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Maths - Matrix to Quaternion - 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

atan2(0.0,0.0)=0.0 (although Math.atan2 returns 0 this is abitary and if both x and y is zero we need to find some other way to get to value)
atan2(0.0,1.0)=0.0
atan2(1.0,1.0)=0.7853981633974483
atan2(1.0,0.0)=1.5707963267948966
atan2(1.0,-1.0)=2.356194490192345
atan2(0.0,-1.0)=3.141592653589793
atan2(-1.0,-1.0)=-2.356194490192345
atan2(-1.0,0.0)=-1.5707963267948966
atan2(-1.0,1.0)=-0.7853981633974483
heading applied first giving 4 possible orientations:

right

reference orientation

1 0 0
0 1 0
0 0 1

trace = m00 + m11 + m22 + 1 = 4

so use:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = 1

back

rotate by 90 degrees about y axis

0 0 1
0 1 0
-1 0 0

trace = m00 + m11 + m22 + 1 = 2

so use:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = 0.7071 + j 0.7071

left

rotate by 180 degrees about y axis

-1 0 0
0 1 0
0 0 -1

trace = m00 + m11 + m22 + 1 = 0

so this may not be accurate:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = j

forward

rotate by 270 degrees about y axis

0 0 -1
0 1 0
1 0 0


trace = m00 + m11 + m22 + 1 = 2

so use:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = 0.7071 - j 0.7071

(equivilant rotation to:
-0.7071 + j 0.7071)

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)

up

0 -1 0
1 0 0
0 0 1

trace = m00 + m11 + m22 + 1 = 2

so use:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = 0.7071 + k 0.7071

 

 

up

0 0 1
1 0 0
0 1 0

trace = m00 + m11 + m22 + 1 = 1

so use:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = 0.5 + i 0.5 + j 0.5 + k 0.5

 

 

up

0 1 0
1 0 0
0 0 -1

trace = m00 + m11 + m22 + 1 = 0

so this may not be accurate:

  • qw= sqrt (1 + m00 + m11 + m22) /2 =0
  • qx = (m21 - m12)/( 4 *qw) gives divide by zero
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = i 0.7071 +j 0.7071

 

up

0 0 -1
1 0 0
0 -1 0

trace = m00 + m11 + m22 + 1 = 1

so use:


  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = 0.5 - i 0.5 - j 0.5 + k 0.5

 

 

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

down

0 1 0
-1 0 0
0 0 1

trace = m00 + m11 + m22 + 1 = 2

so use:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = 0.7071 - k 0.7071

(equivilant rotation to:
-0.7071 + k 0.7071)

down

0 0 1
-1 0 0
0 -1 0


trace = m00 + m11 + m22 + 1 = 1

so use:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)


q = 0.5 - i 0.5 + j 0.5 - k 0.5

down

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

trace = m00 + m11 + m22 + 1 = 0

so this may not be accurate:


  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = -i 0.7071 + j 0.7071

(equivilant rotation to:
i 0.7071 - j 0.7071)

down

0 0 -1
-1 0 0
0 1 0

trace = m00 + m11 + m22 + 1 = 1

so use:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)


q = 0.5 + i 0.5 - j 0.5 - k 0.5

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

right

1 0 0
0 0 -1
0 1 0

trace = m00 + m11 + m22 + 1 = 2

so use:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = 0.7071 + i 0.7071

 

 

back

0 1 0
0 0 -1
-1 0 0

trace = m00 + m11 + m22 + 1 = 1

so use:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = 0.5 + i 0.5 + j 0.5 - k 0.5

 

left

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

trace = m00 + m11 + m22 + 1 = 0

so this may not be accurate:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = j 0.7071 - k 0.7071

 

forward

0 -1 0
0 0 -1
1 0 0

trace = m00 + m11 + m22 + 1 = 1

so use:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = 0.5 + i 0.5 - j 0.5 + k 0.5

 

Apply bank +180 degrees:

right

1 0 0
0 -1 0
0 0 -1

trace = m00 + m11 + m22 + 1 = 0

so this may not be accurate:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = i

back

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

trace = m00 + m11 + m22 + 1 = 0

so this may not be accurate:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = i 0.7071 - k 0.7071

 

left

-1 0 0
0 -1 0
0 0 1

trace = m00 + m11 + m22 + 1 = 0

so this may not be accurate:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = k

forward

0 0 1
0 -1 0
1 0 0

trace = m00 + m11 + m22 + 1 = 0

so this may not be accurate:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = i 0.7071 + k 0.7071

 

Apply bank -90 degrees:

right

1 0 0
0 0 1
0 -1 0

trace = m00 + m11 + m22 + 1 = 2

so use:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = 0.7071 - i 0.7071

(equivilant rotation to:
-0.7071 + i 0.7071)

 

back

0 -1 0
0 0 1
-1 0 0

trace = m00 + m11 + m22 + 1 = 1

so use:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = 0.5 - i 0.5 + j 0.5 + k 0.5

 

left

-1 0 0
0 0 1
0 1 0

trace = m00 + m11 + m22 + 1 = 0

so this may not be accurate:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = j 0.7071 + k 0.7071

 

 

forward

0 1 0
0 0 1
1 0 0

trace = m00 + m11 + m22 + 1 = 1

so use:

  • qw= sqrt (1 + m00 + m11 + m22) /2
  • qx = (m21 - m12)/( 4 *qw)
  • qy = (m02 - m20)/( 4 *qw)
  • qz = (m10 - m01)/( 4 *qw)

q = 0.5 - i 0.5 - j 0.5 - k 0.5

 
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Correspondence about this page

Book Shop - Further reading.

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cover us uk de jp fr caVisualizing Quaternions by Andrew J. Hanson

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Where I can, I have put links to Amazon for commercial software, not directly related to the software project, but related to the subject being discussed, click on the appropriate country flag to get more details of the software or to buy it from them.
cover Dark Basic Professional Edition - It is better to get this professional edition

cover This is a version of basic designed for building games, for example to rotate a cube you might do the following:
make object cube 1,100
for x=1 to 360
rotate object 1,x,x,0
next x

cover Game Programming with Darkbasic - book for above software

 

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.

 

progam

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/

This site may have errors. Don't use for critical systems.

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