/** Frame-rate-independent exponential decay multiplier: `exp(−rate·dt)`. */
export declare function expDecay(rate: number, dt: number): number;
/**
 * Invert a center/scale/rotation placement: screen-UV pointer → shape-local xy (y up).
 * The forward transform is `screen = center + R(rot)·(local·scale)/aspectLift`; this is its
 * exact inverse, with x lifted by aspect so the local frame is isotropic.
 */
export declare function pointerToShapeLocal(opts: {
    pointerX: number;
    pointerY: number;
    centerX: number;
    centerYv: number;
    scale: number;
    rotC: number;
    rotS: number;
    aspect: number;
}): {
    x: number;
    y: number;
};
/**
 * Body-frame angular velocity from consecutive euler angles (radians), for a field sampled at
 * `R·p`: features move at `−(Rᵀω)×r`, so the euler rates are conjugated by Rᵀ and negated.
 * Returns null on the first frame (no previous sample).
 */
export declare function omegaFromRotationDeltas(prev: {
    x: number;
    y: number;
    z: number;
} | null, next: {
    x: number;
    y: number;
    z: number;
}, dt: number): {
    x: number;
    y: number;
    z: number;
} | null;
/** Cap an omega vector's magnitude and derive the entrainment gain from it. */
export declare function entrainmentFromOmega(om: {
    x: number;
    y: number;
    z: number;
}, opts: {
    cap: number;
    gainRate: number;
    gainMax: number;
}): {
    omegaX: number;
    omegaY: number;
    omegaZ: number;
    entrain: number;
};
/**
 * A stateful smoothed-motion-direction tracker: the axis eases toward the pointer's travel
 * direction with a teleport guard — a jump wider than `teleport` world units in one frame is
 * the pointer re-entering the canvas, not a stroke, so the axis holds its previous heading.
 */
export declare function createMotionAxis(opts?: {
    smoothing?: number;
    teleport?: number;
}): {
    update(x: number, y: number): {
        x: number;
        y: number;
    };
};
/**
 * Fit ~`count` roughly-square cells over an overscanned `[−ov, domainX+ov] × [−ov, 1+ov]`
 * domain (the jittered home grid — off-screen agents backfill pulled-in edges).
 */
export declare function fitJitteredGrid(count: number, domainX: number, overscan: number): {
    cols: number;
    cellW: number;
    cellH: number;
};
/** Fit an isotropic W×H agent grid to a count and aspect, each axis clamped to [min, max]. */
export declare function fitIsoGrid(count: number, aspect: number, min: number, max: number): {
    w: number;
    h: number;
};
/**
 * Camera rotation matrix rows (Rz·Ry·Rx), conjugated for screen space (y down): the y-up
 * matrix with the off-diagonal y terms sign-flipped, so positive angles read as the natural
 * orbit directions in the editor. Angles in radians.
 */
export declare function cameraRowsYDown(rx: number, ry: number, rz: number): {
    rowX: [number, number, number];
    rowY: [number, number, number];
    rowZ: [number, number, number];
};
/**
 * The R3 sub-cell dither for a voxel lattice: frame `i`'s offset, scaled by one cell — evenly
 * covers the sub-cell cube over frames, so a grid can never stand in coherent moiré with a
 * boundary. (See `R3_ALPHA` on the harness.)
 */
export declare function r3SubCellOffset(frameIdx: number, cell: number): {
    x: number;
    y: number;
    z: number;
};
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