<< Chapter < Page Chapter >> Page >

Finding an arc-length parameterization

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

  1. r ( t ) = 4 cos t i + 4 sin t j , t 0
  2. r ( t ) = t + 3 , 2 t 4 , 2 t , t 3
  1. First we find the arc-length function using [link] :
    s ( t ) = a t r ( u ) d u = 0 t −4 sin u , 4 cos u d u = 0 t ( −4 sin u ) 2 + ( 4 cos u ) 2 d u = 0 t 16 sin 2 u + 16 cos 2 u d u = 0 t 4 d u = 4 t ,

    which gives the relationship between the arc length s and the parameter t as s = 4 t ; so, t = s / 4. Next we replace the variable t in the original function r ( t ) = 4 cos t i + 4 sin t j with the expression s / 4 to obtain
    r ( s ) = 4 cos ( s 4 ) i + 4 sin ( s 4 ) j .

    This is the arc-length parameterization of r ( t ) . Since the original restriction on t was given by t 0 , the restriction on s becomes s / 4 0 , or s 0.
  2. The arc-length function is given by [link] :
    s ( t ) = a t r ( u ) d u = 3 t 1 , 2 , 2 d u = 3 t 1 2 + 2 2 + 2 2 d u = 3 t 3 d u = 3 t 9.

    Therefore, the relationship between the arc length s and the parameter t is s = 3 t 9 , so t = s 3 + 3. Substituting this into the original function r ( t ) = t + 3 , 2 t 4 , 2 t yields
    r ( s ) = ( s 3 + 3 ) + 3 , 2 ( s 3 + 3 ) 4 , 2 ( s 3 + 3 ) = s 3 + 6 , 2 s 3 + 2 , 2 s 3 + 6 .

    This is an arc-length parameterization of r ( t ) . The original restriction on the parameter t was t 3 , so the restriction on s is ( s / 3 ) + 3 3 , or s 0.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the arc-length function for the helix

r ( t ) = 3 cos t , 3 sin t , 4 t , t 0.

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

s = 5 t , or t = s / 5 . Substituting this into r ( t ) = 3 cos t , 3 sin t , 4 t gives

r ( s ) = 3 cos ( s 5 ) , 3 sin ( s 5 ) , 4 s 5 , s 0.

Got questions? Get instant answers now!


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 ( [link] ). The curvature of the graph at that point is then defined to be the same as the curvature of the inscribed circle.

This figure is the graph of a curve. The curve rises and falls in the first quadrant. Along the curve, where the curve changes from decreasing to increasing there is a circle. The bottom of the circle curves the same as the graph of the curve. There is also a second smaller circle where the curve goes from increasing to decreasing. Part of the circle falls on the curve. Both circles have the radius r represented.
The graph represents the curvature of a function y = f ( x ) . The sharper the turn in the graph, the greater the curvature, and the smaller the radius of the inscribed circle.


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     κ at s is

κ = d T d s = T ( s ) .

Visit this website for more information about the curvature of a space curve.

The formula in the definition of curvature is very useful in terms of calculation. In particular, recall that T ( t ) represents the unit tangent vector to a given vector-valued function r ( t ) , and the formula for T ( t ) is T ( t ) = r ( t ) r ( t ) . To use the formula for curvature, it is first necessary to express r ( t ) in terms of the arc-length parameter s , then find the unit tangent vector T ( s ) for the function r ( s ) , then take the derivative of T ( s ) with respect to s. This is a tedious process. Fortunately, there are equivalent formulas for curvature.

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
I'm not good at math so would you help me
what is the problem that i will help you to self with?
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?