real valued impulse response

On Thu, 28 Aug 2014 20:21:08 -0400, Randy Yates wrote:

> Tim Wescott <seemywebsite@myfooter.really> writes:
>> [...]
>> Your confusion confuses me.  Rather, I'm not quite sure where your
>> confusion lies.
>>
>> Quaternions are, as close as you can come, the next higher dimensional
>> "complex numbers".  They have a complete algebra that is more than 4-
>> vectors.  They're like complex numbers in that they have imaginary
>> parts, but unlike complex numbers in that they have three imaginary
>> parts.  You can add them, subtract them, and multiply them. 
>> Multiplication is not commutative -- a * b does not necessarily equal b
>> * a, in quaternion-land.
>>
>> They're exceedingly useful as a compact way of representing rotations
>> is 3D space, however there are so many rules and exceptions that you
>> have to remember that they'll make your brain twitch.
> 
> I did a little reading on quaternions and division rings. What a rich
> set of mathematics this algebra! I surely wish I had the time and money
> to study this area of mathematics intead of writing my umpteenth
> thousandth line of C/C++ code...

Rich and weird.

Actually, if you do an "ordinary" vector dot or cross product, you're just 
doing part of a quaternion multiply: if a and b are quaternions 
representing positions in 3D space (meaning their real parts are zero), 
then the real part of a * b is the negative of the dot product of the 
corresponding 3D vectors, and the imaginary part of a * b is either the 
vector cross product or the negative thereof.

Basically, people extracted an easier to work with subset out of 
quaternions, called it 3D vectors, and proceeded to do useful (and easier 
to understand) mathematics with it.

-- 

Tim Wescott
Wescott Design Services
http://www.wescottdesign.com

Thanks, everyone. 

I will probably go with 2 dimension to make my things simple, though, I
have often wondered that if just 2 dimensional value would suffice then why
bring an additional complex variable (i/j = sqrt(-1)) at all ...

	 

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