Catenary

catenary[s_]:={Log[s+Sqrt[1+s^2]],Sqrt[1+s^2]-1}

bound=6;
ParametricPlot[catenary[s],{s,-bound,bound},AspectRatio->{1,1},PlotRange->All]

catenaryGraph[x_]:=Cosh[x]-1

bound=3;
Plot[catenaryGraph[x],{x,-bound,bound},AspectRatio->{1,1}]

Problem 1-1-2

curve[s_]:={2 ArcTan[s]-s, Log[1+s^2]}

bound=10;
ParametricPlot[curve[s],{s,-bound,bound},
	AspectRatio->{1,1}]

Clothoid

bound=2Pi; num=100;
clx=Table[N[{b,NIntegrate[Cos[u^2/2],{u,0,b}]}],{b,-bound,bound,bound/num}]//Interpolation;
cly=Table[N[{b,NIntegrate[Sin[u^2/2],{u,0,b}]}],
    {b,-bound,bound,bound/num}]//Interpolation;
ParametricPlot[{clx[t],cly[t]},{t,-bound,bound}]

Problem 1 - 2

Clear[a]

bound=4Pi;

per[a_,b_]:=(
perx=Table[N[{u,NIntegrate[Cos[a Sin[t]+b t],{t,0,u},
	AccuracyGoal->5]}],
	{u,0,bound,bound/100}]//Interpolation;
pery=Table[N[{u,NIntegrate[Sin[a Sin[t]+b t],{t,0,u},
	AccuracyGoal->5]}],
	{u,0,bound,bound/100}]//Interpolation;
ParametricPlot[{perx[t],pery[t]},{t,0,bound}]
)

per[3,0.2]

b=1;
period=Table[{a,NIntegrate[Cos[a Sin[t]+b t],{t,0,2Pi},AccuracyGoal->5]},{a,0,10,1/10}]//Interpolation;

Plot[period[a],{a,0,10}]

cl=FindRoot[period[a]==0,{a,3,5}]

per[a/.cl,1]

cl=FindRoot[period[a]==0,{a,6,8}]

per[a/.cl,1]