Maths - AxisAngle to Quaternion

Prerequisites

Definition of terms:

Equations

qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)

where:

Code

Java code to do conversion:

// assumes axis is already normalised
public void set(AxisAngle4d a1) {
  double s = Math.sin(a1.angle/2);
  x = a1.x * s;
  y = a1.y * s;
  z = a1.z * s;
  w = Math.cos(a1.angle/2);
}

Derivation of Equations

see quaternion representation of rotations.

This can be proved as follows:

from trig formula we get

cos(angle/2)2 + sin(angle/2)2 = 1

multiplying th sine part by 1 = ax*ax + ay*ay + az*az will have no effect so we can write:

cos(angle/2)2 + (ax*ax+ ay*ay + az*az) * sin(angle/2)2 = 1

expanding out gives:

cos(angle/2)2 + ax*ax * sin(angle/2)2 + ay*ay * sin(angle/2)2+ az*az * sin(angle/2)2 = 1

This shows that the quaternion is normalised since it is in the form:

qw2 + qx2 + qy2 +qz2 =1

and if we take the square root of each of these terms we get the parts of the quaternion:

qw = cos(angle/2)
qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)

That last part was not exactly a rigourous proof, but we can easily check that it is correct by checking rotations about each axis seperately as is done here.

Issues

Most maths libraries use radians instead of degrees (apart from OpenGL).

Example

we take the 90 degree rotation from this: to this:

As shown here the axis angle for this rotation is:

angle = 90 degrees
axis = 1,0,0

So using the above result:

cos(45 degrees) = 0.7071

sin(45 degrees) = 0.7071

qx= 0.7071

qy = 0

qz = 0

qw = 0.7071

this gives the quaternion (0.7071+ i 0.7071) which agrees with the result here

Angle Calculator and Further examples

I have put a java applet here which allows the values to be entered and the converted values shown along with a graphical representation of the orientation.

Also further examples in 90 degree steps here

 


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