Maths - Axis Angle examples 90 degree steps

Maths - Axis-Angle - Sample Orientations

In order to try to explain things and give some examples we can try I thought it might help to show the rotations for a finite subset of the rotation group. We will use the set of rotations of a cube onto itself, this is a permutation group which gives 24 possible rotations as explaned on this page.

Heading applied first giving 4 possible orientations:

reference orientation

angle = 0 degrees
axis = 1,0,0

rotate by 90 degrees about y axis

angle = 90 degrees
axis = 0,1,0

rotate by 180 degrees about y axis

angle = 180 degrees
axis = 0,1,0

rotate by 270 degrees about y axis

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

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 straight down orientations are known as singularities because they can be fully defined without using the bank value)

angle = 90 degrees
axis = 0,0,1

aeroplane

angle = 120 degrees
axis = 0.5774,0.5774,0.5774

angle = 180 degrees
axis = 0.7071,0.7071,0

aeroplane

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

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

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

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

aeroplane

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

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

aeroplane

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

Normally we don't go beyond attitude + or - 90 degrees because these are singularities, instead apply bank +90 degrees:

aeroplane

angle = 90 degrees
axis = 1,0,0

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

aeroplane

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

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

Apply bank +180 degrees:

angle = 180 degrees
axis = 1,0,0

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

angle = 180 degrees
axis = 0,0,1

angle = 180 degrees
axis = 0.7071,0,0.7071

Apply bank -90 degrees:

aeroplane

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

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

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

aeroplane

angle = 180 degrees
axis = 0,0.7071,0.7071

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

encoding of these rotations in quaternions is shown here.
encoding of these rotations in matrices is shown here.
encoding of these rotations in euler angles is shown here.

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see also:

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

3D Math Primer - Aimed at complete beginners to vector and matrix algebra.

Other Math Books

Commercial Software Shop

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.

Dark Basic Professional Edition - It is better to get this professional edition

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

Game Programming with Darkbasic - book for above software

Discussion about using Darkbasic for physics simulation

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/

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