"Any exact answer you can get using this, you can also get using trig." -- Well, of course! But it's a pain in the ass calculating the *exact* values for COS and SIN, and Taylor Series approximations are time consuming. In practice, one can interpolate along tables (which is fast and easy), but these new tools also provide a fast method (relative to Taylor series approximations) for computing the same values without a table.
"Secondly, if a quantity is irrational then computing it a different way will not make it rational." -- No kidding. That's what I said. Reread my first and second posts. The point is that if you start with rational quantites and apply SINs and COSs, you'll end up truncating your results and losing accuracy. This result does not require that. He does NOT state that the operations map irrational values to rational values -- of course that's wrong. However, these operations are closed under the rationals.
"Next, this will not extend easily into complex numbers" -- I never said it would. What I did say was that it will be interesting to see how it does if, in fact, it does. Because Fourier transforms are already quickly computable by FFTs, I can't imagine it will revolunize the world of systems engineering. Analysis via classic Fourier/LaPlace transforms is well studied and not likely to go anywhere.
It may be interesting, however, to see how it lends itself to computer graphics where speed is king. I'm not so quick to discount the new tools he provides in this arena.
"Lastly, measurement is a physical act, taking a limit is a theoretical act. Measuring an angle certainly does not require limits. Now it does require limits to develop a rigorous theoretical definition of angle." -- If I used "measure" in describing the definition of an angle, then I apologize for the confusion, I don't normally reread my Slashdot posts. However, to have a function that describes the distance between two lines without the use limits is conceptually interesting.
"Leave the rigor to the mathematicians, it's what we do, we're good at it, you're not." -- HA! This is where Slashdotters inevitably go! You might be surprised to learn that many engineers do take courses in rigourous mathematics, even graduate math courses (oh my!). Pick up an IEEE journal on Information Theory, Control Systems, or Stochastic Signal Processing -- theory is just as important to the researchers in the field.
In your next response (because you WILL respond), avoid attacks and concentrate on the facts -- I'll respond more quickly then;)
In the world of irrational numbers, you get back irrational numbers. The point is that our practical tools for doing trig (advanced calculators) don't necessarily give us a rational number when our measurements are rational.
However, that doesn't take away from the computational advantages. Accurate Taylor series approximations for cos and sin are more intense than the few multiplications and divisions required by applying this method to compute spreads and quadrances.
The point about limits: to compute the _exact_ angle between two lines, you can't use a protractor. Geometrically, it requires limits since arc length of a circle is used.
As for the conceptual difficulty, its just because we're not used to it. If any of us had learned this starting in high school, it would become fairly intuitive.
The concept of spread is actually pretty straight-forward. Basically, given any two lines L1 and L2 that intersect at a single point O (parrallel lines are too trivial), spread is, informally, a function of their 'shortest quadrance (distance^2) apart'. Formally: (1) take any point A on the first line L1. Denote qudrance between O and A is Q. (2) compute the shortest distance between A and the other line L2 and square it for quadrance (call the quadrance R) (3) spread between L1 and L2 = s(L1,L2)=R/Q
Calculation of (1) and (3) is trivial. Calculation of (2) isn't so bad either (if you have a coordinate system -- but you can always add one). I believe that it basically involves a vector dot-product for a projection and then an application of the Pyth. Thm. using quadances.
The beauty is that you can do this by hand! In classical trigonometry, you practically need a calculator to handle angles and you'll likely end up with an irrational number somewhere that you'll approximate to a rational one. In a world of rational numbers, quadrance and spread give you rational numbers back! Now THAT's accuracy. In fact, you get rationals of polynomials with rational coefficients.
Basically, we've been spoiled by the advent of calculus and computers. Classical trigonometry is hard. The mesurement of an angle actually requires the computation of limits, and our modern calculations of COS, SIN,... use, I believe, Taylor series expansions.
For purposes of surveying (though IANA Surveyor so I'm sorry if this sounds ignorant), a machine that measures spread instead of angle and a calculator that inputs distances (and converts to quadrances) is the biggest change. As two lines become more separated, spread increases just like angle, though not at the same rate (probably at a rate of something like cos or sin).
Of course you can express all of it using SINs and COSs, but that's not the point. The real question for us in the engineering discplines is how it will effect our use of complex numbers. What we have now is fairly convenient, but I wonder what this has to offer? Unfortunately, they didn't provide the PDF for *that* chapter.
Slashdot Mirror
"Any exact answer you can get using this, you can also get using trig." -- Well, of course! But it's a pain in the ass calculating the *exact* values for COS and SIN, and Taylor Series approximations are time consuming. In practice, one can interpolate along tables (which is fast and easy), but these new tools also provide a fast method (relative to Taylor series approximations) for computing the same values without a table.
;)
"Secondly, if a quantity is irrational then computing it a different way will not make it rational." -- No kidding. That's what I said. Reread my first and second posts. The point is that if you start with rational quantites and apply SINs and COSs, you'll end up truncating your results and losing accuracy. This result does not require that. He does NOT state that the operations map irrational values to rational values -- of course that's wrong. However, these operations are closed under the rationals.
"Next, this will not extend easily into complex numbers" -- I never said it would. What I did say was that it will be interesting to see how it does if, in fact, it does. Because Fourier transforms are already quickly computable by FFTs, I can't imagine it will revolunize the world of systems engineering. Analysis via classic Fourier/LaPlace transforms is well studied and not likely to go anywhere.
It may be interesting, however, to see how it lends itself to computer graphics where speed is king. I'm not so quick to discount the new tools he provides in this arena.
"Lastly, measurement is a physical act, taking a limit is a theoretical act. Measuring an angle certainly does not require limits. Now it does require limits to develop a rigorous theoretical definition of angle." -- If I used "measure" in describing the definition of an angle, then I apologize for the confusion, I don't normally reread my Slashdot posts. However, to have a function that describes the distance between two lines without the use limits is conceptually interesting.
"Leave the rigor to the mathematicians, it's what we do, we're good at it, you're not." -- HA! This is where Slashdotters inevitably go! You might be surprised to learn that many engineers do take courses in rigourous mathematics, even graduate math courses (oh my!). Pick up an IEEE journal on Information Theory, Control Systems, or Stochastic Signal Processing -- theory is just as important to the researchers in the field.
In your next response (because you WILL respond), avoid attacks and concentrate on the facts -- I'll respond more quickly then
let's call the whole thing off
In the world of irrational numbers, you get back irrational numbers. The point is that our practical tools for doing trig (advanced calculators) don't necessarily give us a rational number when our measurements are rational.
However, that doesn't take away from the computational advantages. Accurate Taylor series approximations for cos and sin are more intense than the few multiplications and divisions required by applying this method to compute spreads and quadrances.
The point about limits: to compute the _exact_ angle between two lines, you can't use a protractor. Geometrically, it requires limits since arc length of a circle is used.
As for the conceptual difficulty, its just because we're not used to it. If any of us had learned this starting in high school, it would become fairly intuitive.
The concept of spread is actually pretty straight-forward. Basically, given any two lines L1 and L2 that intersect at a single point O (parrallel lines are too trivial), spread is, informally, a function of their 'shortest quadrance (distance^2) apart'. Formally:
... use, I believe, Taylor series expansions.
(1) take any point A on the first line L1. Denote qudrance between O and A is Q.
(2) compute the shortest distance between A and the other line L2 and square it for quadrance (call the quadrance R)
(3) spread between L1 and L2 = s(L1,L2)=R/Q
Calculation of (1) and (3) is trivial. Calculation of (2) isn't so bad either (if you have a coordinate system -- but you can always add one). I believe that it basically involves a vector dot-product for a projection and then an application of the Pyth. Thm. using quadances.
The beauty is that you can do this by hand! In classical trigonometry, you practically need a calculator to handle angles and you'll likely end up with an irrational number somewhere that you'll approximate to a rational one. In a world of rational numbers, quadrance and spread give you rational numbers back! Now THAT's accuracy. In fact, you get rationals of polynomials with rational coefficients.
Basically, we've been spoiled by the advent of calculus and computers. Classical trigonometry is hard. The mesurement of an angle actually requires the computation of limits, and our modern calculations of COS, SIN,
For purposes of surveying (though IANA Surveyor so I'm sorry if this sounds ignorant), a machine that measures spread instead of angle and a calculator that inputs distances (and converts to quadrances) is the biggest change. As two lines become more separated, spread increases just like angle, though not at the same rate (probably at a rate of something like cos or sin).
Of course you can express all of it using SINs and COSs, but that's not the point. The real question for us in the engineering discplines is how it will effect our use of complex numbers. What we have now is fairly convenient, but I wonder what this has to offer? Unfortunately, they didn't provide the PDF for *that* chapter.