Complex tangency locus (phantom curves)

Sweep a family of parallel lines across a real curve and follow all of its intersections in ℂ²: the real curve and its phantom branches form one connected locus, glued at the tangency points.

Inigo Quilez — “Intersections are always there” (original Short) Carlo Perassi — reply Short Source repository Theory (PDF) Phantom loci (PDF) 16 worked examples (PDF) Input × mirror classification (PDF)

Explorer

Floor = the real plane; height = Im t. Drag the picture to rotate.

Real curve (Im t = 0)Phantom branches (Im t ≠ 0)Tangency points

Real plane

Dashed: the current line. Hollow rings mark the real part of each complex intersection, the “shadow” of the phantom point on the line.

Intersections at current c

#	t	x	y	Type

Setup

Fix a direction d = (cos θ, sin θ) and its normal n = (−sin θ, cos θ). Every line of the family is n·p = c with c real, parametrised by t:

p(t) = c n + t d, P_c(t) = F(c n + t d) = a_kt^k + … + a₀(c)

For a real polynomial F of degree k, this is a degree-k polynomial in t with real coefficients, and the leading coefficient depends only on d. Allowing t ∈ ℂ gives exactly k intersections in ℂ² for every c (counted with multiplicity; fewer if d is an asymptotic direction), and sweeping c traces a real 1-dimensional locus on the complex curve. The picture plots it as (c, Re t, Im t), with the floor being the real plane itself.

Why conjugate pairs

P_c(t̄) = P_c(t) ⇒ roots are real or a ± ib

The coefficients are real, so complex roots come in conjugate pairs. That is why the phantom branches are mirror-symmetric about the floor, and why the ℂ² points come as (x, y) and (x̄, ȳ).

Branches are born at tangencies

Disc_t(P_c) = 0 ⇔ F = 0 and ∇F · d = 0

A conjugate pair can only leave the real plane where two real roots collide, i.e. where the discriminant vanishes. That is where the line is tangent to the real curve (or passes through a singular point): the critical points of the height c = n·p restricted to the curve. The tangency dots are found by bisecting on the number of real roots.

Perpendicular departure, same curvature

c ≈ c₀ + (κ/2) t² ⇒ t = ± √(2(c − c₀)/κ); t = i s: c = c₀ − (κ/2) s²

On one side of c₀ the two roots are real (the curve), on the other they are purely imaginary to second order. The phantom branch leaves at a right angle to the real curve with the same curvature κ, bending the other way. At an inflection tangent the local form is t³ instead, and three branches meet (try the inflection example: cubic, θ = −45°, c = 0).

Circle and hyperbola

t² + c² = r², t = i s ⇒ c² − s² = r²

For the circle, lines beyond the tangency points (|c| > r) meet the complex circle at t = ± i √(c² − r²). In the vertical plane through the tangency points that is the hyperbola c² − s² = r²: the circle and the hyperbola are two real slices of one complex conic, glued at the tangency points. The parabola y = x² with horizontal lines gives the phantom parabola y = −s² in the same way.

Reading the picture. Changing θ rotates the family of lines, so the phantom branches swing around the curve and always attach at the current tangency points.