public sealed class Quaternion : IMessage<Quaternion>, IEquatable<Quaternion>, IDeepCloneable<Quaternion>, IBufferMessage, IMessage
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 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.
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.
[[["Easy to understand","easyToUnderstand","thumb-up"],["Solved my problem","solvedMyProblem","thumb-up"],["Other","otherUp","thumb-up"]],[["Hard to understand","hardToUnderstand","thumb-down"],["Incorrect information or sample code","incorrectInformationOrSampleCode","thumb-down"],["Missing the information/samples I need","missingTheInformationSamplesINeed","thumb-down"],["Other","otherDown","thumb-down"]],["Last updated 2025-08-28 UTC."],[[["\u003cp\u003eQuaternions are used for three-dimensional rotations, offering robustness by avoiding gimbal lock issues encountered with Euler angles.\u003c/p\u003e\n"],["\u003cp\u003eQuaternions are represented in the form \u003ccode\u003ew + xi + yj + zk\u003c/code\u003e, and the \u003ccode\u003e(x, y, z, w)\u003c/code\u003e naming convention is chosen for clarity in 3D Cartesian space, following the Hamilton convention.\u003c/p\u003e\n"],["\u003cp\u003eKey quaternion definitions include: norm (magnitude), unit (normalized) quaternion, pure quaternion, rotation quaternion, and orientation quaternion.\u003c/p\u003e\n"],["\u003cp\u003eNormalization of a quaternion, by dividing by its norm, is essential for rotation and orientation to maintain accuracy and avoid rounding errors.\u003c/p\u003e\n"],["\u003cp\u003eThe \u003ccode\u003e(x, y, z, w)\u003c/code\u003e and \u003ccode\u003e(-x, -y, -z, -w)\u003c/code\u003e pairs represent the same rotation; however, keeping \u003ccode\u003ew\u003c/code\u003e positive is recommended for a unique representation upon normalization.\u003c/p\u003e\n"]]],[],null,["# Class Quaternion (2.17.0)\n\nVersion latestkeyboard_arrow_down\n\n- [2.17.0 (latest)](/dotnet/docs/reference/Google.Api.CommonProtos/latest/Google.Type.Quaternion)\n- [2.15.0](/dotnet/docs/reference/Google.Api.CommonProtos/2.15.0/Google.Type.Quaternion)\n- [2.10.0](/dotnet/docs/reference/Google.Api.CommonProtos/2.10.0/Google.Type.Quaternion)\n- [2.2.0](/dotnet/docs/reference/Google.Api.CommonProtos/2.2.0/Google.Type.Quaternion) \n\n public sealed class Quaternion : IMessage\u003cQuaternion\u003e, IEquatable\u003cQuaternion\u003e, IDeepCloneable\u003cQuaternion\u003e, IBufferMessage, IMessage\n\nA quaternion is defined as the quotient of two directed lines in a\nthree-dimensional space or equivalently as the quotient of two Euclidean\nvectors (\u003chttps://en.wikipedia.org/wiki/Quaternion\u003e).\n\nQuaternions are often used in calculations involving three-dimensional\nrotations (\u003chttps://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation\u003e),\nas they provide greater mathematical robustness by avoiding the gimbal lock\nproblems that can be encountered when using Euler angles\n(\u003chttps://en.wikipedia.org/wiki/Gimbal_lock\u003e).\n\nQuaternions are generally represented in this form: \n\n w + xi + yj + zk\n\nwhere x, y, z, and w are real numbers, and i, j, and k are three imaginary\nnumbers.\n\nOur naming choice `(x, y, z, w)` comes from the desire to avoid confusion for\nthose interested in the geometric properties of the quaternion in the 3D\nCartesian space. Other texts often use alternative names or subscripts, such\nas `(a, b, c, d)`, `(1, i, j, k)`, or `(0, 1, 2, 3)`, which are perhaps\nbetter suited for mathematical interpretations.\n\nTo avoid any confusion, as well as to maintain compatibility with a large\nnumber of software libraries, the quaternions represented using the protocol\nbuffer below *must* follow the Hamilton convention, which defines `ij = k`\n(i.e. a right-handed algebra), and therefore: \n\n i^2 = j^2 = k^2 = ijk = −1\n ij = −ji = k\n jk = −kj = i\n ki = −ik = j\n\nPlease DO NOT use this to represent quaternions that follow the JPL\nconvention, or any of the other quaternion flavors out there.\n\nDefinitions:\n\n- Quaternion norm (or magnitude): `sqrt(x^2 + y^2 + z^2 + w^2)`.\n- Unit (or normalized) quaternion: a quaternion whose norm is 1.\n- Pure quaternion: a quaternion whose scalar component (`w`) is 0.\n- Rotation quaternion: a unit quaternion used to represent rotation.\n- Orientation quaternion: a unit quaternion used to represent orientation.\n\nA quaternion can be normalized by dividing it by its norm. The resulting\nquaternion maintains the same direction, but has a norm of 1, i.e. it moves\non the unit sphere. This is generally necessary for rotation and orientation\nquaternions, to avoid rounding errors:\n\u003chttps://en.wikipedia.org/wiki/Rotation_formalisms_in_three_dimensions\u003e\n\nNote that `(x, y, z, w)` and `(-x, -y, -z, -w)` represent the same rotation,\nbut normalization would be even more useful, e.g. for comparison purposes, if\nit would produce a unique representation. It is thus recommended that `w` be\nkept positive, which can be achieved by changing all the signs when `w` is\nnegative. \n\nInheritance\n-----------\n\n[object](https://learn.microsoft.com/dotnet/api/system.object) \\\u003e Quaternion \n\nImplements\n----------\n\n[IMessage](https://cloud.google.com/dotnet/docs/reference/Google.Protobuf/latest/Google.Protobuf.IMessage-1.html)[Quaternion](/dotnet/docs/reference/Google.Api.CommonProtos/latest/Google.Type.Quaternion), [IEquatable](https://learn.microsoft.com/dotnet/api/system.iequatable-1)[Quaternion](/dotnet/docs/reference/Google.Api.CommonProtos/latest/Google.Type.Quaternion), [IDeepCloneable](https://cloud.google.com/dotnet/docs/reference/Google.Protobuf/latest/Google.Protobuf.IDeepCloneable-1.html)[Quaternion](/dotnet/docs/reference/Google.Api.CommonProtos/latest/Google.Type.Quaternion), [IBufferMessage](https://cloud.google.com/dotnet/docs/reference/Google.Protobuf/latest/Google.Protobuf.IBufferMessage.html), [IMessage](https://cloud.google.com/dotnet/docs/reference/Google.Protobuf/latest/Google.Protobuf.IMessage.html) \n\nInherited Members\n-----------------\n\n[object.Equals(object, object)](https://learn.microsoft.com/dotnet/api/system.object.equals#system-object-equals(system-object-system-object)) \n[object.GetType()](https://learn.microsoft.com/dotnet/api/system.object.gettype) \n[object.ReferenceEquals(object, object)](https://learn.microsoft.com/dotnet/api/system.object.referenceequals)\n\nNamespace\n---------\n\n[Google.Type](/dotnet/docs/reference/Google.Api.CommonProtos/latest/Google.Type)\n\nAssembly\n--------\n\nGoogle.Api.CommonProtos.dll\n\nConstructors\n------------\n\n### Quaternion()\n\n public Quaternion()\n\n### Quaternion(Quaternion)\n\n public Quaternion(Quaternion other)\n\nFields\n------\n\n### WFieldNumber\n\n public const int WFieldNumber = 4\n\nField number for the \"w\" field.\n\n### XFieldNumber\n\n public const int XFieldNumber = 1\n\nField number for the \"x\" field.\n\n### YFieldNumber\n\n public const int YFieldNumber = 2\n\nField number for the \"y\" field.\n\n### ZFieldNumber\n\n public const int ZFieldNumber = 3\n\nField number for the \"z\" field.\n\nProperties\n----------\n\n### Descriptor\n\n public static MessageDescriptor Descriptor { get; }\n\n### Parser\n\n public static MessageParser\u003cQuaternion\u003e Parser { get; }\n\n### W\n\n public double W { get; set; }\n\nThe scalar component.\n\n### X\n\n public double X { get; set; }\n\nThe x component.\n\n### Y\n\n public double Y { get; set; }\n\nThe y component.\n\n### Z\n\n public double Z { get; set; }\n\nThe z component.\n\nMethods\n-------\n\n### CalculateSize()\n\n public int CalculateSize()\n\n### Clone()\n\n public Quaternion Clone()\n\n### Equals(Quaternion)\n\n public bool Equals(Quaternion other)\n\n### Equals(object)\n\n public override bool Equals(object other)\n\n**Overrides** \n[object.Equals(object)](https://learn.microsoft.com/dotnet/api/system.object.equals#system-object-equals(system-object))\n\n### GetHashCode()\n\n public override int GetHashCode()\n\n**Overrides** \n[object.GetHashCode()](https://learn.microsoft.com/dotnet/api/system.object.gethashcode)\n\n### MergeFrom(CodedInputStream)\n\n public void MergeFrom(CodedInputStream input)\n\n### MergeFrom(Quaternion)\n\n public void MergeFrom(Quaternion other)\n\n### ToString()\n\n public override string ToString()\n\n**Overrides** \n[object.ToString()](https://learn.microsoft.com/dotnet/api/system.object.tostring)\n\n### WriteTo(CodedOutputStream)\n\n public void WriteTo(CodedOutputStream output)"]]