/**
 *  A quaternion is defined as the quotient of two directed lines in a
 *  three-dimensional space or equivalently as the quotient of two Euclidean
 *  vectors (https://en.wikipedia.org/wiki/Quaternion).
 * 
 *  Quaternions are often used in calculations involving three-dimensional
 *  rotations (https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation),
 *  as they provide greater mathematical robustness by avoiding the gimbal lock
 *  problems that can be encountered when using Euler angles
 *  (https://en.wikipedia.org/wiki/Gimbal_lock).
 * 
 *  Quaternions are generally represented in this form:
 * 
 *      w + xi + yj + zk
 * 
 *  where x, y, z, and w are real numbers, and i, j, and k are three imaginary
 *  numbers.
 * 
 *  Our naming choice `(x, y, z, w)` comes from the desire to avoid confusion for
 *  those interested in the geometric properties of the quaternion in the 3D
 *  Cartesian space. Other texts often use alternative names or subscripts, such
 *  as `(a, b, c, d)`, `(1, i, j, k)`, or `(0, 1, 2, 3)`, which are perhaps
 *  better suited for mathematical interpretations.
 * 
 *  To avoid any confusion, as well as to maintain compatibility with a large
 *  number of software libraries, the quaternions represented using the protocol
 *  buffer below *must* follow the Hamilton convention, which defines `ij = k`
 *  (i.e. a right-handed algebra), and therefore:
 * 
 *      i^2 = j^2 = k^2 = ijk = −1
 *      ij = −ji = k
 *      jk = −kj = i
 *      ki = −ik = j
 * 
 *  Please DO NOT use this to represent quaternions that follow the JPL
 *  convention, or any of the other quaternion flavors out there.
 * 
 *  Definitions:
 * 
 *    - Quaternion norm (or magnitude): `sqrt(x^2 + y^2 + z^2 + w^2)`.
 *    - Unit (or normalized) quaternion: a quaternion whose norm is 1.
 *    - Pure quaternion: a quaternion whose scalar component (`w`) is 0.
 *    - Rotation quaternion: a unit quaternion used to represent rotation.
 *    - Orientation quaternion: a unit quaternion used to represent orientation.
 * 
 *  A quaternion can be normalized by dividing it by its norm. The resulting
 *  quaternion maintains the same direction, but has a norm of 1, i.e. it moves
 *  on the unit sphere. This is generally necessary for rotation and orientation
 *  quaternions, to avoid rounding errors:
 *  https://en.wikipedia.org/wiki/Rotation_formalisms_in_three_dimensions
 * 
 *  Note that `(x, y, z, w)` and `(-x, -y, -z, -w)` represent the same rotation,
 *  but normalization would be even more useful, e.g. for comparison purposes, if
 *  it would produce a unique representation. It is thus recommended that `w` be
 *  kept positive, which can be achieved by changing all the signs when `w` is
 *  negative.
 */
export interface Quaternion {
    /**  The x component. */
    x?: number

    /**  The y component. */
    y?: number

    /**  The z component. */
    z?: number

    /**  The scalar component. */
    w?: number
}

export namespace Quaternion {
    export const name = 'google.type.Quaternion'
}