c={((-1+(t^3))/(1+(t^2))),(t/2)-2 (t^2)}

ParametricPlot[c,  {t,-1,1}, AspectRatio->Automatic]

dc=Simplify[D[c,t]]

dc/.{t->1/8}

r=FullSimplify[Sqrt[dc.dc]]

Plot[r,  {t,-1,1}]

delka=NIntegrate[r,{t, -1,1}]

cs=c/.{t->Sin[s]}

ParametricPlot[cs, {s,-Pi/2,Pi/2}, PlotRange->All,AspectRatio->Automatic]

dcs=Factor[D[cs,s]]

rs=Sqrt[dcs.dcs]

Plot[rs, {s,-Pi/2,Pi/2}]

delkas=NIntegrate[rs,{s,-Pi/2,Pi/2}]

(* zacina novy priklad *)

k={Cos[α],Sin[α]}
p={-1,0}+u{1,t}

res=Solve[p[[1]]^2+p[[2]]^2==1,u]

kt=Together[p/.res[[2]]]

ParametricPlot[k,{α, -Pi,Pi}]

k
kt

Simplify[kt/.{t->Tan[α/2]}]

FullSimplify[(k/.{α->2ArcTan[t]})-kt]

Integrate[Sqrt[1+t^3],t]

dkt=D[kt, t]
rkt=FullSimplify[Sqrt[dkt.dkt], t∈Reals]

Plot[rkt, {t, -10,10}, PlotRange->All]

(* studujme ctvrt kruznice *)

kt

ParametricPlot[kt, {t,0,1}]

Plot[rkt, {t,0,1}, AxesOrigin->{0,0}]

kt/.{t->1/2}
FullSimplify[kt/.{t->-1+(Sqrt[2])}]

ks=Together[FullSimplify[Factor[kt/.{t->s/((1-(Sqrt[2]))*s+1-(1-(Sqrt[2])))}]]]

ParametricPlot[ks,{s,0,1}]

Simplify[ks/.{s->1/2}]

rks=FullSimplify[Sqrt[Factor[D[ks,s].D[ks,s],Extension->(Sqrt[2])]],s>0]

Plot[rks, {s,0,1}, AxesOrigin->{0,0}]

Together[Expand[N[ks]]]