Prerequisites
You may want to review vectors:
Multiply Matrix by Vector
A matrix can convert a vector into another vector by multiplying it by a matrix as follows:
|
|
= |
| m00 |
m01 |
m02 |
| m10 |
m11 |
m12 |
| m20 |
m21 |
m22 |
|
|
We can think of this as transforming (v_in0,v_in1,v_in2) into (v_out0,v_out1,v_out2).
If we apply this to every point in the 3D space we can think of the matrix as transforming the whole vector field.
This can represent any linear transform, including scaling and rotation (translation can be done by adding vectors).
Applying the Transform
The result of this multiplication can be calculated by treating the vector as a n x 1 matrix, so in this case we multiply a 3x3 matrix by a 3x1 matrix we get:
|
|
= |
| m00*v_in0 + m01*v_in1
+ m02*v_in2 |
| m10*v_in0 + m11*v_in1
+ m12*v_in2 |
| m20*v_in0 + m21*v_in1
+ m22*v_in2 |
|
So any vector v_in can be converted into another vector v_out.
We can show all vectors as a vector field as
shown here.
Scaling
We can scale using the following matrix:
If we want to scale equally in all dimensions then m00 =m11=m22
Translation
We can translate a vector by adding an offset vector, not a linear transform as shown above. Sometimes we want to combine translation with rotation so that we can do both in one operation, but we cant do 3D translation by multiplication with a 3x3 matrix . but we can with a 4x4 matrix as
defined here.
Rotation
We can use a 3x3 matrix to represent rotation in 3 dimensions as
defined here. Rotations in any arbitrary number of dimensions are discussed in group theory.
Sample 3D 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 multiples 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:
heading applied first giving 4 possible orientations:
|
reference orientation
heading = 0
attitude = 0
bank = 0
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=1
ca=1
cb=1
sh=0
sa=0
sb=0
|
rotate by 90 degrees about y axis
heading = 90 degrees
attitude = 0
bank = 0
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=0
ca=1
cb=1
sh=1
sa=0
sb=0
|
rotate by 180 degrees about y axis
heading = 180 degrees
attitude = 0
bank = 0
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=-1
ca=1
cb=1
sh=0
sa=0
sb=0
|
rotate by 270 degrees about y axis
heading = -90 degrees
attitude = 0
bank = 0
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=0
ca=1
cb=1
sh=-1
sa=0
sb=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)
|
heading = 0
attitude = 90 degrees
bank = 0
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=1
ca=0
cb=1
sh=0
sa=1
sb=0
|
heading = 90 degrees
attitude = 90 degrees
bank = 0
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=0
ca=0
cb=1
sh=1
sa=1
sb=0
|
heading = 180 degrees
attitude = 90 degrees
bank = 0
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=-1
ca=0
cb=1
sh=0
sa=1
sb=0
|
heading = -90 degrees
attitude = 90 degrees
bank = 0
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=0
ca=0
cb=1
sh=-1
sa=1
sb=0
|
Or instead apply attitude -90 degrees (also a singularity):
|
heading = 0
attitude = -90 degrees
bank = 0
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=1
ca=0
cb=1
sh=0
sa=-1
sb=0
|
heading = 90 degrees
attitude = -90 degrees
bank = 0
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=0
ca=0
cb=1
sh=1
sa=-1
sb=0
|
heading = 180 degrees
attitude = -90 degrees
bank = 0
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=-1
ca=0
cb=1
sh=0
sa=-1
sb=0
|
heading = -90 degrees
attitude = -90 degrees
bank = 0
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=0
ca=0
cb=1
sh=-1
sa=-1
sb=0
|
Normally we dont go beond attitude + or - 90 degrees because thes are singularities,
instead apply bank +90 degrees:
|
heading = 0
attitude = 0
bank = 90 degrees
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=1
ca=1
cb=0
sh=0
sa=0
sb=1
|
heading = 90 degrees
attitude = 0
bank = 90 degrees
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=0
ca=1
cb=0
sh=1
sa=0
sb=1
|
heading = 180 degrees
attitude = 0
bank = 90 degrees
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=-1
ca=1
cb=0
sh=0
sa=0
sb=1
|
heading = -90 degrees
attitude = 0
bank = 90 degrees
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=0
ca=1
cb=0
sh=-1
sa=0
sb=1
|
Apply bank +180 degrees:
|
heading = 0
attitude = 0
bank = 180 degrees
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=1
ca=1
cb=-1
sh=0
sa=0
sb=0
|
heading = 90 degrees
attitude = 0
bank = 180 degrees
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=0
ca=1
cb=-1
sh=1
sa=0
sb=0
|
heading = 180 degrees
attitude = 0
bank = 180 degrees
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=-1
ca=1
cb=-1
sh=0
sa=0
sb=0
|
heading = -90 degrees
attitude = 0
bank = 180 degrees
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=0
ca=1
cb=-1
sh=-1
sa=0
sb=0
|
Apply bank -90 degrees:
|
heading = 0
attitude = 0
bank = -90 degrees
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=1
ca=1
cb=0
sh=0
sa=0
sb=-1
|
heading = 90 degrees
attitude = 0
bank = -90 degrees
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=0
ca=1
cb=0
sh=1
sa=0
sb=-1
|
heading = 180 degrees
attitude = 0
bank = -90 degrees
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=-1
ca=1
cb=0
sh=0
sa=0
sb=-1
|
heading = -90 degrees
attitude = 0
bank = -90 degrees
| ch*ca |
-ch*sa*cb + sh*sb |
ch*sa*sb + sh*cb |
| sa |
ca*cb |
-ca*sb |
| -sh*ca |
sh*sa*cb + ch*sb |
- sh*sa*sb + ch*cb |
ch=0
ca=1
cb=0
sh=-1
sa=0
sb=-1
|
| |
|
|
reference orientation
|
rotate by 90 degrees about x axis
|
rotate by 180 degrees about x axis
|
rotate by 270 degrees about x axis
|
| |
|
rotate by 90 degrees about z axis
|
rotate by 90 degrees about y axis
|
|
|
|
| |
rotate by 180 degrees about z axis
|
|
rotate by 180 degrees about y axis
|
|
|
|
|
rotate by 270 degrees about z axis
|
|
|
rotate by 270 degrees about y axis
|
|
|
|
When we combine these rotations about the x,y and z axies in 90 degree multiples
there are 24 possible orientations as shown here:
encoding of these rotations in quaternions
is shown here.
encoding of these rotations in axis-angle is
shown here.
encoding of these rotations in euler angles is
shown here.
|
metadata block |
|
|
| 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. |
Mathematics for 3D game Programming - Includes introduction to Vectors, Matrices,
Transforms and Trigonometry. (But no euler angles or quaternions). Also includes
ray tracing and some linear & rotational physics also collision detection
(but not collision response).
|
|
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. |
Matlab.
|
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. |
|
|
Terminology and Notation
Specific to this page here:
|
|
|
program
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.
Copyright (c) 1998-2008 Martin John Baker - All rights reserved.
