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// Copyright 2019 Google LLC.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
//

syntax = "proto3";

package google.type;

option cc_enable_arenas = true;
option go_package = "google.golang.org/genproto/googleapis/type/quaternion;quaternion";
option java_multiple_files = true;
option java_outer_classname = "QuaternionProto";
option java_package = "com.google.type";
option objc_class_prefix = "GTP";

// 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.
//
message Quaternion {
  // The x component.
  double x = 1;

  // The y component.
  double y = 2;

  // The z component.
  double z = 3;

  // The scalar component.
  double w = 4;
}