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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 ) = R cos ( 2 π N t h ) i + R sin ( 2 π N t h ) j + t k , 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 [link] . First of all,

r ( t ) = 2 π N R h sin ( 2 π N t h ) i + 2 π N R h cos ( 2 π N t h ) j + k .

Therefore,

s = a b r ( t ) d t = 0 h ( 2 π N R h sin ( 2 π N t h ) ) 2 + ( 2 π N R h cos ( 2 π N t h ) ) 2 + 1 2 d t = 0 h 4 π 2 N 2 R 2 h 2 ( sin 2 ( 2 π N t h ) + cos 2 ( 2 π N t h ) ) + 1 d t = 0 h 4 π 2 N 2 R 2 h 2 + 1 d t = [ t 4 π 2 N 2 R 2 h 2 + 1 ] 0 h = h 4 π 2 N 2 R 2 + h 2 h 2 = 4 π 2 N 2 R 2 + h 2 .

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 ) = a t ( f ( u ) ) 2 + ( g ( u ) ) 2 + ( h ( u ) ) 2 d u .

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 ) in [link] , we can differentiate both sides of the equation:

s ( t ) = d d t [ a t ( f ( u ) ) 2 + ( g ( u ) ) 2 + ( h ( u ) ) 2 d u ] = d d t [ a t r ( u ) d u ] = 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 ) = a t r ( u ) d u = a t 1 d u = t a ,

which means that t represents the arc length as long as a = 0.

Arc-length function

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

s ( t ) = a t r ( u ) d u .

Furthermore, d s d t = 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 ) = 3 cos t , 3 sin t , 0 t 2 π that parameterizes a circle of radius 3, we can change the parameter from t to 4 t , obtaining a new parameterization r ( t ) = 3 cos 4 t , 3 sin 4 t . The new parameterization still defines a circle of radius 3, but now we need only use the values 0 t π / 2 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 .

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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