ClearAll["Global`*"]

B[n_,i_]:=Binomial[n,i]*t^i*(1-t)^(n-i)

n=4

BB=Table[B[n,i],{i,0,n}]

Plot[BB,{t,0,1}]

P={{0,0}, {1,-2},{1, 3},{-2,3}, {-4,0}}

pl1=Graphics[Line[P]];

kk=Sum[P[[i]]*BB[[i]],{i,1, Length[P]}]

pl2=ParametricPlot[kk,{t,0,1}];

Show[pl1,pl2]

speed=Simplify[Norm[D[kk, t]], t>0]

L=N[Integrate[speed,{t,0,1}],40]

m=100

samply=Table[i/m, {i,0,m}]

body=Table[kk/.{t->samply[[i]]}, {i,1, Length[samply]}];

Lm=N[Sum[Sqrt[(body[[i]]-body[[i+1]]).(body[[i]]-body[[i+1]])], {i, 1, Length[body]-1}], 30]

er100=L-Lm

er20=0.0191096637529507475193127090852672598127082285658301612758

er20/er100

kk1=D[kk,t]
kk2=D[kk1,t]

tv=Simplify[kk1/Sqrt[kk1.kk1]]
nv={{0,-1},{1,0}}.tv

kz=Simplify[Det[{kk1, kk2}]/Sqrt[kk1.kk1]^3]

Plot[kz, {t,0,1}, PlotRange->All]

AT={{1-T^2,-2T},{2T,1-T^2}}/(1+T^2)

A=AT/.{T->7}

MatrixForm[A]

kk

kkP=A.kk+{3,-7}

pl3=ParametricPlot[kkP,{t,0,1}];

Show[pl2, pl3, PlotRange->All]

tt=1/5

kktt=kk/.{t->tt}
pl4=Graphics[{PointSize[0.02],Red,Point[kktt]}];
kztt=kz/.{t->tt}
Rtt=1/kztt
nvtt=nv/.{t->tt}
Stt=kktt+Rtt *nvtt
pl5=Graphics[{PointSize[0.02],Magenta,Point[Stt]}];
Oktt=Stt+Rtt*{Cos[α], Sin[α]}
pl6=ParametricPlot[Oktt, {α, -Pi,Pi}, PlotStyle->Gray];

Show[pl2, pl4,pl5, pl6, PlotRange->All]

nvtt

IOktt=(x-Stt[[1]])^2+(y-Stt[[2]])^2-Rtt^2

Series[IOktt/.{x->kk[[1]], y->kk[[2]]},{t, 1/5,5}]