# 04: Elliptic Concentrated Liquidity Pool (E-CLP)

## Source
- **Protocol**: Gyroscope
- **Files**: `GyroECLPMath.sol` (53KB), `eclp_float.py`, `eclp_prec_implementation.py`
- **Repos**: `gyrostable/gyro-pools`, `balancer/balancer-v3-monorepo`
- **Paper**: `gyrostable/technical-papers/E-CLP/E-CLP Mathematics.pdf`

## Rationale for MYCO

The E-CLP is the most mathematically sophisticated AMM curve in production. It adds two degrees of freedom beyond the 2-CLP:

1. **Rotation (φ)**: Tilts the price curve, useful when the target price ratio isn't 1:1
2. **Stretching (λ)**: Controls how elongated the ellipse is — higher λ = more concentrated near the peg, like a flattened oval

For MYCO:
1. The A-matrix transformation is the **key abstraction** that generalizes to N dimensions
2. Rotation handles non-unit price targets (e.g., ETH/USDC at $3000)
3. Stretching controls how much slippage increases as reserves deviate from target
4. Production-tested on Balancer V3 with formal security review and technical paper
5. The 5 parameters (α, β, c, s, λ) provide fine-grained curve shaping

## Invariant

$$|A(v - \text{offset})|^2 = r^2$$

where:
- $v = (x, y)$ are reserve balances
- $\text{offset} = (a, b)$ are virtual offsets (functions of $r$)
- $A$ is the 2×2 transformation matrix mapping ellipse → unit circle
- $r$ is the scalar invariant (liquidity parameter)

**A-matrix:**

$$A = \begin{pmatrix} c/\lambda & -s/\lambda \\ s & c \end{pmatrix}$$

**5 Parameters:**

| Param | Meaning | Range |
|-------|---------|-------|
| $\alpha$ | Lower price bound | (0, 1) typically |
| $\beta$ | Upper price bound | (1, ∞) typically |
| $c$ | $\cos(-\phi)$ — rotation | [0, 1] |
| $s$ | $\sin(-\phi)$ — rotation | [0, 1] |
| $\lambda$ | Stretching factor | [1, 10^8] |

Constraint: $c^2 + s^2 = 1$

**Derived parameters** (computed once, stored as immutables):
- $\tau(\alpha), \tau(\beta)$ — unit circle endpoints at 38-decimal precision
- $u, v, w, z$ — decomposition of $A \cdot \chi$ (center direction)
- $d_{Sq}$ — error correction for $c^2 + s^2$

## Swap Math

Given new x, solve for y on the ellipse via quadratic:

$$(A_{01}^2 + A_{11}^2) \cdot v^2 + 2(A_{00} A_{01} + A_{10} A_{11}) \cdot u \cdot v + (A_{00}^2 + A_{10}^2) \cdot u^2 = r^2$$

where $u = x - a$, $v = y - b$, solved with standard quadratic formula.

## Properties

- **Elliptical iso-invariant curves**: Level sets are ellipses (not circles or hyperbolas)
- **Concentrated + rotated**: Liquidity concentrated along the ellipse's long axis
- **Homogeneous degree 1**: $r(k \cdot v) = k \cdot r(v)$ — compatible with BPT math
- **Degenerates to 2-CLP**: When $\lambda = 1$ and $\phi = 0$, the ellipse becomes a circle

## MYCO Application

The E-CLP is the **2-asset specialization** of the full MYCO bonding surface. For any pair of reserve assets within the N-D surface, the local geometry is essentially an ellipse — the N-D surface is an N-D generalization of this.

The A-matrix pattern is the key insight: by encoding geometry as a linear transform, we can:
1. Compute invariants efficiently (quadratic formula, not iterative)
2. Parameterize curves with intuitive knobs (rotation, stretch)
3. Generalize to N dimensions naturally (N×N matrix)

## Implementation

See `src/primitives/elliptical_clp.py`