In the previous article, we compared the slippage difference between curve and uniswap V3 through data. We can find that under comparable conditions, the slippage of uniswap V3 is smaller than that of curve, but some people want to know why. So today we will start from the technical principles of curve and uniswap to analyze why the slippage of the curve is larger than that of uniswap V3. As we all know, curve adopts the mixed curve equation of constant product and constant sum. As for the constant sum market making curve, the exchange price has no slippage with limited liquidity, but as for the constant product market making curve, the exchange price exists slippage with unlimited liquidity. To achieve lower slippage of exchange for AMM, curve combines the above two curves and proposes a hybrid market making curve:

A(P0x+y)+xy=k

Wherein P0 is the balanced price

y=(k-P0Ax)/(X+A)

Derivation of the above formula can get the price

As parameter A approaches infinity

Therefore, the larger A , the closer the exchange price to P0, that is, the smaller the slippage,

According to the market making curve of uniswap V3,

Derivation of the above formula we can get the price

m is the magnification factor of x

as parameter m approaches infinity

It can be seen from the above that the exchange price of curve is strongly related to A, while the exchange price of uniswap V3 is strongly related to m, and we know that the amplification factor A in curve is determined internally by the protocol and cannot be changed by external users, while the amplification factor m of uniswap V3 is completely determined by the price range set by the external liquidity provider. In other words, the amplification factor m can theoretically be infinitely. However the amplification factor A of curve cannot be infinitely in order to ensure infinite liquidity. So the slippage of the curve must be larger than that of uniswap V3.