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.
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