The bounded circle (β² + 1/γ² = 1) and the unbounded hyperbola (γ² − (βγ)² = 1) share the same vertex and the same angle. The circle stays finite; the hyperbola runs off — which is why rapidity, not angle, composes additively.
β = sin α = 0.00γ = 1.00φ = 0.00α = 0°
circular sector Φ/2 = ½α — the circular angle (bounded) hyperbolic sector Ψ/2 = ½φ — the rapidity (unbounded) compass arc — carries the long vector (length γ) onto the y-axis
The Gudermannian bridge. The circular angle α and the hyperbolic rapidity φ measure the same physical velocity through two different geometries, linked by the Gudermannian function: α = gd φ, equivalently sin α = tanh φ, cos α = sech φ, sec α = cosh φ = γ. The circular angle saturates at 90° as vₓ → c, while the rapidity grows without bound — which is exactly why φ, not α, adds when velocities are composed. The circle keeps the invariant finite; the hyperbola carries the additive parameter.
The point on the circle is (β, 1/γ); extend the same ray to the tangent line and rise to the hyperbola to reach (βγ, γ). As vₓ → c, the circle point stays on the unit circle while the hyperbola point escapes to infinity.