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Arc Length

Main Concept

The arc length is defined as the length of a curve. It can be calculated by subdividing the curve into smaller pieces, each of which is approximated by a line segment that connects two points on the curve. Then the sum of the lengths of all of these line segments will approximate the length of the curve; the approximation will be more precise as the size of the line segments decreases. If the endpoints of these approximating line segments are xk, k=0..n, then their total length is given by:

 

Length  = k=1nxkxk12+ykyk12.

 

If you then take the limit as the size of the line segments decreases to 0, the formula above becomes an integral. The form of the integral varies according to the type of curve:

 

• 

y=fx over the interval x a,b:

Arc Length  = ab1+dydx2ⅆx

• 

x=gy over the interval y c,d:

Arc Length  =  cd1+dxdy2ⅆy

• 

parametric curve x,y = ft,gt over the interval t 0,1:

Arc Length  = 01dxdt2+dydt2ⅆt

 

 

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