09-02-2003 14:01									#1
	Quaternions and other mathy doohickies Inflate your Upload Space

Continued from a threadjack here: http://apolyton.net/forums/showthre...&threadid=76602

Matricies in 3D transforms (as we use them in compsci land) require a full 3x3 matrix, or 9 floats.

The unit quaternion:
q = [ a b c d ] (so: a^2 + b^2 + c^2 + d^2 = 1)

is equivalent to this 3x3 rotation matrix:

code:

M(q) = [ (a^2+b^2-c^2-d^2)  (2bc-2ad)          (2bd+2ac)         ]
       [ (2bc+2ad)          (a^2-b^2+c^2-d^2)  (2cd-2ab)         ]
       [ (2bd-2ac)          (2cd+2ab)          (a^2-b^2-c^2+d^2) ]

	09-02-2003 14:11									#2
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Hmmmm.. I guess all you need are four numbers for a rotation in R³ (3 for the vector around which you're rotating and one for the angle by which you rotate around it).

	09-02-2003 14:13									#3
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Quaternions are spiffy 'cause the composition of two rotations is simply the quaternion product. So if we wanted to rotate a vector a by a quaternion p, and then again by quaternion q, it's simply M(qp)a.

IIRC, interpolations are also very easy. I don't remember those off the top of my head so we'll avoid those.

	09-02-2003 14:29									#4
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I've never dealt with quaternions personally. Just heard 'em described superficially a while back.

On the subject of mathy doohickies, the group defined by the set of 1-1 maps of a set of 3 elements to itself where the product is defined as composition is isomorphic to the the group defined by the set of all symmetries of an equalateral triangle.

	09-02-2003 14:32									#5
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quote: Originally posted by Ramo On the subject of mathy doohickies, the group defined by the set of 1-1 maps of a set of 3 elements to itself where the product is defined as composition is isomorphic to the the group defined by the set of all symmetries of an equalateral triangle.

I'll keep that in mind.

	09-02-2003 14:33									#6
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You better.

	09-02-2003 14:38									#7
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A complex function is differentiable if and only if the partial of the real part of the function with respect to real part of the variable is the partial of the imaginary part of the fuction with respct to the imaginary part of the variable and the partial of the imaginary part of the function with respect to the real part of the variable is the negative of the partial of the real part of the function with respect to the imaginary part of the variable.

Words to live by...

	09-02-2003 14:43									#8
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(this is somewhat related to the topic)
I'm still kinda blown away at the performance of the NV30 chip -- it play with about 500 million polygons/second. 500,000,000. That's a staggering number.

	09-02-2003 17:41									#9
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quote: Originally posted by Ramo On the subject of mathy doohickies, the group defined by the set of 1-1 maps of a set of 3 elements to itself where the product is defined as composition is isomorphic to the the group defined by the set of all symmetries of an equalateral triangle.

Why equilateral? Would this not be true for any triangle? (Although then the 3 elements would have to be non-identical presumably.)

	09-02-2003 18:09									#10
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quote: Originally posted by Rogan Josh Why equilateral? Would this not be true for any triangle? (Although then the 3 elements would have to be non-identical presumably.)

Nope. If the triangle isn't equilateral, it doesn't have as many symmetries.

	09-02-2003 18:13									#11
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I thoght english and finnish were the only allowed languages here

	09-02-2003 19:03									#12
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The interesting thing about quadrature is that even though a high-order interpolating function will often wildly oscillate such that the infinity-norm increases as the function's order increases, these oscillations do not decrease the accuracy of quadrature, hence the reason that Runge-Kutta methods are still accurate for a high order so long as the function's high-order derivatives are well-behaved.

	09-02-2003 19:07									#13
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3d transformations

thank god we don't need them!


*quietly reminds himself he's about to learn differential equations next simester*

	09-02-2003 19:18									#14
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quote: Originally posted by Zero-Tau Nope. If the triangle isn't equilateral, it doesn't have as many symmetries.

Yes - that was what I was meaning about the 3 elements being non-identical. Or in other words, it would only be equilateral if the three elements are identical. Do you agree?

	10-02-2003 01:38									#15
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quote: Originally posted by Ramo I've never dealt with quaternions personally. Just heard 'em described superficially a while back. On the subject of mathy doohickies, the group defined by the set of 1-1 maps of a set of 3 elements to itself where the product is defined as composition is isomorphic to the the group defined by the set of all symmetries of an equalateral triangle.

Also called the Dihedral group of order 6 (D6)
Also called the Symmetric Group of order 6 (S6)
Is by the way the smallest non-Abelian (non-commutative group)
for example, flipping a triangle over and then turning it counterclockwise 120 degree is not the same at turning it then flipping it....

	10-02-2003 01:38									#16
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quote: Originally posted by Ramo A complex function is differentiable if and only if the partial of the real part of the function with respect to real part of the variable is the partial of the imaginary part of the fuction with respct to the imaginary part of the variable and the partial of the imaginary part of the function with respect to the real part of the variable is the negative of the partial of the real part of the function with respect to the imaginary part of the variable. Words to live by...

Cauchy Riemman Equations
What do you study Ramo?

	10-02-2003 01:43									#17
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No Rogan
in a set all elements are non-identical
if a set of 3 elements has 2 that are identical, its cardinality is defined as being less than 2 ....
So suppose we have two sets : A {1,2,3} and B {4,5,6}
there are 6 bijections (one to one functions) beetween them
1 - > 4
2 - > 5
3 - > 6

1 - > 4
2 - > 6
3 - > 5

1 - > 5
2 - > 4
3 - > 6

1 - > 5
2 - > 6
3 - > 4

1 - > 6
2 - > 4
3 - > 5

1 - > 6
2 - > 5
3 - > 4

And they form a group, of order 6, usually known as the Symmetric group of order 6 (the group of permutations of 3 elements) which is isomorphic (the same up to notation) as the group of symmetries of an equilateral triangle.
Suppose we have two equilateral triangles, with the vertices labeled as the sets before
so one trinagle with vertices 1,2,3 and the other 4,5,6
there are 6 ways to send one triangle so its exactly on top of the other which correspond to the 6 ways listed above.
If the triangle is not equilateral there will be less (2 ways actually for an isoceles triangle and one way for a scalen (sp?) )
even though the numbers in the set are all different (they have to be, they are used as labelling)

	10-02-2003 01:50									#18
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LT, I'm an undergrad in physics and math. Primarily physics though (that's what I plan on going to grad school in).

	10-02-2003 01:54									#19
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quote: Originally posted by Ramo A complex function is differentiable if and only if the partial of the real part of the function with respect to real part of the variable is the partial of the imaginary part of the fuction with respct to the imaginary part of the variable and the partial of the imaginary part of the function with respect to the real part of the variable is the negative of the partial of the real part of the function with respect to the imaginary part of the variable. Words to live by...

You should only be dealing with harmonic functions anyhow. Simply analytic and differentiable finctions aren't very interesting.

	10-02-2003 02:07									#20
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True. We've just started power series and the like.

	10-02-2003 06:56									#21
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Have you done any algebra?
jsut how much algebra is useful in physics anyway?
I think some basic group theory for quantum mech or something ?

	10-02-2003 07:00									#22
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I'm taking an algebra class right now. True, group theory is pretty essential in quantum mechanics, but it's not that useful at the undergrad level. I won't need it for another year.

	10-02-2003 10:00									#23
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This thread reminds me why I hated differential equations. Those bastards at the uni made me take it before they'd give me my science degree.

	10-02-2003 10:35									#24
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As well they should have.

	10-02-2003 10:47									#25
	Inflate your Upload Space

That doesn't mean I can't loath them for it.

	10-02-2003 10:52									#26
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Admittedly I feel the same way about my PDE course.