[Insight-users] quaternion integration

[Laura Igual]

Thank you John.
I had found the same solution.
However, I see a problem here. If we use the spherical coordinates to
represent quaternion rotations, the radium 'a' of the sphere has to be equal
to the angel of rotation 'alpha' of the quaternion. But, the axis of
rotation has to be unitary, so a has to be a=1. Is it that right?

I mean:
w = cos(alpha/2);
x = sin(alpha/2) *u_x;
y = sin(alpha/2) *u_y;
z = sin(alpha/2) *u_z;
and
u_x = sin(theta)*cos(phi);
u_y = sin(theta)*sin(phi);
u_z = cos(theta);
with a=1.

I am loosing a variable here!
What am I doing wrong?

Laura



On Wed, Jun 1, 2011 at 12:28 AM, John Drozd <john.drozd at gmail.com> wrote:

> Hi again,
>
> I forgot the bottom line of partial derivatives for z in the Jacobian
> expression on the first page.
> See attached revised page 1.
>
> John
>
>
> On Tue, May 31, 2011 at 6:19 PM, John Drozd <john.drozd at gmail.com> wrote:
>
>> Hi Laura,
>>
>> I calculated: dq = -0.5 a^2 sin^4(alpha/2) sin(theta) d(alpha) da d(theta)
>> d(phi)
>>
>> See my attached derivation for dq using Jacobians.
>>
>> Take care,
>> John
>>
>> On Sat, May 28, 2011 at 1:37 PM, Laura Igual <lauraigual at gmail.com>wrote:
>>
>>> Dear all,
>>>
>>> I am working with quaternion, and I have some questions about integration
>>> of a function of quaternions.
>>> Looking for information in the web, I have end up in itk website and I
>>> have read the interesting tutorials of Luis Ibañez, but I have still some
>>> doubts.
>>>
>>> I would be very gratefully if you could answer the following question:
>>>
>>> I have to compute the integral of a function of quaternions.
>>> A quaternion q=(w,x,y,z) can be written depending on angel of rotation
>>> alpha and the rotation axis u.
>>> w = cos(alpha/2);
>>> x = sin(alpha/2) *u_x;
>>> y = sin(alpha/2) *u_y;
>>> z = sin(alpha/2) *u_z;
>>>
>>>  I use spherical representation of the quaternions, where
>>>  t = theta = colatitude
>>>  t = phi = inclination angel
>>>  a = radius
>>> then:
>>> u_x = a *sin(theta)*cos(phi);
>>> u_y = a *sin(theta)*sin(phi);
>>> u_z = a *cos(theta);
>>>
>>> I want to solve integral between angel1 and angel2 of f(q), but I don't
>>> know the expression for differential of q!
>>> Is dq = sin(theta)?
>>>
>>> Could you give me any clue?
>>>
>>> Thank you very much in advance,
>>>
>>> Laura
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>>
>>
>> --
>> John Drozd
>> Post-Doctoral Fellow, Robarts Research Institute
>> The University of Western Ontario
>> London, ON, Canada
>>
>>
>
>
> --
> John Drozd
> Post-Doctoral Fellow, Robarts Research Institute
> The University of Western Ontario
> London, ON, Canada
>
>
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