S=2{Cos[t], Sin[t]}
v=-{Cos[t]^2-Sin[t]^2, 2Sin[t]Cos[t]}

X=S+v

ParametricPlot[X, {t,-Pi,Pi}]

dX=D[X,t]
ddX=D[dX,t]

r=Sqrt[FullSimplify[dX.dX]]
kz=FullSimplify[Det[{dX, ddX}]/r^3]

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

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

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

{x,y}==Xs

rov1=Numerator[Factor[x-Xs[[1]]]]
rov2=Numerator[Factor[y-Xs[[2]]]]

Ximpl=Factor[Resultant[rov1, rov2, s]]/4096

ContourPlot[Ximpl==0, {x,-4,4},{y,-4,4}]

Solve[Factor[Grad[Ximpl, {x,y}]]=={0,0}]

Plot3D[{Ximpl, 0}, {x,-3.1,3.1},{y,-3,3}]

X

m=2

samply=N[Table[{t->-Pi+i *2Pi/m}, {i,0,m}]]

Body=X/.samply

delky=Sum[Norm[Body[[i+1]]-Body[[i]]], {i, 1, m}]

X

A={{Cos[g], -Sin[g]}, {Sin[g], Cos[g]}}/.{g->Pi/4}
p={1, 2}

Xzobr=A.X+p

pl1=ParametricPlot[X, {t,-Pi,Pi}]
pl2=ParametricPlot[Xzobr, {t,-Pi,Pi}, PlotStyle->Magenta]

Show[pl1,pl2, PlotRange->All]

par={t, t^2}

Show[ParametricPlot[par, {t, -2,2}],ParametricPlot[offset, {t, -2,2}, PlotStyle->Magenta], PlotRange->All]

dpar=D[par, t]
rpar=Sqrt[dpar.dpar]

T=dpar/rpar

offset=par+{{0,1},{-1,0}}.T/2

X
r
kz

bod={t->Pi/2}

Xb=X/.bod

R=1/(kz/.bod)

n=({{0,-1},{1,0}}.dX/r)/.bod

N[n.n]

S=Xb+R*n

oskKr=S+R{Cos[u], Sin[u]}

pl3=ParametricPlot[oskKr, {u, -Pi,Pi}, PlotStyle->Darker[Green]]

Show[pl1, pl3, Graphics[{PointSize[0.02],Point[Xb]}]]

X

oskKr

Series[X, {t, Pi/2, 4}]

vetve=Solve[Ximpl==0, y]

vetve/.{x->1}

Xfu=y/.vetve[[4]]

Plot[Xfu,{x, -3, 1.5}]

oskKrIm=({x,y}-S).({x,y}-S)-R^2

OskFu=y/.Solve[oskKrIm==0, y][[2]]

Plot[{Xfu, OskFu},{x, -3, 1.5}]

Series[Xfu, {x, 1, 5}]
Series[OskFu, {x, 1, 5}]