I am trying to learn quaternions rather than just relying on euler angles. Here''s my problem. I''ve been to several websites concerning quaternions, and there are some properties I don''t get, and make no sense (I must not be looking at it right, or the pages I''m reading aren''t explaining everything). I read some here, and about 20 pages worth of google search links. Here''s the deal. I understand imaginary numbers (we covered them in a college trig class a coupla years ago). pretty simple. i=sqrt(-1). ok, so quaternions have one real part and three imaginary parts. Here''s where the problem comes in. I read the property:
i^2=j^2=k^2=-1.
according to this, i=j=k=sqrt(-1) (unless I''m missing something). But then there is the property that:
i*j*k=-1
shouldn''t that be -i???? and then lastly, this property makes no sense to me either.
i * j = k
There are several of them that take this form, but umm...if i = j = sqrt(-1), then isn''t this basically i^2? which should be -1? What am I missing here?
properties of quaternions
I did a paper on these last year and I still didn't totally understand them, but think of i, j, and k as orthonormal vectors. And the multiplication used as a cross product.
i*j*k = (i*j) * k = k *k = -1
when he says i * j = k, like I said think of it as a cross product.
I may be innacurate about this stuff, but hopefully that clears it up a little
- Andrew
one more thing, you saying i = j = k = sqrt(-1) is incorrect. I remember when I was reading what Hamiltion said, he said that you can't establish a linear relationship between the three imaginary components, since they're in different directions in imaginary space (or something like that).
[edited by - atcdevil on November 3, 2002 9:33:09 PM]
i*j*k = (i*j) * k = k *k = -1
when he says i * j = k, like I said think of it as a cross product.
I may be innacurate about this stuff, but hopefully that clears it up a little
- Andrew
one more thing, you saying i = j = k = sqrt(-1) is incorrect. I remember when I was reading what Hamiltion said, he said that you can't establish a linear relationship between the three imaginary components, since they're in different directions in imaginary space (or something like that).
[edited by - atcdevil on November 3, 2002 9:33:09 PM]
Author
Thanks andrew..but here another thing then, if you explanation is indeed the correct one. If you think of it as a cross-product (I am familiar with the properties of the cross product as well) then
i x j = k, j x i = -k
j x k = i, k x j = -i
k x i = j, i x k = -j
okay the properties look the same BUT what are you crossing? Looking at i, j, and k as vectors of the form:
i = sqrt(-1)i + 0j + 0k
j = 0i + sqrt(-1)j + 0k
k = 0i + 0j + sqrt(-1)k
but if you cross i x j, you get -k, so again...where am i going wrong here...and I don''t understand then if i=j=k=sqrt(-1), then how can you use them as vectors?
i x j = k, j x i = -k
j x k = i, k x j = -i
k x i = j, i x k = -j
okay the properties look the same BUT what are you crossing? Looking at i, j, and k as vectors of the form:
i = sqrt(-1)i + 0j + 0k
j = 0i + sqrt(-1)j + 0k
k = 0i + 0j + sqrt(-1)k
but if you cross i x j, you get -k, so again...where am i going wrong here...and I don''t understand then if i=j=k=sqrt(-1), then how can you use them as vectors?
You''re right that
i == sqrt( -1 )
j == sqrt( -1 )
k == sqrt( -1 )
but, note that
i != j
j != k
k != i
so i = j = k = sqrt( -1 ) is FALSE.
however, i*i = j*j = k*k.
It''s true that the cross product is similar to quat product, but with 3d vectors you have proper divisors of zero, i.e.
vectors a,b can have a X b = 0 even although a != 0 and b != 0. ( a and b are collinear).
The quat product rectifies this, i.e. a * b = 0 => a = 0 or b = 0, making it a much "nicer" mathematical object to study (apart from the non-commutativity).
Note that if quats were commutative, they would form a field (like what the complex numbers do) and could be used to form a vector space, a well studied phenomena. But they don''t commute. However, every non-zero quat has a quat inverse.
"Most people think, great God will come from the sky, take away everything, and make everybody feel high" - Bob Marley
i == sqrt( -1 )
j == sqrt( -1 )
k == sqrt( -1 )
but, note that
i != j
j != k
k != i
so i = j = k = sqrt( -1 ) is FALSE.
however, i*i = j*j = k*k.
It''s true that the cross product is similar to quat product, but with 3d vectors you have proper divisors of zero, i.e.
vectors a,b can have a X b = 0 even although a != 0 and b != 0. ( a and b are collinear).
The quat product rectifies this, i.e. a * b = 0 => a = 0 or b = 0, making it a much "nicer" mathematical object to study (apart from the non-commutativity).
Note that if quats were commutative, they would form a field (like what the complex numbers do) and could be used to form a vector space, a well studied phenomena. But they don''t commute. However, every non-zero quat has a quat inverse.
"Most people think, great God will come from the sky, take away everything, and make everybody feel high" - Bob Marley
One way to look at this is via a multiplication table
x | 1 i j k
-----------
1 | 1 i j k
i | i -1 -k j
j | j k -1 -i
k | k -j i -1
Where the product is of the column heading x the row heading in that order.
This describes all the possible products of real numbers and the three imaginaries i, j and k. As well as this you also have normal addition and multiplication, e.g.
2 * i = i + i = 2i
k - k - k - k - k = -3k = -3 * k
All other rules can be derived from this, e.g.
ijk = (ij) * k = k * k = -1
x | 1 i j k
-----------
1 | 1 i j k
i | i -1 -k j
j | j k -1 -i
k | k -j i -1
Where the product is of the column heading x the row heading in that order.
This describes all the possible products of real numbers and the three imaginaries i, j and k. As well as this you also have normal addition and multiplication, e.g.
2 * i = i + i = 2i
k - k - k - k - k = -3k = -3 * k
All other rules can be derived from this, e.g.
ijk = (ij) * k = k * k = -1
If you want a decent understanding, and actually the only correct one, look into geometric algebra. Quaternions have _nothing_ to do with imaginary number of 4 dimensional hyperspheres. Instead, a quaternion is a combination of a scalar and a concept known as a bivector which represents the dual of the rotation axis.
Great sources can be found on the following links:
http://carol.wins.uva.nl/~leo/clifford/index.html
http://modelingnts.la.asu.edu/
And I suggest you start with the following two papers:
http://www.science.uva.nl/~leo/clifford/dorst-mann-I.pdf
http://www.science.uva.nl/~leo/clifford/dorst-mann-II.pdf
Email me at J.Suter@student.utwente.nl if you have any questions, or alternatively check out http://jaap.flipcode.com/ga/
Cheers,
Jaap Suter
Great sources can be found on the following links:
http://carol.wins.uva.nl/~leo/clifford/index.html
http://modelingnts.la.asu.edu/
And I suggest you start with the following two papers:
http://www.science.uva.nl/~leo/clifford/dorst-mann-I.pdf
http://www.science.uva.nl/~leo/clifford/dorst-mann-II.pdf
Email me at J.Suter@student.utwente.nl if you have any questions, or alternatively check out http://jaap.flipcode.com/ga/
Cheers,
Jaap Suter
Author
Hey thanks! That''s alot of reading material to cover, so I''ll get started right away. I appreciate it.
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