Rotation

template<IsFloatingPoint T>
class Rotation

Represents a 3D rotation using a 3x3 matrix, with various construction and conversion utilities.

Provides static methods for constructing rotations from matrices, quaternions, axis-angle, Euler angles, and basis vectors. Supports conversion to/from quaternions, matrix, and axis representations, as well as composition and application to vectors.

Template Parameters:

T – Floating point type (float or double)

Public Functions

Rotation() = default
template<IsFloatingPoint U>
explicit operator Rotation<U>() const

Convert this Rotation to another floating point type.

Template Parameters:

U – Target floating point type

Returns:

Rotation

std::string to_string() const

Get a string representation of the rotation matrix.

Returns:

std::string String representation

Rotation inverse() const

Get the inverse rotation.

Returns:

Rotation<T> Inverse rotation

Quaternion<T> local_to_parent_quaternion() const

Get the local-to-parent rotation as a Hamilton quaternion.

Returns:

Quaternion<T> Hamilton quaternion

ShusterQuaternion<T> local_to_parent_shuster_quaternion() const

Get the local-to-parent rotation as a Shuster quaternion.

Returns:

ShusterQuaternion<T> Shuster quaternion

Quaternion<T> parent_to_local_quaternion() const

Get the parent-to-local rotation as a Hamilton quaternion.

Returns:

Quaternion<T> Hamilton quaternion

ShusterQuaternion<T> parent_to_local_shuster_quaternion() const

Get the parent-to-local rotation as a Shuster quaternion.

Returns:

ShusterQuaternion<T> Shuster quaternion

Mat3<T> local_to_parent_matrix() const

Get the local-to-parent rotation matrix.

Returns:

Mat3<T> Rotation matrix

Mat3<T> parent_to_local_matrix() const

Get the parent-to-local rotation matrix.

Returns:

Mat3<T> Rotation matrix

Vec3<T> x_axis() const

Get the X axis of the rotation (first column of the matrix).

Returns:

Vec3<T> X axis vector

Vec3<T> y_axis() const

Get the Y axis of the rotation (second column of the matrix).

Returns:

Vec3<T> Y axis vector

Vec3<T> z_axis() const

Get the Z axis of the rotation (third column of the matrix).

Returns:

Vec3<T> Z axis vector

Rotation operator*(const Rotation &b) const

Compose two rotations (this * b).

Parameters:

b – Other rotation

Returns:

Rotation<T> Composed rotation

Rotation &operator*=(const Rotation &b)

Compose this rotation with another (in-place).

Parameters:

b – Other rotation

Returns:

Rotation<T>& Reference to this rotation

Vec3<T> operator*(const Vec3<T> &b) const

Apply the rotation to a vector.

Parameters:

b – Vector to rotate

Returns:

Vec3<T> Rotated vector

Public Static Functions

static Rotation from_local_to_parent(Mat3<T> matrix)

Construct a Rotation from a local-to-parent rotation matrix.

Parameters:

matrix – 3x3 rotation matrix

Returns:

Rotation<T> Rotation object

static Rotation from_local_to_parent(Quaternion<T> quaternion)

Construct a Rotation from a local-to-parent quaternion.

Parameters:

quaternion – Hamilton quaternion

Returns:

Rotation<T> Rotation object

static Rotation from_local_to_parent(ShusterQuaternion<T> shuster_quaternion)

Construct a Rotation from a local-to-parent Shuster quaternion.

Parameters:

shuster_quaternion – Shuster quaternion

Returns:

Rotation<T> Rotation object

static Rotation from_local_to_parent(Vec3<T> axis, units::Radian angle)

Construct a Rotation from an axis and angle (Rodrigues’ formula).

Parameters:

  • axis – Rotation axis (will be normalized)

  • angle – Rotation angle in radians

Returns:

Rotation<T> Rotation object

static Rotation from_parent_to_local(Mat3<T> matrix)

Construct a Rotation from a parent-to-local rotation matrix.

Parameters:

matrix – 3x3 rotation matrix

Returns:

Rotation<T> Rotation object

static Rotation from_parent_to_local(Quaternion<T> quaternion)

Construct a Rotation from a parent-to-local quaternion.

Parameters:

quaternion – Hamilton quaternion

Returns:

Rotation<T> Rotation object

static Rotation from_parent_to_local(ShusterQuaternion<T> shuster_quaternion)

Construct a Rotation from a parent-to-local Shuster quaternion.

Parameters:

shuster_quaternion – Shuster quaternion

Returns:

Rotation<T> Rotation object

static Rotation from_parent_to_local(Vec3<T> axis, units::Radian angle)

Construct a Rotation from a parent-to-local axis and angle.

Parameters:

  • axis – Rotation axis (will be normalized)

  • angle – Rotation angle in radians

Returns:

Rotation<T> Rotation object

static Rotation extrinsic_euler_angles(units::Radian angle1, units::Radian angle2, units::Radian angle3, std::string sequence = "XYZ")

Construct a Rotation from extrinsic Euler angles.

Parameters:

  • angle1 – First rotation angle (radians)

  • angle2 – Second rotation angle (radians)

  • angle3 – Third rotation angle (radians)

  • sequence – Axis sequence (e.g., “XYZ”)

Returns:

Rotation<T> Rotation object

static Rotation intrinsic_euler_angles(units::Radian angle1, units::Radian angle2, units::Radian angle3, std::string sequence = "XYZ")

Construct a Rotation from intrinsic Euler angles.

Parameters:

  • angle1 – First rotation angle (radians)

  • angle2 – Second rotation angle (radians)

  • angle3 – Third rotation angle (radians)

  • sequence – Axis sequence (e.g., “XYZ”)

Returns:

Rotation<T> Rotation object

static Rotation from_basis_vectors(Vec3<T> x_axis, Vec3<T> y_axis, Vec3<T> z_axis)

Construct a Rotation from three basis vectors.

Parameters:

  • x_axis – X axis vector

  • y_axis – Y axis vector

  • z_axis – Z axis vector

Returns:

Rotation<T> Rotation object

static Mat3<T> local_to_parent_x(units::Radian angle)

Get a rotation matrix for a rotation about the X axis.

Parameters:

angle – Rotation angle in radians

Returns:

Mat3<T> Rotation matrix

static Mat3<T> local_to_parent_y(units::Radian angle)

Get a rotation matrix for a rotation about the Y axis.

Parameters:

angle – Rotation angle in radians

Returns:

Mat3<T> Rotation matrix

static Mat3<T> local_to_parent_z(units::Radian angle)

Get a rotation matrix for a rotation about the Z axis.

Parameters:

angle – Rotation angle in radians

Returns:

Mat3<T> Rotation matrix

static Mat3<T> parent_to_local_x(units::Radian angle)

Get a parent-to-local rotation matrix about the X axis.

Parameters:

angle – Rotation angle in radians

Returns:

Mat3<T> Rotation matrix

static Mat3<T> parent_to_local_y(units::Radian angle)

Get a parent-to-local rotation matrix about the Y axis.

Parameters:

angle – Rotation angle in radians

Returns:

Mat3<T> Rotation matrix

static Mat3<T> parent_to_local_z(units::Radian angle)

Get a parent-to-local rotation matrix about the Z axis.

Parameters:

angle – Rotation angle in radians

Returns:

Mat3<T> Rotation matrix