Standard DBM predicts when the CME nose reaches 1 AU. When Earth sits at angular offset θ from the CME axis it intercepts the side of the expanding front, which arrives later and moves more slowly.
What changed in v3.4. Versions up to v3.3 stacked a Vršnak cosine projection t_nose(1/cosθ − 1) underneath an ellipsoid shape term and added the two together. That sum had no closed-form justification, could not be reproduced from the printed report, and systematically under-predicted the delay. It has been replaced by the standard self-similar circular ("ice-cream cone") front, solved exactly.
R_apex / d = (1 + sin λ) / (cos Δ + √(sin²λ − sin²Δ))
A sphere of radius r = h·sinλ centred at distance h on the cone axis, apex at R = h(1+sinλ). Solving |d·û − h·n̂| = h sinλ gives the relation above: the radial distance the apex has reached at the moment the flank first touches a target at distance d and off-axis angle Δ.
Checks: Δ=0 gives R_apex = d (direct hit, zero delay). Δ=λ gives R_apex = d(1+sinλ)/cosλ — finite, no divergence at the cone edge. Δ>λ has no solution: Earth is outside the cone.
The delay is then the DBM travel time for the apex to cover R_apex − 1 AU, using the same drag parameter and wind as the main integration — not a geometric multiplier applied to the nose transit time.
v_front(Δ) = v_apex · d / R_apex
Because the front is self-similar in h, the local radial speed where it touches the target follows directly. A flank encounter is always slower than the apex. The simulator now uses this local front speed for shock formation, impact speed and geomagnetic coupling; earlier versions reused the nose speed, which was not geometrically self-consistent.
Not modelled: front flattening / pancaking. An axis ratio is computed and printed in the debug report as a diagnostic, but it is not applied to the arrival time. Treat wide, fast halos as having additional unmodelled delay uncertainty.