For any given arrangement of points, there is only one region in which an unbounded curve can meet all four tangent lines. It is shaded in blue.

The curve is oriented. The direction of increasing t can be changed by continuous manipulation (crossing the curve over itself) or pushing the curve to another region. But if the curve remains in the same region as two lines cross, the locus point does not flip sides.

Let the tangent points fall in a sequence α, β,γ δ. (There are two directions. Choose one). The order of dummy points P, Q, R, S, T, X determines the order α, β,γ δ, and changes dynamically to maintain it.

The formula describes the curve as Archimedes ΔαQδ, which encloses the (smallest) parabola section containing all four tangent points