A={-1,0}
B={0,t}

p=A+s*(B-A)

res=Solve[Factor[p.p-1]==0, s][[2]]

k=Together[p/.res]

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

k/.{t->1/2}

dk=Together[D[k,t]]

speed=Simplify[Sqrt[dk.dk], t>0]

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

k

T=FullSimplify[ToRadicals[Tan[Pi/8]]]

FullSimplify[k/.{t->T}]

G=(AA*s+BB)/(CC*s+DD)

sys={(G/.{s->0})==0,(G/.{s->1})==1,(G/.{s->1/2})==T}

res=Solve[sys][[1]]

Φ=Factor[G/.res]

Plot[Φ,{s,0,1}, AspectRatio->Automatic]

kk=Simplify[Factor[k/.{t->Φ}]]

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

FullSimplify[kk/.{s->1/2}]

speed=Sqrt[Factor[D[kk, s].D[kk, s]]]

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

kk

L=N[Pi/2,30]

m=32

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

body=Table[kk/.{s->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]

L-Lm