import { MatN, type PlaneRotation } from './matn.js';
import { VecN } from './vecn.js';
/**
 * An element of so(n) — the Lie algebra of N-dimensional rotations —
 * stored as coefficients over the coordinate-plane basis.
 *
 * Coefficient order: (0,1), (0,2), …, (0,n−1), (1,2), …, (n−2,n−1).
 * A coefficient θ on plane (i,j) generates rotation of axis i toward
 * axis j (matching `MatN.rotationInPlane`); there are n(n−1)/2 of them.
 *
 * Bivectors are the dimension-correct replacement for axis-angle vectors:
 * in 3D the plane count happens to equal the axis count (3), in 4D there
 * are 6 planes, and no "rotation axis" exists.
 */
export declare class BivectorN {
    readonly n: number;
    readonly coeffs: Float64Array;
    constructor(n: number, coeffs?: ArrayLike<number>);
    /** Flat index of plane (i,j) with i < j. */
    static planeIndex(n: number, i: number, j: number): number;
    static fromPlanes(n: number, planes: readonly PlaneRotation[]): BivectorN;
    /** Coefficient on plane (i,j); antisymmetric, so get(j,i) = −get(i,j). */
    get(i: number, j: number): number;
    set(i: number, j: number, value: number): this;
    clone(): BivectorN;
    scale(s: number): this;
    /**
     * The corresponding skew-symmetric matrix Ω, acting as dx = Ω·x:
     * Ω[j][i] = +θ_ij and Ω[i][j] = −θ_ij for i < j.
     */
    toSkewMatrix(): MatN;
}
/** Exterior product of two vectors: (a ∧ b)_ij = a_i b_j - a_j b_i. */
export declare function wedgeVectors(left: VecN, right: VecN): BivectorN;
/**
 * The exponential map so(n) → SO(n): converts an infinitesimal rotation
 * (or an angular-velocity bivector times a timestep) into a rotation
 * matrix.
 *
 * Implementation: scaling-and-squaring with a Taylor series — scale the
 * skew matrix down by 2^s until its norm is small, sum the series to
 * convergence, square s times, then re-orthonormalize to shed the last
 * bits of floating-point drift. Exact for the n=3 axis-angle case and the
 * single-plane case (verified against `MatN.rotationInPlane` in tests).
 */
export declare function expBivector(b: BivectorN): MatN;
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