The document discusses converting between quaternions and axis-angle representations of rotation. A quaternion contains 4 variables (qw, qx, qy, qz) that define a rotation. An axis-angle representation uses an axis vector (ax, ay, az) and a rotation angle. The document provides equations to convert between the two representations and works through examples, such as a 90 degree rotation around the x-axis yielding the quaternion (0.7071, 0.7071, 0, 0). It also derives the axis-angle to quaternion conversion equations from trigonometric identities.

1. QUATERNION TO AXIS ANGLE AXIS ANGLE TO QUATERNION

2. WHAT IS A QUATERNION4 scalar variables (sometimes known as Euler Parameters not to be confused with Euler angles)Similar with algebra process but in quaternion multiplication is not commutative.Ex. xy * zw != zw * xy

3. QUATERNION TO AXIS ANGLEEquations:angle = 2 * acos(qw)x = qx / sqrt(1 – qw * qw)y = qy / sqrt(1 – qw * qw)z = qz / sqrt(1 – qw * qw)

4. QUATERNION TO AXIS ANGLEQuaternion in terms of axis angleq = cos(angle/2) + i ( x * sin(angle/2)) + j (y * sin(angle/2)) + k ( z * sin(angle/2))

5. QUATERNION TO AXIS ANGLEAt angle 0 degrees?Let’s first use the quaternion to axis formulaq = cos(angle/2) + i ( x * sin(angle/2)) + j (y * sin(angle/2)) + k ( z * sin(angle/2))

6. QUATERNION TO AXIS ANGLEAt angle 0 degrees?So…q = cos(0/2) + i ( x * sin(0/2)) + j (y * sin(0/2)) + k ( z * sin(0/2))Then...q = 1 + i (x * 0) + j (y * 0) + k (z * 0)

7. QUATERNION TO AXIS ANGLEAt angle 0 degrees?q = 1working back from above equation qw = 1Angle = 2 * sqrt (1 – qw*qw)Angle = 2 * sqrt (1 – 1 * 1)Angle = 0 degrees

8. QUATERNION TO AXIS ANGLETry solving….A.) 180 degreesB.) 0.7071 + i0.7071

9. AXIS ANGLE TO QUATERNIONEquations:qx = ax * sin(angle/2)qy = ay * sin(angle/2)qz = az * sin(angle/2)qw = cos(angle/2)WHERE DID THESE EQUATIONS CAME FROM?!

10. AXIS ANGLE TO QUATERNIONDerivation of the equationFrom trigonometry formulacos(angle/2)2 + sin(angle/2)2 = 1

11. AXIS ANGLE TO QUATERNIONmultiplying the 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

12. AXIS ANGLE TO QUATERNIONExpanding the equation…cos(angle/2)2 + ax*ax * sin(angle/2)2 + ay*ay * sin(angle/2)2+ az*az * sin(angle/2)2 = 1

13. AXIS ANGLE TO QUATERNIONThis shows that the quaternion is normalized since it is in the form:qw2 + qx2 + qy2 +qz2 =1 is like…cos(angle/2)2 + ax*ax * sin(angle/2)2 + ay*ay * sin(angle/2)2+ az*az * sin(angle/2)2 = 1

14. AXIS ANGLE TO QUATERNIONqw2 + qx2 + qy2 +qz2 =1qw2 = cos(angle/2)2qx2= ax*ax * sin(angle/2)2qy2 = ay*ay * sin(angle/2)2qz2 = az*az * sin(angle/2)2

15. AXIS ANGLE TO QUATERNIONWhen we square the equations, the equations turned into…qw = cos(angle/2)qx= ax * sin(angle/2)qy = ay * sin(angle/2)qz= az * sin(angle/2)

16. AXIS ANGLE TO QUATERNIONLets try solving it…A.) Angle = 90 degrees, axis = 1, 0 , 0

17. AXIS ANGLE TO QUATERNIONLets try solving it…A.) Angle = 90 degrees, axis = 1, 0 , 0Use these equations:qw = cos(angle/2)qx= ax * sin(angle/2)qy = ay * sin(angle/2)qz= az * sin(angle/2)

18. AXIS ANGLE TO QUATERNIONSolving these…qw = cos(90/2),qx= 1 * sin(90/2)qy = 0 * sin(90/2)qz= 0 * sin(90/2)

19. AXIS ANGLE TO QUATERNIONqw = 0.7071qx= 0.7071qy = 0qz= 0This gives a quaternion (0.7071 + i0.7071)

20. QUATERNION TO AXIS ANGLE AXIS ANGLE TO QUATERNIONGroup 4:Jameson Chu ™Robin Marcelo ™Arric Tan™

21. QUATERNION TO AXIS ANGLE AXIS ANGLE TO QUATERNIONSources: http://www.euclideanspace.com/maths/geometry/rotations/conversions/angleToQuaternion/index.htmhttp://www.euclideanspace.com/maths/geometry/rotations/conversions/quaternionToAngle/index.htmhttp://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/index.htm