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# Cross-pair Swaps

An additional disadvantage of the pair-based structure is its lack of capital efficiency. Liquidity provided to one pair is restricted to usage within that pool alone. Even if an asset is staked in large quantities within the AMM, disproportionate distribution across the pairs results in some pairs having low liquidity and higher slippage.

Centaur Swap is designed to enable asset pools to trade with each other in a seamless manner. This means that once an asset pool is funded, it is effectively paired against every other existing asset pool. Based on the curve described in the previous segment, the price for each token against USDT or ETH can be calculated at any point in time, while taking into account the

**Current Balance**of each individual pool. The price of any two tokens can then be used as a reference to determine an exchange rate for a direct swap.Consider a swap from

**Token A**, which is currently in**Deficit**to**Token B**, which is currently in**Surplus**.Curve of Token A with its current balance at A1

Due to the

**Deficit**of**Token A**, its price at point**is above***A1**, the***i****External Price**and traders are incentivised to sell**Token A**. The intended quantity of tokens sold by the trader would increase the**Current Balance**to point*and the value would be represented by the shaded area under the curve ("***A2****|A|"**).By integrating the curve

$y = -kx^3 + i$

, the value for**|A|**can be calculated as:$\begin{aligned}
|A| &= \int_{x_1}^{x_2}(i-kx^3)dx \\
&= (ix_2 - \frac{k{x_2}^4} {4}) - (ix_1 - \frac{k{x_1}^4} {4}) \\
&=i(x_2 - x_1) - \frac{k}{4}({x_2}^4 - {x_1}^4)
\end{aligned}$

where the

$x_1$

and $x_2$

represent the values of *and***A1***.***A2**Curve of Token B with its current balance at B1

After deriving

**|A|**, the its value can be projected onto the curve of**Token B**, which is currently in**Surplus**at point*. As token swaps must result in an equivalent exchange of value, the shaded area between points***B1***and***B1***("***B2****|B|**") will be exactly the same as**|A|**. With that, we can solve for**B2**to calculate the new**Current Balance**after the trade is executed, and after dividing the quantity of**Token B**received by the quantity of**Token A**sold, we can determine the exchange rate for any two tokens liquidity pool in a manner that does not rely on pairing.Last modified 2yr ago