Arclength as a function

Dr. Therese Shelton
Southwestern University
Georgetown, TX

Caution:  remember that you may have to tweak some things if you change functions or windows.

Set-up

Remember, the curve should be a simple smooth path.

Clear[x,y,tmin,tmax, xmin,xmax,ymin,ymax];
x[t_]:=Cos[t];
 y[t_]:=Sin[t];
tmin=0.0; tmax=6.0;
xmin = -1.25; xmax=1.25;
ymin=-1.25;  ymax=1.25;   
Clear[colorr]
colorr=RGBColor[1.,0.,0.];

Clear[plotAll];
plotAll =ParametricPlot[     {x[t],y[t]},           {t,tmin,tmax},
 PlotRange->{    {xmin,xmax},{ymin,ymax}    }  ,
AspectRatio->Automatic,PlotStyle->{colorr}   ]

arclength, as a function

Clear[arclength,speed]

arclength[t_]:= (∫_tmin^t)(Sqrt[(((x'[u]))^2) + (((y'[u]))^2)])\[DifferentialD]u

(∫_tmin^t)(Sqrt[(((x'[u]))^2) + (((y'[u]))^2)])\[DifferentialD]u

numerical and visual aid

Clear[numberIntervals,deltat];

numberIntervals = 10;

deltat=(tmax-tmin)/numberIntervals

Table[{t,arclength[t]}, {t, tmin, tmax,deltat}]//TableForm

Since arclength is a non-decreasing function, its largest value will occur at the largest t-value

Clear[ybig];
ybig = arclength[tmax]

Clear[p1,p2]
p1 = Plot[
arclength[t],{t,tmin,tmax},
PlotRange-> {{tmin,tmax},{0,ybig}},Axes->True,AxesLabel->{" t"," arclength[t]"}
        ]

p2 = Plot[
arclength[t],{t,tmin,tmax},
PlotRange-> {{tmin,tmax},{0,ybig}},Axes->True
        ]

Manipulate[Show[GraphicsRow[{
Show[p1,
Graphics[{  {colorr,PointSize[.04],Point[{q,arclength[q]}] }    }],
PlotRange-> {{tmin,tmax},{0,ybig}},Axes->True,AxesLabel->{" t"," arclength[t]"}],
ParametricPlot[ 
{x[t],y[t]},{t,tmin,q},
PlotStyle-> { { colorr,Thickness[.02] } },
PlotRange-> {{xmin,xmax},{ymin,ymax}},Axes->True,AspectRatio->Automatic,
AxesLabel->{" x[t]","y[t]"}]
}]],{q,tmin+deltat,tmax,deltat}]

Do the circle first, since most of us know the circumference.  Then use other functions that we are less familiar with and are more interesting, such as {Cos[t], 3/50t*Sin[t]} and {(((t^3) - t)/((t^4) + 1)), (( t)/((t^4) + 1))} .

relate graph of the arclength function to its derivative, speed

Barr, Vector Calculus, 2nd Ed., Prentice Hall, Ch 5, Sec. 1, p276.
 "For each smooth path C there is a special parameterization that is occassionally important.  Given [vector valued function] f[t], a≤t≤b for C, we define the pathlength function to be L[t]= (∫_a^t)||f'[t]||dt, a≤t≤b.  Since f is smooth, the integral is continuous, and by the Fundamental Theorem of Calculus, L'[t]=||f'[t]||. ... The letter s is often used to represent the arclength function: s=(∫_a^t)||f'[t]||dt.  This means that (ds/dt)= ||f'[t]||, which can be interpreted physically as saying that the rate of change in the distance traveled by a particle with position function f is its speed."

D[arclength[t],t]

speed[t_]:=(Sqrt[(((x'[t]))^2) + (((y'[t]))^2)])

(Sqrt[(((x'[t]))^2) + (((y'[t]))^2)])

TrigExpand[((Cos[t])^2)+((Sin[t])^2)]

Clear[p3];
p3=Plot[
speed[t],{t,tmin,tmax},
PlotRange-> {{tmin,tmax},{0,ybig}},Axes->True,
PlotStyle->{{Dashing[{.05,.05}]}}
        ]

Use Cal I concepts to relate arclength to its dashed derivative, speed.  For instance, the arclength is non-decreasing, so the speed is always positive.

Show[p2,p3,PlotLabel->"solid is arclength; dashed is speed"]