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Maths - Matrix to Axis-Angle - Sample Orientations

Sample Rotations

In order to try to explain things I thought it might help to work out a simple case where rotations are only allowed in mutiples of 90 degrees. This should make it easier to illustrate the orientation with a simple aeroplane figure, we can rotate this either about the x,y or z axis as shown here:

When we combine these rotations about the x,y and z axies in 90 degree multiples there are 24 possible orientations as in the table shown below.

In the following table we will need to know what quadrant the results are in, so I have taken some sample results from Math.atan2

heading applied first giving 4 possible orientations:

rightUp

reference orientation

1 0 0
0 1 0
0 0 1

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(4) / 2.0) =0
x = 0
y = 0
z = 0

angle = 0 degrees
axis = 1,0,0

backUp

rotate by 90 degrees about y axis

0 0 1
0 1 0
-1 0 0

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(2) / 2.0) = 90deg
x = 0
y = 2
z = 0

angle = 90 degrees
axis = 0,1,0

leftUp

rotate by 180 degrees about y axis

-1 0 0
0 1 0
0 0 -1

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 180 degrees
axis = 0,1,0

forwardUp

rotate by 270 degrees about y axis

0 0 -1
0 1 0
1 0 0

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 90 degrees
axis = 0,-1,0

or

angle = -90 degrees
axis = 0,1,0

Then apply attitude +90 degrees for each of the above: (note: that if we went on to apply bank to these it would just rotate between these values, the straight up and streight down orientations are known as singularities because they can be fully defined without using the bank value)

upLeft

0 -1 0
1 0 0
0 0 1

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 90 degrees
axis = 0,0,1

 

upForward

0 0 1
1 0 0
0 1 0

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0) / 2.0) =2*60deg = 120 deg
x = 1
y = 1
z = 1


angle = 120 degrees
axis = 0.5774,0.5774,0.5774

 

 

upRight

0 1 0
1 0 0
0 0 -1

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01


angle = 180 degrees
axis = 0.7071,0.7071,0

 

upBack

0 0 -1
1 0 0
0 -1 0

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01


angle = 120 degrees
axis = -0.5774,-0.5774,0.5774

 

 

Or instead apply attitude -90 degrees (also a singularity):

downRight

0 1 0
-1 0 0
0 0 1

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 90 degrees
axis = 0,0,-1

(equivilant rotation to:
angle = -90 degrees
axis = 0,0,1)

 

downBack

0 0 1
-1 0 0
0 -1 0

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01


angle = 120 degrees
axis = -0.5774,0.5774,-0.5774

 

downLeft

0 -1 0
-1 0 0
0 0 -1

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01


angle = 180 degrees
axis = -0.7071,0.7071,0

 

 

downForward

0 0 -1
-1 0 0
0 1 0

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01


angle = 120 degrees
axis = 0.5774,-0.5774,-0.5774

 

 

Normally we dont go beond attitude + or - 90 degrees because thes are singularities, instead apply bank +90 degrees:

rightForward

1 0 0
0 0 -1
0 1 0

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01


angle = 90 degrees
axis = 1,0,0

 

backRight

0 1 0
0 0 -1
-1 0 0

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01


angle = 120 degrees
axis = 0.5774,0.5774,-0.5774

 

leftBack

-1 0 0
0 0 -1
0 -1 0

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01


angle = 180 degrees
axis = 0,0.7071,-0.7071

 

 

forwardLeft

0 -1 0
0 0 -1
1 0 0

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01


angle = 120 degrees
axis = 0.5774,-0.5774,0.5774

 

Apply bank +180 degrees:

rightDown

1 0 0
0 -1 0
0 0 -1

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01


angle = 180 degrees
axis = 1,0,0

 

backDown

0 0 -1
0 -1 0
-1 0 0

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01


angle = 180 degrees
axis = 0.7071,0,-0.7071

 

leftDown

-1 0 0
0 -1 0
0 0 1

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01


angle = 180 degrees
axis = 0,0,1

 

forwardDown

0 0 1
0 -1 0
1 0 0

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01


angle = 180 degrees
axis = 0.7071,0,0.7071

 

Apply bank -90 degrees:

rightBack

1 0 0
0 0 1
0 -1 0

 

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01


angle = 90 degrees
axis = -1,0,0

(equivilant rotation to:
angle = -90 degrees
axis = 1,0,0)

 

backLeft

0 -1 0
0 0 1
-1 0 0

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01


angle = 120 degrees
axis = -0.5774,0.5774,0.5774

 

leftForward

-1 0 0
0 0 1
0 1 0

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01


angle = 180 degrees
axis = 0,0.7071,0.7071

forwardRight

0 1 0
0 0 1
1 0 0

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01

angle = 2 * acos(sqrt(1.0 + m00 + m11 + m22) / 2.0)
x = m21 - m12
y = m02 - m20
z = m10 - m01


angle = 120 degrees
axis = -0.5774,-0.5774,-0.5774

 
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make object cube 1,100
for x=1 to 360
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next x

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Can you help?

Please send me any improvements to here. I would appreciate ideas to make the pages more useful including error correction, ideas for new pages, improvements to wording. It helps if you quote the full URL of the page.

 

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I am working on a project which uses these principles, if you would like to help me with this you are welcome to join in, here:

http://sourceforge.net/projects/mjbworld/

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