ω² = c²(kₓ² + kθ²). A massive particle keeps kθ fixed at 2π/λ₀ (the fundamental internal mode), so its state runs along the hyperbola where the plane kθ = const cuts the cone.
At vₓ = 0 the point sits at the vertex of the hyperbola, on the ω axis at height ω₀ = m₀c² — pure rest energy. As vₓ increases it climbs the hyperbola (ω = γω₀, kₓ = βγ·kθ), approaching the cone's edge — the photon line ω = c·kₓ — as vₓ → c. The faint horizontal circle is the ω = const section of the cone the point also lies on; drag to rotate the view.