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
real valued impulse response
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