kruz={Cos[t], Sin[t]}
kruzI={-Pi, Pi}

cycl={t-Sin[t],1-Cos[t]}
cyclI={-4Pi, 4Pi}

card={2Cos[t]-(Cos[t]^2-Sin[t]^2),2Sin[t]-2*Sin[t]*Cos[t]}
cardI={-Pi, Pi}

pl1=ParametricPlot[kruz,{t,kruzI[[1]], kruzI[[2]]}]

ParametricPlot[cycl,{t,cyclI[[1]], cyclI[[2]]}]

pl2=ParametricPlot[card,{t,cardI[[1]], cardI[[2]]}, PlotStyle->Magenta]

Show[pl1, pl2, PlotRange->All]

kruz2=kruz/.{Cos[t]->(1-s^2)/(1+s^2), Sin[t]->(2s)/(1+s^2)}

ParametricPlot[kruz2, {s, -100, 100}, PlotRange->All]

card2=Factor[card/.{Cos[t]->(1-s^2)/(1+s^2), Sin[t]->(2s)/(1+s^2)}]

ParametricPlot[card2, {s, -100, 100}, PlotRange->All]

temp=Numerator[Factor[{x,y}-card2]]

cardImpl=Factor[Resultant[temp[[1]], temp[[2]], s]]/4096

ContourPlot[cardImpl==0, {x,-3,3},{y,-3,3}]

speedCard2=Sqrt[Factor[Factor[D[card2,s]].Factor[D[card2,s]]]]

speedCard=Sqrt[Factor[Factor[D[card,t]].Factor[D[card,t]]]]

Integrate[speedCard, {t, -Pi, Pi}]

card

m=200
tBody=N[Table[cardI[[1]]+i/m*(cardI[[2]]-cardI[[1]]), {i,0,m}]];
cardBody=Table[card/.{t->tBody[[i]]}, {i,1,Length[tBody]}];
Sum[EuclideanDistance[cardBody[[i]], cardBody[[i+1]]], {i,1, Length[cardBody]-1}]

er2=16-15.9985

er1=16-15.9941

pl3=Graphics[Line[cardBody]];

Show[pl2, pl3]

Dcard=D[card, t]
speedCard=Simplify[Sqrt[Dcard.Dcard]]
TT=Dcard/speedCard
NN={{0,-1}, {1,0}}.TT

TT/.{t->Pi/2}
NN/.{t->Pi/2}

krCard=FullSimplify[Det[{Dcard, D[Dcard,t]}]/speedCard^3]

Plot[krCard, {t, -Pi, Pi}, AxesOrigin->{0,0}]

krCard2=FullSimplify[Det[{D[card2,s], D[card2,{s,2}]}]/(D[card2,s].D[card2,s])^(3/2)]

Plot[krCard2, {s, -10, 10}, AxesOrigin->{0,0}]

krCard/.{t->Pi/2}

Tan[Pi/4]

krCard2/.{s->1}

Plot[FullSimplify[(krCard2/.{s->Tan[t/2]})-krCard], {t,-1,1}]

kr0=N[krCard/.{t->-Pi}]

m=200
tBody=N[Table[cardI[[1]]+i/m*(cardI[[2]]-cardI[[1]]), {i,0,m}]];
cardBody=Table[card/.{t->tBody[[i]]}, {i,1,Length[tBody]}];
v2=cardBody[[2]]-cardBody[[1]]
v1=cardBody[[m+1]]-cardBody[[m]]
krApr=ArcSin[Det[{v1,v2}]/(Norm[v1]*Norm[v2])]*2/(Norm[v1]+Norm[v2])

0.3750424107830154`

er2=krApr-kr0

er1

er1/er2