rotatepoint

Quaternion point rotation

Syntax

rotationResult = rotatepoint(quat,cartesianPoints)

Description

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

Examples

Rotate Point Using Quaternion Vector

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

x = 0.5;
y = 0.5;
z = 0;

plot(x,y,"ko")
hold on
axis([-1 1 -1 1])

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

quat = quaternion([0,0,pi/4; ...
                   0,0,-pi/2],"euler","XYZ","point");
               
rotatedPoint = rotatepoint(quat,[x,y,z])

rotatedPoint = 2×3

   -0.0000    0.7071         0
    0.5000   -0.5000         0

Plot the rotated points.

plot(rotatedPoint(1,1),rotatedPoint(1,2),"bo")
plot(rotatedPoint(2,1),rotatedPoint(2,2),"go")

Rotate Group of Points Using Quaternion

Define two points in three-dimensional space. Define a quaternion to rotate the point by first rotating 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 rotatepoint to rotate both points using the quaternion rotation operator. Display the result.

rP = rotatepoint(quat,[a;b])

rP = 2×3

    0.6124    0.5000   -0.6124
   -0.3536    0.8660    0.3536

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

plot3(a(1),a(2),a(3),"bo");

hold on
grid on
axis([-1 1 -1 1 -1 1])
xlabel("x")
ylabel("y")
zlabel("z")

plot3(b(1),b(2),b(3),"ro")
plot3(rP(1,1),rP(1,2),rP(1,3),"bd")
plot3(rP(2,1),rP(2,2),rP(2,3),"rd")

plot3([0;rP(1,1)],[0;rP(1,2)],[0;rP(1,3)],"k")
plot3([0;rP(2,1)],[0;rP(2,2)],[0;rP(2,3)],"k")
plot3([0;a(1)],[0;a(2)],[0;a(3)],"k")
plot3([0;b(1)],[0;b(2)],[0;b(3)],"k")

Input Arguments

quat — Quaternion that defines rotation

quaternion object | vector of quaternion objects

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

cartesianPoints — Three-dimensional Cartesian points

1-by-3 numeric vector | N-by-3 numeric matrix

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

rotationResult — Repositioned Cartesian points

1-by-3 numeric vector | numeric matrix

Rotated Cartesian points defined using the quaternion rotation, 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

Algorithms

Quaternion point rotation rotates a point specified in R³ according to a specified quaternion:

$$L_q(u)=qu{q}^{*}$$

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

For convenience, the rotatepoint function takes in 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], for example,

rereferencedPoint = rotatepoint(q,[x,y,z])

the rotatepoint function performs the following operations:

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

   $$u_q=0+xi+yj+zk$$

2. Normalizes the quaternion, q:

   $$q_n=\frac{q}{\sqrt{a^2+b^2+c^2+d^2}}$$

3. Applies the rotation:

   $$v_q=q{u}_q{q}^{*}$$

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

Extended Capabilities

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

Version History

Introduced in R2018a