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Quaternion frame rotation

Since R2020a



rotationResult = rotateframe(quat,cartesianPoints) rotates the frame of reference for the Cartesian points using the quaternion, quat. The elements of the quaternion are normalized before use in the rotation.

Frame Rotation


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Define a point in three dimensions. The coordinates of a point are always specified in the order x, y, and z. For convenient visualization, define the point on the x-y plane.

x = 0.5;
y = 0.5;
z = 0;
hold on
axis([-1 1 -1 1])

Create a quaternion vector specifying two separate rotations, one to rotate the frame 45 degrees and another to rotate the point -90 degrees about the z-axis. Use rotateframe to perform the rotations.

quat = quaternion([0,0,pi/4; ...
rereferencedPoint = rotateframe(quat,[x,y,z])
rereferencedPoint = 2×3

    0.7071   -0.0000         0
   -0.5000    0.5000         0

Plot the re-referenced points.


Define two points in three-dimensional space. Define a quaternion to re-reference the points by first rotating the reference frame about the z-axis 30 degrees and then about the new y-axis 45 degrees.

a = [1,0,0];
b = [0,1,0];
quat = quaternion([30,45,0],"eulerd","ZYX","point");

Use rotateframe to reference both points using the quaternion rotation operator. Display the result.

rP = rotateframe(quat,[a;b])
rP = 2×3

    0.6124   -0.3536    0.7071
    0.5000    0.8660   -0.0000

Visualize the original orientation and the rotated orientation of the points. Draw lines from the origin to each of the points for visualization purposes.


hold on
grid on
axis([-1 1 -1 1 -1 1])



Input Arguments

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Quaternion that defines rotation, specified as a quaternion object or a vector of quaternion objects. quat and cartesianPoints must have compatible sizes:

  • length(quat) == size(cartesianPoints,1), or

  • length(quat) == 1, or

  • size(cartesianPoints,1) == 1

Three-dimensional Cartesian points, specified as a 1-by-3 numeric vector representing a single point or an N-by-3 numeric matrix representing N points. quat and cartesianPoints must have compatible sizes:

  • length(quat) == size(cartesianPoints,1) or

  • length(quat) == 1, or

  • size(cartesianPoints,1) == 1

Data Types: single | double

Output Arguments

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Cartesian points defined in reference to rotated reference frame, returned as a 1-by-3 numeric vector or a numeric matrix.

rotationResult is a 1-by-3 vector when quat is a scalar quaternion object and cartesianPoints is a 1-by-3 vector representing a single point. Otherwise, rotationResult is an M-by-3 matrix, where M is the maximum of length(quat) and size(cartesianPoints,1).

Data Types: single | double


Quaternion frame rotation re-references a point specified in R3 by rotating the original frame of reference according to a specified quaternion:


where q is the quaternion, * represents conjugation, and u is the point to rotate, specified as a quaternion.

For convenience, the rotateframe function takes a point in R3 and returns a point in R3. Given a function call with some arbitrary quaternion, q = a + bi + cj + dk, and arbitrary coordinate, [x,y,z],

point = [x,y,z];
rereferencedPoint = rotateframe(q,point)
the rotateframe function performs the following operations:

  1. Converts point [x,y,z] to a quaternion:


  2. Normalizes the quaternion, q:


  3. Applies the rotation:


  4. Converts the quaternion output, vq, back to R3

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced in R2020a