About

maxon::Quaternion is a Maxon API class used to represent quaternions and deliver standard mathematical operations. Unit quaternions, also known as versors, provide a convenient mathematical notation for representing orientations and rotations of objects in three dimensions. Compared to Euler angles they are simpler to compose and avoid the problem of gimbal lock. Compared to rotation matrices they are more compact, more numerically stable, and may be more efficient.

    // This example shows how to use a quaternion to rotate a point in a left-handed coordinate system.
  
    // define the point to rotate
    const maxon::Vector point (1, 0, 0);
    DiagnosticOutput(diagIDString + "start point: @", point);
    
    // define the rotation angle
    const maxon::Float rotAngle(90);
    
    // define the rotation axis
    const maxon::Vector rotAxis(0, 0, 1);
    
    // create a quaternion representing the point
    const maxon::Quaternion<maxon::Float> quaternionPointA(point, 0);
    DiagnosticOutput(diagIDString + "quaternionPointA: @", quaternionPointA);
    
    // create the quaternion represention the rotation (q)
    const maxon::Float a = rotAxis.x * sin(maxon::DegToRad(rotAngle/2));
    const maxon::Float b = rotAxis.y * sin(maxon::DegToRad(rotAngle/2));
    const maxon::Float c = rotAxis.z * sin(maxon::DegToRad(rotAngle/2));
    const maxon::Float d = cos(maxon::DegToRad(rotAngle/2));
    const maxon::Quaternion<maxon::Float> quaternion(a, b, c, d);
    DiagnosticOutput(diagIDString + "quaternion: @", quaternion);
    
    // get the conjugated quaternion (q')
    const maxon::Quaternion<maxon::Float> quaternionCnj = quaternion.GetConjugate();
    DiagnosticOutput(diagIDString + "quaternionCnj: @", quaternionCnj);
    
    // get the rotated point p1 = q * p * q'
    const maxon::Quaternion<maxon::Float> quanterionPointARot = quaternion * quaternionPointA * quaternionCnj;
    DiagnosticOutput(diagIDString + "quanterionPointARot: @", quanterionPointARot);
    
    // extract the rotation point
    const maxon::Vector rotPoint = quanterionPointARot.v;
    DiagnosticOutput(diagIDString + "rotated point: @", rotPoint);

Creation and initialization

A maxon::Quaternion instance can be created on the stack and initialized through the proper method using a numerical data type as template:

maxon::Quaternion::Quaternion(): creates a zero-valued quaternion instance on the stack;
maxon::Quaternion::Quaternion(const VectorType& v, const ValueType w): creates a quaternion instance whose vector part and real part are valued respectively with the passed parameters' values;
maxon::Quaternion::Quaternion(const ValueType x, const ValueType y, const ValueType z, const ValueType w): creates a quaternion instance whose four components are valued with the respectively passed parameters' values.

    // This example allocates a few maxon::Quaternion objects showing the different allocation methods.
    
    // just allocate an zero-valued quaternion object
    const maxon::Quaternion<maxon::Float> zeroQuaternion;
    
    // just allocate a quaternion object defining the vectorial part and the real part
    const maxon::Vector vectorVal(1.0, 1.0, 0);
    const maxon::Float floatVal(1.5);
    const maxon::Quaternion<maxon::Float> vectorRealQuaternion(vectorVal, floatVal);
    
    // just allocate a quaternion object defining the four components used to value vectorial and real part
    const maxon::Float a = 0.2; // 1st vectorial component
    const maxon::Float b = 0.4; // 2nd vectorial component
    const maxon::Float c = 0.6; // 3rd vectorial component
    const maxon::Float d = 0.8; // this the real part
    const maxon::Quaternion<maxon::Float> fourComponentsQuaternion(a, b, c, d);
    

Standard operations

The maxon::Quaternion class delivers a set of standard mathematical operators to perform calculation with quaternion. The operators deliver are:

maxon::Quaternion::operator+=(): right-assigns the sum of the current value and of the maxon::Quaternion used in the expression;
maxon::Quaternion::operator-=(): right-assigns the difference of the current value and of the maxon::Quaternion used in the expression;
maxon::Quaternion::operator*=(): right-assigns the product of the current value and of the maxon::Quaternion used in the expression;
maxon::Quaternion::operator+(): returns the sum of two maxon::Quaternion instances;
maxon::Quaternion::operator-(): returns the difference of two maxon::Quaternion instances;
maxon::Quaternion::operator*(): returns the product of two maxon::Quaternion instances;
maxon::Quaternion::operator*(const ValueType s) const: returns a maxon::Quaternion instance scaled by "s";
maxon::Quaternion::operator*(const ValueType s, const QuaternionType& q): scales the passed maxon::Quaternion instance "q" by the passed parameter's value "s".

    // This example shows how to perform simple mathematical operations between quaternions.
    
    // just allocate a constant quaternion using the four components
    const maxon::Quaternion<maxon::Float> quaternionA(1, 0, 0, 12);
    DiagnosticOutput(diagIDString + "quaternionA: @", quaternionA);
    
    // just allocate a quaternion equal to the previous one
    maxon::Quaternion<maxon::Float> quaternionB = quaternionA;
    DiagnosticOutput(diagIDString + "quaternionB: @", quaternionB);
    
    // sum A to B
    quaternionB += quaternionA;
    DiagnosticOutput(diagIDString + "quaternionB += quaternionA: @", quaternionB);
    
    // subtract A to B
    quaternionB -= quaternionA;
    DiagnosticOutput(diagIDString + "quaternionB -= quaternionA: @", quaternionB);
    
    // sum A and B
    const maxon::Quaternion<maxon::Float> quaternionAsumB = quaternionA + quaternionB;
    DiagnosticOutput(diagIDString + "quaternionAsumB : @", quaternionAsumB);
    
    // subtract A and B
    const maxon::Quaternion<maxon::Float> quaternionAdifB = quaternionA - quaternionB;
    DiagnosticOutput(diagIDString + "quaternionAdifB: @", quaternionAdifB);
    
    // multiply A and B
    const maxon::Quaternion<maxon::Float> quaternionAprdB = quaternionA * quaternionB;
    DiagnosticOutput(diagIDString + "quaternionAprdB: @", quaternionAprdB);
    
    // multiply B and A
    const maxon::Quaternion<maxon::Float> quaternionBprdA = quaternionB * quaternionA;
    DiagnosticOutput(diagIDString + "quaternionBprdA: @", quaternionBprdA);
    
    // scale A by scalar s
    const maxon::Float scalar = 2;
    const maxon::Quaternion<maxon::Float> quaternionAscaled = quaternionA * scalar;
    DiagnosticOutput(diagIDString + "quaternionAscaled: @", quaternionAscaled);

Get and Set operations

The maxon::Quaternion class is provided with a number of get methods to retrieve useful data from a maxon::Quaternion instance:

maxon::Quaternion::GetMatrix(): returns the rotation matrix computed from the quaternion instance;
maxon::Quaternion::GetAxisRotation(): calculate axis and rotation angle from quaternion and returns an error if the quaternion is not normalized or the direction of the axis is undefined;
maxon::Quaternion::GetSquaredLength(): gets the squared length of the quaternion;
maxon::Quaternion::GetLength(): gets the length of the quaternion;
maxon::Quaternion::GetNormalized(): gets the quaternion with normalized length and returns an error if the quaternion length is zero;
maxon::Quaternion::GetInverse(): gets the inverse of a quaternion and returns an error if the quaternion's length is zero;
maxon::Quaternion::GetConjugate(): gets the conjugated quaternion;
maxon::Quaternion::GetLog(): gets the natural logarithm of a quaternion and returns an error if the quaternion's length is zero;
maxon::Quaternion::GetExp(): gets the exponential of a quaternion and returns an error if the quaternion's length is zero;
maxon::Quaternion::GetPow(): gets the power of quaternion by the passed value and returns an error if the quaternion's length is zero;
maxon::Quaternion::GetDot(): gets the "dot" product of two quaternions;
maxon::Quaternion::GetLerp(): gets the linear interpolation of two quaternions using the passed blend parameter;
maxon::Quaternion::GetSlerp(): gets the spherical linear interpolation of two quaternions using the passed blend parameter;
maxon::Quaternion::GetSquad(): gets the spherical quadrangle interpolation of a sequence of quaternions qi and qi+1 using the passed blend parameter;
maxon::Quaternion::GetInnerQuaternion(): gets the inner quaternion from the series of quaternions qi-1 and qi+1;
maxon::Quaternion::GetSpline(): gets the spherical spline interpolation of two quaternions qi and qi+1 using the passed blend parameter.

    // This example shows how to use the "get" methods provided with maxon::Quaternion.
    
    // just allocate a constant quaternion using the four components
    // representing a rotation of 90 deg around Y-axis in left-handed c.sys
    const maxon::Quaternion<maxon::Float> quatY90 (0, sin(PI05/2), 0, cos(PI05/2));
    
    // get the rotation matrix associated with the quaternion
    const maxon::Matrix rotMat = quatY90.GetMatrix();
    
    // get the axis and rotation angle associated with the quaternion
    maxon::Vector rotAxis;
    maxon::Float rotAngle;
    quatY90.GetAxisRotation(rotAxis, rotAngle) iferr_return;
    
    // get the squared length of the quaternion
    const maxon::Float valSqLength = quatY90.GetSquaredLength();
    
    // get the length of the quaternion
    const maxon::Float valLength = quatY90.GetLength();
    
    // get the normalized quaternion
    const maxon::Quaternion<maxon::Float> quatNormalized = quatY90.GetNormalized() iferr_return;
    
    // get the inverse quaternion
    const maxon::Quaternion<maxon::Float> quatInverted = quatY90.GetInverse() iferr_return;
    
    // get the conjugate quaternion
    const maxon::Quaternion<maxon::Float> quatConjugated = quatY90.GetConjugate();
    
    // get the natural log quaternion
    const maxon::Quaternion<maxon::Float> quatLog = quatY90.GetLog() iferr_return;
    
    // get the exponential equaternion
    const maxon::Quaternion<maxon::Float> quatExp = quatY90.GetExp() iferr_return;
    
    // get the quaternion elevated to the given power
    const maxon::Quaternion<maxon::Float> quatPow = quatY90.GetPow(3.0) iferr_return;
 
    // get the dot product of two quaternion
    const maxon::Float valDotProduct = maxon::Quaternion<maxon::Float>::GetDot(quatY90, quatConjugated);
    
    // get the linear interpolation of the given quaternion and the given blend ratio
    const maxon::Quaternion<maxon::Float> quatX90 (sin(PI05/2), 0, 0, cos(PI05/2));
    const maxon::Quaternion<maxon::Float> quatLERP = maxon::Quaternion<maxon::Float>::GetLerp(quatY90, quatX90, 0.5) iferr_return;
    
    // get the spherical linear interpolation of the given quaternion and the given blend ratio
    const maxon::Quaternion<maxon::Float> quatSLERP = maxon::Quaternion<maxon::Float>::GetSlerp(quatY90, quatX90, 0.5) iferr_return;
    
    // get the spherical quadrangle interpolation of a sequence of quaternions qi and qi+1 using the passed blend parameter
    const maxon::Quaternion<maxon::Float> quatA (1, 0, 0, 1);
    const maxon::Quaternion<maxon::Float> quatAp1 (1, 1, 0, 1);
    const maxon::Quaternion<maxon::Float> quatB (0, 0, 1, 1);
    const maxon::Quaternion<maxon::Float> quatBp1 (0, 1, 1, 1);
    const maxon::Quaternion<maxon::Float> quatSQUAD = maxon::Quaternion<maxon::Float>::GetSquad(quatA, quatAp1, quatB, quatBp1, 0.4) iferr_return;
    
    // get the inner quaternion from a sequance of quaternions qi-1, qi and qi+1
    const maxon::Quaternion<maxon::Float> quatCm1 (-1, 1, 0, 1);
    const maxon::Quaternion<maxon::Float> quatC (1, 1, 0, 1);
    const maxon::Quaternion<maxon::Float> quatCp1 (2, 1, 0, 1);
    const maxon::Quaternion<maxon::Float> quatInner = maxon::Quaternion<maxon::Float>::GetInnerQuaternion(quatCm1, quatC, quatCp1) iferr_return;
    
    // get the spline quaternion from a sequence of quaternions
    const maxon::Quaternion<maxon::Float> quatDm1 (0, 0, 0, 1);
    const maxon::Quaternion<maxon::Float> quatD (1, 0, 0, 1);
    const maxon::Quaternion<maxon::Float> quatDp1 (2, 0, 0, 2);
    const maxon::Quaternion<maxon::Float> quatDp2 (3, 0, 0, 2);
    const maxon::Quaternion<maxon::Float> quatSpline = maxon::Quaternion<maxon::Float>::GetSpline(quatDm1, quatD, quatDp1, quatDp2, 0.75) iferr_return;

The maxon::Quaternion class is provided with a number of "set" methods to define a maxon::Quaternion instance:

maxon::Quaternion::SetMatrix(): sets the quaternion based on the passed rotation matrix;
maxon::Quaternion::SetRotation(): sets the quaternion based on the rotation of order X -> Y -> Z;
maxon::Quaternion::SetAxisRotation(): sets the quaternion based on the given rotation axis and rotation value.

    // This example shows how to use the "set" methods provided with maxon::Quaternion.
    
    // allocate a constant transformation matrix representing
    // a rotation of 90 deg around Y-axis in left-handed c.sys
    const maxon::SqrMat3<maxon::Vector> mat3x3(maxon::Vector(0, 0, 1), maxon::Vector(0, 1, 0), maxon::Vector(-1.0, 0.0, 0.0));
    
    // allocate a quaternion and set the values accordingly to the matrix defined
    maxon::Quaternion<maxon::Float> quatA;
    quatA.SetMatrix(mat3x3);
    
    // allocate a Eulerian rotation vector (each component of the vector
    // represents a rotation around an axis) represeting a rotation of 90 deg
    // around Y-axis in left-handed c.sys
    const maxon::Vector rotVet (0.0, PI05, 0.0);
    
    // allocate a quaternion and set the values accordingly to the vector defined
    maxon::Quaternion<maxon::Float> quatB;
    quatB.SetRotation(rotVet);
    
    // allocate a vector representing a rotation axis and value representing the rotation amount
    const maxon::Vector rotAxis (0.0, 1.0, 0.0);
    const maxon::Float rotAngle(PI05);
    
    // allocate a quaternion and set the values accordingly to the axis and angle defined
    maxon::Quaternion<maxon::Float> quatC;
    quatC.SetAxisRotation(rotAxis, rotAngle);

The maxon::Quaternion class can be used to easily perform a rotation on a geometrical entity.

    // allocate a point on X-axis
    const maxon::Vector point_on_X (1.0, 0.0, 0.0);
    
    // define the rotation around the Y-axis
    const maxon::Vector axis_Y (0.0, 1.0, 0.0);
    
    // define the rotation amount in radians
    const maxon::Float rot_angle (PI05);
    
    // allocate and define the quaternion used to rotate the point
    maxon::Quaternion<maxon::Float> rotate_on_Y;
    rotate_on_Y.SetAxisRotation(axis_Y, rot_angle);
    
    // the new point should now be on the Z-axis
    const maxon::Vector point_on_Z = rotate_on_Y.GetMatrix() * point_on_X;

Conversion

A maxon::Quaternion instance can be converted to maxon::String using:

maxon::Quaternion::ToString(): returns a maxon::String representation of the maxon::Quaternion object.

    // This example shows how to convert a maxon::Quaternion to maxon::String.
    
    // just allocate a complex object
    const maxon::Quaternion<maxon::Float> quatA(1, 1, 0, 1);
    const maxon::String stringQuaternion = quatA.ToString(nullptr);
    DiagnosticOutput(diagIDString + "quatA: @", quatA);
    DiagnosticOutput(diagIDString + "quatA.ToString(): @", stringQuaternion);

Utilities

A maxon::Quaternion instance can be read from and written to disk by serializing the data contained using the conventional functions.

    // This example shows how to store and retrieve a maxon::Quaternion from a file.
    
    const maxon::Quaternion<maxon::Float> savedQuat(1, 1, 0, 1);
    
    // file URL
    const maxon::Url url = (targetFolder + "quaternion.txt"_s)iferr_return;
    const maxon::Id  fileID("net.maxonexample.quaternion");
    
    // save to file
    maxon::WriteDocument(url, maxon::OPENSTREAMFLAGS::NONE, fileID, savedQuat, maxon::IOFORMAT::JSON) iferr_return;
    
    // read from file
    maxon::Quaternion<maxon::Float> loadedQuat;
    maxon::ReadDocument(url, fileID, loadedQuat) iferr_return;

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