Switch the liquidity shape to a triangle

[0xOmarA]

I would recommend not reviewing the diff of packages/caviarnine-v1-adapter-v1/src/lib.rs but just review the latest implementation of open_liquidity_position. For this case, the change is quite large that the diff might not be a good thing to look at.

For tests it would be good to look at the diff. I've added tests to ensure that the liquidity (L = sqrt(K)) in the bins is equal by calculating the standard deviation since they're not strictly equal.

[iamyulong]

Do we have some visualization for curve y = sqrt(tick_to_price(x + 1)) - sqrt(tick_to_price(x))? Trying to relate this to the triangular shape.

[0xOmarA]

I have graphed it out here: https://www.desmos.com/calculator/krrqdfifeg and here https://docs.google.com/spreadsheets/d/1LiD_XfjvgXuGStU2KPWhMJglLYYnuy6zJq2fgvg84Mg/edit?usp=sharing

Let me provide more context on the triangle shape and why it was chosen: Caviar based their calculations around the fact that at every price point along an xy = k curve the k is equal and doesn't change. So, they started with this premise and started working backwards to determine how much should be added to each bin such that the k in each bin is equal. They also used the fact that all bins at a lower price than the current price contain only Y and all bins above the current price contain only X.

They chose an arbitrary K to be base their calculation on, let's say 500 and used the equations below to determine how much should each bin have:

$$ L = \sqrt{k} $$

$$ x = L \frac{ \sqrt{p_b} - \sqrt{p_a} }{ \sqrt{p_a} \times \sqrt{p_b} } $$

$$ y = L ( \sqrt{p_b} - \sqrt{p_a} ) $$

In the end, after calculating the amount they got that they should be a triangle.

I follow a very similar approach in the code here where the code doesn't actually say "get me a triangle formation", rather the code is contributing the amount of resources that we need to get an equal K in all of the bins and doesn't care what shape that gives us (but it results in a triangle as seen in the second link I shared above)

[iamyulong]

Wonder if it's possible to get some test vectors from Caviarnine and assert_eq the bin contributions?

[0xOmarA]

They've already confirmed that that graphs here are what they expect.

[iamyulong]

Any documentation for the division above?

[0xOmarA]

Yup, this is derived from the equation that I named $L_x$ in the comments which is as follows:

$$ L_x = x\dfrac{\sqrt{p_a}\sqrt{p_b}}{\sqrt{p_b} - \sqrt{p_a}} $$

We can modify the equation above and use it to find $x$ which will be:

$$ x = L \frac{ \sqrt{p_b} - \sqrt{p_a} }{ \sqrt{p_a} \times \sqrt{p_b} } $$

You can also find these equations here as equations 4 and 5.

[0xOmarA]

Merging this and will try to get a Caviarnine test case in later on.