How the main AMM families differ mathematically: constant product, concentrated liquidity, and stable curves

Automated market makers replace order books with curves, and each design's choice of curve determines how swap outputs are computed, how depth is measured, and where it can fail. The constant-product model keeps the product of two reserves fixed, offering a closed-form output that depends only on balances and fees, with liquidity spread across all prices. Concentrated-liquidity pools instead place capital within chosen price bands, behaving like constant-product pools with larger virtual reserves inside those bands. That approach has a hazard: a single-band formula assumes liquidity continues beyond the band, so it can quote outputs larger than the pool actually holds. Stable-asset curves are designed for assets that should trade near parity, concentrating depth around the balance point and steepening away from it. Because these families are not interchangeable, integrators should treat their formulas and depth measures as distinct.