Chapter 2: Space Curves
Section 2.2: Arc Length as Parameter
For curves given in different formats, Table 2.2.1 lists formulas for the arc-length function, which measures the length of the curve as a function of the curve's parameter. In each case, the integrand for the arc-length function is recognized as ρ=∥R′∥. Hence, by the fundamental theorem of calculus, dsdp=ρ, and again by elementary calculus, dpds=1/ρ.
Table 2.2.1 The arc-length function
If the parameter for a curve is s, the curve's arc length, then (by the chain rule)
dRds=dRdpdpds=dRdp1ρ ⇒ dRds = dRdp1ρ=ρρ=1
Calculate the length of the helix defined in Example 2.1.4.
Calculate the length of the curve defined in Example 2.1.5.
Calculate the length of the curve defined parametrically by xt=t cost, yt=t sint, zt=t3/6 for 0≤ t≤2 π.
Obtain st, the arc-length function for the helix in Example 2.1.4.
Obtain st, the arc-length function for the curve in Example 2.1.3.
Obtain the arc-length function for the curve x=p2−p/2,y=4/3 p3/2, where p∈0,∞.
Invert s=sp to obtain p=ps and reparametrize the curve with the arc length s as the parameter.
Show that dRds=1.
<< Previous Section Table of Contents Next Section >>
© Maplesoft, a division of Waterloo Maple Inc., 2021. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
What kind of issue would you like to report? (Optional)