Both technologies follow logistic ("S-curve") performance-improvement trajectories over time t (years), the standard shape used in technology-maturation and disruptive-innovation modeling:
P(t) = P_start + (P_max − P_start) / (1 + e^(−k·(t − t_mid)))
The incumbent starts already partway up its curve — it has been sustaining-innovating for years, often adding performance beyond what most customers actually use (visible once its curve blows past the high-end demand plane). The entrant starts far below all three demand thresholds — cheap, simple, and "good enough" only for the least-demanding segment — but with a steeper rate kent, per Christensen's theory of disruptive innovation.
- Low-end / Mainstream / High-end planes — fixed performance thresholds each market segment actually needs. A curve piercing a plane means that segment's demand is now satisfied.
- kent vs kinc — when the entrant's improvement rate clearly exceeds the incumbent's, it eventually crosses every plane the incumbent already occupies: the disruption is complete.
- Crossover year — the first year t at which entrant performance ≥ incumbent performance, solved numerically by scanning the sampled curves.
- Play / Year slider — scrubs current time t; both curves are always fully plotted (dim), with the solid, bright segment showing "so far" and a marker ball at the current year.
Real-world relevance: this is the same qualitative pattern behind minicomputers → PCs, film → digital photography, and landline → mobile-first internet access — a low-end, low-margin entrant nobody in the incumbent's mainstream market took seriously improves faster than customer needs shift, and eventually captures the whole market.