5
Notes and Links:
1. Function notation
Recall the notation that R stands for the real numbers. Similarly, R2 is a twodimensional vector, and R3 is a three-dimensional vector.
Scalar-valued functions
In one-variable calculus, you worked a lot with one-variable functions, i.e.,
functions from R onto R. If f(x) is such a one-variable functions, we can
write f:R→R as a shorthand way of expressing that f is a function from R onto R.
A function like f(x,y)=x+y is a function of two variables. It takes an element of R2,
like (2,1), and gives a value that is a real number (i.e., an element of R),
like f(2,1)=3. Since f maps R2 to R, we write f:R2→R. We can also use this
“mapping” notation to define the actual function. We could define the
above f(x,y) by writing f:(x,y)↦x+y.
To contrast a simple real number with a vector, we refer to the real number as
a scalar. Hence, we can refer to f:R2→R as a scalar-valued function of two
variables or even just say it is a real-valued function of two variables.
Everything works the same for scalar valued functions of three or more variables.
For example, f(x,y,z), which we can write f:R3→R, is a scalar-valued function of
three variables.
Vector-valued functions
In contrast, a vector-valued function takes on values that are vectors. First, let's
talk about vector-valued functions of a single variable.
A vector-valued function in two dimensions can be written f:R→R2. An example
is f(t)=(3t,−t). For a given real number, which we'll denote by ♣ for fun, f(♣) is the
two-dimensional vector (3♣,−♣). Similarly, a vector-valued function in three
dimensions can be written f:R→R3. For example, if f(s)=(1−s,s3,coss),
then f(0)=(1,0,1). We sometimes write vector-valued functions using the standard
unit vector i, j, and k, as in f(s)=(1−s)i+s3j+(coss)k.
Lastly, we can have vector-valued functions of multiple variables. For example, a
function could take values in R3, say (x,y,z), and map them to R2, such
as f(x,y,z)=(x−y,x22/z). We can write a function from R3 to R2 as f:R3→R2. You
get the idea.
The domain of a function
The function f(t)=(t,t2) is defined over all real numbers R, i.e., the domain of the
function is R. Sometimes a function of one variable may be defined over a subset
of real numbers, say some set U⊂R; in this case, the domain of the function is U.
(Note, the symbol “⊂” just means “is subset of”.) In three dimensions, for
example, we can specify the domain by writing f:U⊂R→R3, or simply f:U→R3.
Source URL: http://mathinsight.org/parametrized_curve_arc_length
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
www.saylor.org
Page 4 of 5
Example: since logt isn't a real number for t≤0, the domain of f(t)=(logt)i+tj, is the
set D=(0,∞). We could write this f:(0,∞)→R2. What would the domain be if we
replaced logt with log(t−3) or log(2−t)? You have to think
where log(t−3) or log(2−t) is a real number, i.e., where t−3>0 or where 2−t>0.
We use the same notation for functions of multiple variables. If we
wrote f:U⊂R2→R3, we would mean a function maps values in a
subset U of R2 to values in R3.
2. http://mathinsight.org/parametrized_curve_introduction
3. http://mathinsight.org/parametrized_curve_derivative#limit_definition
4. http://mathinsight.org/parametrized_curve_arc_length#mjx-eqn-total_length
5. http://mathinsight.org/double_integral_introduction
6. http://mathinsight.org/parametrized_curve_derivative_location_velocity
7. http://mathinsight.org/parametrized_curve_arc_length_examples#lengthind
8. http://mathinsight.org/parametrized_curve_arc_length_examples
Source URL: http://mathinsight.org/parametrized_curve_arc_length
Saylor URL: http://www.saylor.org/courses/ma103/
Attributed to: [Duane Q. Nykamp]
www.saylor.org
Page 5 of 5