equation solving - Intersection of two similar parametric curves

I need to do something similar to this link down here, but i was not able to adapt it to my situation.

How to get intersection values from a parametric graph?

I have this as a "inicialization cell".

Di = 20;  (*mm*)
R = Di/2;  (*mm*)

nd = 2;
ϕ = 2*Pi/nd;
Vcor = 10;  (*m/min*)
Vava = 0.5;  (*m/min*)
Vava2 = 0.5*1000/60;  (*mm/s*)
f = N[varn /. Solve[Vcor == Pi*Di*varn/1000, varn][[1]]];  (*rpm*)
ω = 2*Pi*f/60;  (*rad/s*)
fz = varfz /. Solve[Vava == varfz*nd*f, varfz][[1]];  (*mm*)

My equations are:

xi[t_, i_] := R*Cos[-ω*t + i*ϕ]
yi[t_, i_] := Vava2*t + R*Sin[-ω*t + i*ϕ]

I want to intersect:

{ xi[t,0], yi[t,0] } == { xi[t + dt,1], yi[t + dt,1] }

I tried this:

Reduce[xi[t, 0] == xi[t + dt, 1] && yi[t, 0] == yi[t + dt, 1] &&
  0 <= t <= (2*Pi/\[Omega]) && 0 <= dt <= (2*Pi/\[Omega])
 , {t, dt}]

but it could give a result after some minutes, i assumed it was wrong.

Any help is appreciated. Thanks in advance

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