Exploring Quaternions: Angles in 3D Space

In summary, quaternions are a generalization of complex numbers in 3D and can be expressed in 4 terms or angles. For unit quaternions, the correct form is e^{(u \theta)} where u is the unit vector with direction cosines.
  • #1
Raparicio
115
0
Dear Friends,

I've read about quaternions, and they can be expresed in 4 terms or in angles like this:

[tex] a + ib + jc + kd = cos \theta + \vec{v} sin \theta [/tex]

Quaternions are a generalization of the complex numbers, but in 3D. My question is about angles. For example, with a complex number, we can write:

[tex] a + ib = sin \theta + i \theta [/tex]

By generalization, is this ok?

[tex] a + ib + jc + kd = cos \theta + i sin \theta + j sin \theta + k sin \theta [/tex]

By other hand, we can take the angles in complex numbers like:

[tex] a + ib = cos \theta + i sin \theta [/tex] --> Only one angle

how much angles has a quaternion? has angle with the real part?

my best reggards.
 
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  • #2
quaternion represents 1 angle...it represents a matrix rotation

you can multiple quaternions to get another quaternion just like you
can multiple M*M
 
  • #3
Raparicio:

I'm not sure you're correct.

Raparicio said:
[tex] a + ib + jc + kd = cos \theta + \vec{v} sin \theta [/tex]

I do not think that would work, since the possible values of [itex]\cos \theta[/itex] are limited to values between 1 and -1, while [itex]a[/itex], for example, has no such limit.

For example, with a complex number, we can write:

[tex] a + ib = sin \theta + i \theta [/tex]

That is incorrect for general a and b, because you can see by inspection that it requires

[tex]a=\sin \theta, b=\theta[/tex]

which would mean [itex]a=\sin b[/itex], which need not be true for a general complex number.

[tex] a + ib + jc + kd = cos \theta + i sin \theta + j sin \theta + k sin \theta [/tex]

Again, that's wrong, because it implies that b=c=d, which need not be true for a general quaternion.

By other hand, we can take the angles in complex numbers like:

[tex] a + ib = cos \theta + i sin \theta [/tex] --> Only one angle

Actually, this isn't general enough. You need

[tex] a + ib = r [cos \theta + i sin \theta] [/tex]

(i.e. an extra parameter, [itex]r[/itex], is requied).
 
  • #4
Raparicio said:
Dear Friends,

I've read about quaternions, and they can be expresed in 4 terms or in angles like this:

[tex] a + ib + jc + kd = cos \theta + \vec{v} sin \theta [/tex]

Quaternions are a generalization of the complex numbers, but in 3D. My question is about angles. For example, with a complex number, we can write:

[tex] a + ib = sin \theta + i \theta [/tex]

By generalization, is this ok?

[tex] a + ib + jc + kd = cos \theta + i sin \theta + j sin \theta + k sin \theta [/tex]

By other hand, we can take the angles in complex numbers like:

[tex] a + ib = cos \theta + i sin \theta [/tex] --> Only one angle

how much angles has a quaternion? has angle with the real part?

my best reggards.
Almost correct.

If [tex]\vec{v}=ib + jc + kd[/tex]
Then [tex]cos \theta + \vec{v} sin \theta=\cos \theta + ib \sin \theta + jc \sin \theta + kd \sin \theta [/tex]
 
  • #5
exponencial

BobG said:
Almost correct.

If [tex]\vec{v}=ib + jc + kd[/tex]
Then [tex]cos \theta + \vec{v} sin \theta=\cos \theta + ib \sin \theta + jc \sin \theta + kd \sin \theta [/tex]

Aha!

And in exponential:

Then [tex]a·cos \theta · e^{ib \theta + jc \theta + kd \theta [/tex] ?
 
  • #6
Raparicio said:
Aha!

And in exponential:

Then [tex]a·cos \theta · e^{ib \theta + jc \theta + kd \theta [/tex] ?
I'm not positive if you're talking about quaternions in general.

However, what you're doing is normally used with unit quaternions. They'll always have a magnitude of 1. As nuerocomp2003 mentioned, they're used for vector transformation. The change the angles associated with the vector they're used on, but dont' change the magnitude of the vector they're used on.

With that in mind, I'm pretty sure you'll wind up simply with:

[tex]e^{(u \theta)}[/tex] where u is the unit vector whose i,j,k components are the direction cosines of the unit vector.

In other words, the [tex]a \cos \theta[/tex] part is not correct for a unit quaternion. The exponent part is correct.
 
  • #7
exactly

BobG said:
I'm not positive if you're talking about quaternions in general.

However, what you're doing is normally used with unit quaternions. With that in mind, I'm pretty sure you'll wind up simply with:

[tex]e^{(u \theta)}[/tex] where u is the unit vector whose i,j,k components are the direction cosines of the unit vector.

Exactly! (im talking about unit quaternions).
 

Related to Exploring Quaternions: Angles in 3D Space

1. What are quaternions and how are they different from other mathematical representations?

Quaternions are a mathematical system that represents rotations in three-dimensional space. Unlike other representations such as Euler angles, quaternions do not suffer from gimbal lock and can avoid numerical problems when interpolating between rotations.

2. How are quaternions used in 3D graphics and computer animation?

Quaternions are commonly used in 3D graphics and computer animation to represent rotations and orientations of objects. They are also used in physics simulations and video games to calculate the rotation of objects in space.

3. How do you convert between quaternions and Euler angles?

Converting between quaternions and Euler angles involves using mathematical formulas to translate the values of one representation into the other. However, it is important to note that this conversion is not always straightforward and can result in inaccuracies.

4. Can quaternions be used to represent any rotation in 3D space?

Yes, quaternions can represent any rotation in three-dimensional space. Unlike other representations, they do not have any singularities or limitations on the range of rotations they can represent.

5. Are there any practical applications of quaternions in real-world scenarios?

Yes, quaternions have various practical applications in the fields of computer graphics, robotics, aerospace engineering, and more. They are also used in GPS systems to accurately determine the orientation of objects in space.

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