TaurusSwapDOCS

Polar Decomposition

Polar decomposition is the mathematical trick that makes O(1) on-chain verification possible. Instead of tracking the full n-dimensional reserve vector, the contract only needs two scalar values.

02 · Polar Decomposition
Reserves decomposed into α (equal-price axis) and w (orthogonal trading component).

The Decomposition

We write the reserve vector as:

x = αv + w

where:

  • v = (1/√n, 1/√n, ..., 1/√n) is the unit vector along the equal-price direction
  • α = (∑ᵢ xᵢ)/√n is the projection of x onto v
  • w = x − αv is the component orthogonal to v

Geometrically, α measures "how far along the equal-price axis" the pool is, while w captures the "imbalance" — how far from equal prices the pool has moved.

Deriving α

The projection of x onto v is:

α = x · v = ∑ᵢ xᵢ(1/√n) = (∑ᵢ xᵢ)/√n

This is just the sum of reserves, scaled by √n. The contract tracks ∑ᵢxᵢ directly, so α is free to compute.

The Orthogonal Component

The squared norm of w is:

‖w‖² = ‖x − αv‖²
= ‖x‖² − 2α(x · v) + α²‖v‖²
= ‖x‖² − 2α² + α² (since ‖v‖² = 1 and x · v = α)
= ‖x‖² − α²

Substituting α = (∑xᵢ)/√n:

‖w‖² = ∑ᵢxᵢ² − (∑ᵢxᵢ)²/n

This is the variance formula! ‖w‖² measures how "spread out" the reserves are from their mean. When all reserves are equal, ‖w‖² = 0.

Why This Matters

The contract stores only two values:

  • sumX = ∑ᵢxᵢ
  • sumXSq = ∑ᵢxᵢ²

From these, it can compute:

α = sumX/√n
‖w‖² = sumXSq − sumX²/n

This is O(1) state — constant size regardless of how many tokens are in the pool. Without this decomposition, the contract would need to store all n reserves explicitly.

Trading in Polar Coordinates

A swap of Δxᵢ tokens (selling token i) changes the state as follows:

sumX' = sumX + Δxᵢ − Δxⱼ
sumXSq' = sumXSq + (xᵢ + Δxᵢ)² − xᵢ² + (xⱼ − Δxⱼ)² − xⱼ²

The contract verifies that the new state satisfies the torus invariant. The actual trade computation (solving for Δxⱼ given Δxᵢ) happens off-chain in the SDK.

Connection to the Sphere

The sphere invariant can be rewritten in polar coordinates:

∑ᵢ(r − xᵢ)² = r²
nr² − 2r·sumX + sumXSq = r²
sumXSq − 2r·sumX + (n−1)r² = 0

This form is what the contract actually checks — it's algebraically equivalent to the sphere equation but uses the tracked values directly.

Key insight: Polar decomposition turns an n-dimensional problem into a 2-dimensional one. The contract never sees the full reserve vector — only its projection (α) and its orthogonal norm (‖w‖).
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