Find out by the real roots of the denominator:

The singularities, equivalent to the points at infinity of, are find out by the real roots of the denominator

ω(u) = (1 - u)²  + 2u (1 - u) ω₁ + u ²  = 0

They are specified by

The curve decrease to

1.      a straight line if space w₁ = 0

2.      an elliptic segment if 0 < w₁ <1

3.      a parabolic segment if w₁ = 1

4.      a hyperbolic segment if w₁ > 1

Geometrically, the curves are governed trhrough the Parameter s = ω₁/1 + ω₁ as

The curve C (u) flows through r₀  (= P₀ ) and r₂  (= P₂ ) with the slope from

Hence

the tangents meets at r₁=P₁