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Trigonometry Redefined without Sines And Cosines
Posted by
CowboyNeal
on Sat Sep 17, 2005 08:32 AM
from the numbers-and-stuff dept.
from the numbers-and-stuff dept.
Spy der Mann writes "Dr. Norman Wildberger, of the South Wales University, has redefined trigonometry without the use of sines, cosines, or tangents. In his book about Rational Trigonometry (sample PDF chapter), he explains that by replacing distance and angles with new concepts: quadrance, and spread, one can express trigonometric problems with simple algebra and fractional numbers. Is this the beginning of a new era for math?"
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No sines and cosines? (Score:5, Funny)
Re:No sines and cosines? (Score:5, Funny)
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Re:No sines and cosines? (Score:5, Funny)
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Re:No sines and cosines? (Score:5, Funny)
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Re:No sines and cosines? (Score:5, Funny)
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Re:No sines and cosines? (Score:4, Interesting)
What did I miss?
Parent
Re:No sines and cosines? (Score:5, Insightful)
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Re:No sines and cosines? (Score:5, Insightful)
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Parent is factually incorrect (Score:4, Informative)
He didn't replace distance with angles, nor is one of the two "the fundamental element of trig" - they are together the fundemental elements of (standard) trigonometry.
What he actually did was that he replaced distance with distance squared and angles with sine squared.
Parent
Re:Read the Article (Score:5, Interesting)
Ironically, you've actually harmed, not helped, anyone who hadn't read the article. The main point of this work is that distance AND angle are the wrong things. The quadrance (square of distance) and spread (effectively, the square of the sine of the angle -- though help points out 'angle' is a handwaving concept to begin with) should be the fundamental elements and makes the trigonometry meaningful and easy. That quadrance and spread are (independently) functions of distance and angle is trivially obvious, but it is exactly the difference that is his point that makes the math so much simpler.
Perhaps a more specific example for those programmers out there. No more need for lookup tables or Taylor expansion for trig functions unless a human at some point needs an angle value at which case it can be converted in the final step before output. (Internally, programs would never need them.)
Parent
Wonderful! (Score:5, Insightful)
I did a stint as a Maths teacher, and it was hard enough trying to convince the kids that it was worth learning Trigonometry then. They'll be even more determined to be ignorant if they hear of this.
Re:Don't worry... (Score:4, Insightful)
Parent
Not just physicists or engineers use trig.... (Score:5, Insightful)
Parent
agreed (Score:4, Insightful)
Parent
Re:Not just physicists or engineers use trig.... (Score:4, Insightful)
Number one: The people using the device still have to know the math. Suppose you want to calculate compound interest using the formula:
[P(1+r/n)^(nt)]-P
Several times I have seen college students fail to produce the correct answer even armed with a textbook, this formula and a calculator. Why? Because they don't understand the math.
Number two: Calculators lie virtually always. Often they produce answers that are "good enough". However, without an understanding of maths, "good enough" typically means "whatever the calculator reported as an answer". Example: what is the sum
10000000 + 0.00000001 ?
The correct answer is 10000000.00000001
The calculator's answer is 10000000.
Parent
Re:Not just physicists or engineers use trig.... (Score:4, Insightful)
The correct answer is 10000000.00000001
No, the correct answer is 10000000. Each term has only one significant figure, so after truncating to the correct precision you get the calculator's displayed answer. Although many calculators will have the inaccurate figure (10000000.00000001) rounded to the nearest base-2 floating point in memory and a long-enough fixed-point format will display it.
Now, if you had said 10000000.00000000 + 0.00000001, then the correct answer would indeed be 10000000.00000001.
Parent
Re:Not just physicists or engineers use trig.... (Score:4, Informative)
So your complaint basically boils down to this: "carpenters don't need to know trignometry, they only need to know Rational Trignometry".
Parent
Re:Don't worry... (Score:5, Insightful)
Luckily its a great store for Physici...
Do you need a cart sir?
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Re:Don't worry... (Score:5, Insightful)
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Re:Don't worry... (Score:5, Insightful)
Simply because you choose a profession does not use it, does not mean it doesn't have value.
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Re:Don't worry... (Score:5, Insightful)
As an aside: I'm always amazed how many people who do sciences and other technical stuff are always interested in many things, like music, politics, aesthetics, social structure...but hardly any political science or sociology student has even a passing interest in the sciences. I'm starting to believe that the latter are just to stupid to realise how much of an impact those things have on their life.
Parent
Re:Don't worry... (Score:5, Informative)
Yes. But then, I live in the Netherlands. Our system is different from the US; instead of (basically) lumping everyone in the same school and making sure they all pass, we send people to different types of high school, based on how they perform on a test at the end of primary school. I went to the highest types of high school, where you get at least 3 years of math IIRC; I took math for the full six years of the program. In the other other types of high school, you get less math because (1) they last shorter, and (2) they tend to focuse more on practical issues than on theoretical ones.
Parent
Re:Don't worry... (Score:4, Insightful)
This sounds like a variant of trig calculations that you often use in computer algorithms, where it's much faster to calculate the square of something than the root. If you do it right, you can avoid roots completely for comparisons, and only do one at the very end of the calculation for actual lengths and distances. Sine and cosine usually appear as the quotient of lengths of sides of a triangle--you rarely calculate sin(x) or cos(x). The one place where roots are unavoidable is normals, which are just so damn handy. But even there you can sometimes get away without normalizing for comparisons in things like backface culling.
Parent
Figures. (Score:5, Funny)
Wow (Score:4, Interesting)
If this book pans out, it would ultimately change Calculus for the better and might allow me to pass my classes- I wonder if I figured out how to apply this book to my classes and came up with the correct answers despite having worked the problems with his methods, if I'd still pass the class?
Re:Wow (Score:5, Insightful)
If you want to be the kind of engineer that implements other engineer's ideas then, by all means, blow off your math classes. But if you want to be someone who your peers turn to when they need help, do yourself a favor and learn the math.
All of the engineering sciences are founded on math (this is espescially true of computer science). If you can out code your instructors, that means you can probably out math them too. What you are interpreting as an inability to memorize functions, is probably really just disinterest.
This disinterest may stem from a feeling that what you are studying has little utility, it may stem from a personal dislike of an instructor, it may stem from the notion that math geeks are all squares and smell funny.
Whatever the reason, you need to get past it. A thorough understanding of the math behind engineering will make your life MUCH easier on down the road.
Parent
Re:Wow (Score:5, Insightful)
And you are wrong about algorithms. Algorithms is math. No ifs, ands or buts about it.
Parent
Re:Wow (Score:4, Interesting)
The only reason people don't realize this more is because most of the really hard stuff is already worked out for them. If you were stuck coding in assembler with no libraries to help you out, you'd realize how much math there is under the hood.
Parent
Re:Wow (Score:5, Interesting)
But first, this small reminder:
sin x (vertical component)
cos x (horizontal component)
tan x = sin x over cos x
sec x = 1 over cos x
csc x = 1 over sin x
cot x = cos x over sin x
-> sctsct
Now we substitute these trig functions with simple symbols:
I = sin x (vertical component)
II = cos x (horizontal component)
III = tan x = sin x over cos x
IV = sec x = 1 over cos x
V = csc x = 1 over sin x
IV = cot x = cos x over sin x
Think of them as one, two, three, four, five, and six. Now it boils down to remembering simple combinations of numbers:
integral{ I } = -II
integral{ II } = I
integral{ III } = ln | III + V |
integral{ IV } = ln | IV + VI |
integral{ V } = ln | III |
integral{ VI } = ln | I |
Once you write down more of these combinations, you'll discover patterns in it and from there on it should be easier than ever to remember trig integrals. And when you know most of the trig integrals, you will know most of the trig derivatives, too!
If anyone is interested in some more documentation on this, then by all means contact me. I am in the process of writing these things down. They should be available some time next week on my homepage.
Parent
Re:Wow (Score:4, Insightful)
I don't know quite how to put this, so I am just going to say it.
The degree doesn't make you an engineer. The MATH makes you an engineer. The degree is just your univerity vouching that you have completed your math and other engineering studies competently.
In my opinion, I think the author of this book is a quack and all I had to see was the first paragraph on the first page of his web site where he states that he has dispensed with (geometric) axioms. You cannot do anything in mathematics without axioms. Period. Math is not capable of proving something from nothing.
Parent
UNSW .. not South Wales (Score:5, Informative)
Just Wait... (Score:4, Interesting)
SOHCAHTOA and abstract survery results (Score:5, Insightful)
Cos = Adj/Hyp
Tan = Op/adjacent.
By as someone who's done some surveying it (which uses angles and distance), its easy to see why people use angles and distance. They are fairly easy to measure, so usefull for plans etc..
Replacing angles and distance with abstract quadrance and spread is exchanging one difficult thing (tan,cos,sin) for another (quad, spread)
Quandrance = distance ^2
Spread hard to see.
Faster calculations ?? (Score:5, Interesting)
in raytracers and 3D engines by using integer numbers.
Yes, for some badly written code (Score:5, Interesting)
If you implement a geometrical function whose input is a bunch of lengths and their ratios and whose output is a bunch of lengths and their ratios then you frequently shouldn't need any trig in your function, instead you should be using algebraic functions (+, -, *, /, nth root etc.). But many people find it easy to solve their problem by converting a bunch of stuff to angles using trig and then converting back. In fact, I optimised someone's rendering code recently by literally looking for every line of code where he'd used trig and replacing it, where possible, with the algebraic version. It worked well and speeded up the code massively. In addition I uncovered what you might call bugs. For example at one point this guy did a 180 degree rotation by converting to polar coordinates, adding M_PI to the angle and converting back. Rotating (x,y) by 180 degrees gives (-x,-y). The guy who'd written it was 'trig happy' and should have stopped to thing about the geometrical significance of what he was doing rather than just doing the obvious thing. If this book encourages people to use trig less it might be a good thing. But you don't need to talk about 'quadrance' and 'spread' to use the principles I'm talking about.
But of course there are times when you need trig. I'd hate to see the guy differentiate rotations without trig.
Parent
Re:Faster calculations ?? (Score:4, Insightful)
Furthermore, a lot of what this guy did is kind of a trick. Using 'spreads' may work when given an explicit triangle, but the part he's skimming over is that spreads are missing a REALLY nice property of angles. They don't add. Angles are a very nice parameter for rotation because a rotation of 10 followed by a rotation by 20 is the same as a rotation by 30. This property is implicitly used all over the place in graphics. So, in the end you probably have to use some angle-like measure when doing computer graphics (which is all about transformations, not measurements of unknowns). And in doing so, I'm sure you end up computing sines and cosines to do projections based on those rotations.
In the end, you just can't cheat your way out of the fact that a projection based on a rotation is a transcendental operation that numerically requires computing a truncated infinite series.
Parent
Interesting - but not entirlely new (Score:4, Insightful)
I read the article and part of hist concept of spread for an angle is the ratio (in a right triangle) of the side opposite the angle to the hypotenuse of the triangle. As I recall from my high-school geometry classes, though, this is exactly how the sine is calculated. (The cosine is the adjacent side divided by the hypotenuse.)
The interesting thing about his approach, though, is that he defines the concept of spread so that there's no need to refer to the underlying sine concept and the calculations are all (relatively) simple algebra - squaring and addition and subtraction. I would like to read some more and play with it a bit - the fact is I still have bad dreams about those damned trig identity tables, which I never really successfully felt comfortable with!
Interesting.
Great for eighth grade, but ... (Score:5, Insightful)
I don't see how this is "easier" (Score:5, Insightful)
Most of you missing the point. (Score:5, Insightful)
I am a high school mathematics teacher and I train students for mathematics competitions. I think most of you are missing the point of Dr. Wildberger.
Dr. Wildberger is not trying to redefine trigonometry, he is simply trying to give it a new perspective and hopefully, the new perspective would allow new insights into new methods of solving trigonometric problems. Protesting that memorizing the trigonometric functions as side adjacent over side opposite, etc., etc., is very easy and intuitive ignores the fact that in analytic geometry, that is not even how the trigonometric functions are defined!
Yes, really! For example, the sine of theta is defined in analysis as the y component of the radial vector from the origin to a point in a circle of unit radius whose arc distance from the x-axis is theta. The cosine of theta is defined similarly but this time taking the x-component. From this two simple definitions, the entire panoply of the trigonometric identities can be usefully derived!
The analytical definition is certainly not intuitive and not easy to memorize for a high school student! The side opposite, side adjacent trick is just that, a trick that is useful sometimes and certainly useful enough for high school mathematics but it is not a very useful definition as far as analysis is concerned.
For example, computing the derivative of the cosine function is not easy to understand if you restrict the definition of cosine to side adjacent over hypotenuse! Not to mention the fact that most students think there is magic involved in the computation of the trigonometric functions because the method of computation is not in their textbooks. It is only when one studies the calculus that the methods for computing the trigonometric functions are explained!
Dr. Wildberger has an idea that he thinks will make trigonometry more intuitive and I hope he is really onto something here. It would certainly help me with my students. I have read only the downloadable first chapter of the book and the idea is intriguing. Waving off Wildberger's new ideas without reading the entire book and without understanding the mathematics of trigonometry is just tragic.
In times like this I always remember the architect (I forgot the name, help me out here please) who refused to accept an architecture medal because the society that was giving out the medal invited Prince Charles to hand out the medal. That architect said, "I refuse to accept a medal from a person who believes that our grandfathers already know everything there is to know about how to build buildings and that there is nothing we can ever add to that knowledge anymore."
Just my two cents.
Very nice. Makes sense to a game programmer (Score:5, Informative)
Wildberger has put a cleaner theory underneath the kind of geometry game engine developers use. This may turn out to be useful.
Lately I've been doing robot motion planning, which has too much unnecessary trig in it. With enough work, it's often possible to derive a trig-free solution to some of the key problems. Better ways to think about trig-free solutions will help.
Re:Now ... (Score:5, Interesting)
Parent
This reminds me of a test in grad school (Score:5, Interesting)
I got no points at all for the question - despite solving the parts relevant to the class - because I didn't know off the top of my head that the integral of that is the arctan function.
I love abstract math but I hate trig.
Parent
Re:Now ... (Score:5, Interesting)
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Re:Now ... (Score:4, Informative)
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Re:Now ... (Score:5, Interesting)
e^(ix)=cos(x)+i*sin(x)
=> cos(x)=(e^(ix)+e^(-ix))/2
=> sin(x)=(e^(ix)-e^(-ix))/(2i)
Once you know this, then integration and derivation of all sin/cos and derived functions boils down to algebra and derivation and integration of e, which is trivial.
I cannot tell you how angry I was with them for not teaching me this until well after integral calculus.
Parent
Re:Now ... (Score:5, Informative)
For example, instead of working on a Euclidean affine coordinate system, by using "Quadrance" as he calls it, the coordinates would not be translation invariant, and you will be forced to attach a non-trivial measure to make integrals work out. So while the integrand might be simplified in the trigonometric identities, you will end up, instead of integrating over "dx", over something like "1/sqrt(x) dx", which hardly makes the integral any more appealing.
Parent
Re:huh? (Score:5, Insightful)
It's not that trigonmetric functions are hard to learn, it's that they rely on transcendal function like sine/cosine which are calculated numerically (as a taylor power series expansion, for example) thus are only approximations to the true values (an accurate number would require the calculation of an infinite series, which isn't practical in given time/space).
The article clearly states that: "Advanced mathematical knowledge, such as linear algebra, number theory and group
theory, is generally not needed." (to use this method)
I think that having a percise, simple (polynomials, rational fractions) alternative to current methods in eucleadian trigonometry, is very welcome.
Wouldn't it be nice to be able to calculate angles and distances without having to use a calculator (for sine/cosine calculations)?
Parent
Re:Redefinition? (Score:5, Insightful)
spread is the square of the sine of an angle.
Parent
Re:Units? (Score:5, Insightful)
I don't usually advocate this kind of behaviour, but the chapter is actually well worth reading. Quadrance is a neat hack - use the square of distance instead of distance to eliminate some nasty square roots, but spread is a much more interesting concept. The notion of spread removes the dependence upon circles to define relative direction, which removes a lot of complexity from trig.
Parent