Trigonometry Redefined without Sines And Cosines 966
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?"
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?
Just Wait... (Score:4, Interesting)
Re:Now ... (Score:5, Interesting)
Faster calculations ?? (Score:5, Interesting)
in raytracers and 3D engines by using integer numbers.
Non-Linear Angles (Score:1, Interesting)
With his method you can't just add angles line that. You have to do an elaborate calculation.
Re:Faster calculations ?? (Score:3, Interesting)
Re:No sines and cosines? (Score:4, Interesting)
What did I miss?
Is this silly? (Score:3, Interesting)
quadrance is the square of the distance
spread is the square of the sin angle
If you want to teach trig this way, it will certainly be easier to learn. But then, your students will not appreciate or understand square roots and negative numbers.
It is highly unlikely to lead to any revolutionary new algorithms. It appears that you are changing an angle to something rational, because all most of the popular sine values have square roots in them. However do not be fooled by this, it is just as hard to calculate spread as it is to calculate the sine of an angle. (except in a few cases).
Galileo got there first (Score:3, Interesting)
In the same way, as I understand it based on the sites referenced in the story, this guy makes use of an angle replacement that is not useful to most people. If I want to divide a circular cake into n portions, using spread will not be helpful. Although degrees are a unit with no rational basis(there are not 360 days in a year, and 480 might be better-divisible by 5,3 and powers of 2) they are relatively easy to use for circle division, which is all the circle math that most people need to do most of the time.
Unfortunately, mathematics lectures at Cambridge left me with the permanent belief that mathematicians' ideas of what is simple and what is complex merely illustrate the fact that physics and engineering use math, but they use a useful subset (which changes with time) and do not necessarily buy into ideas which mathematicians regard as self-evident.
I looked at one of his first examples (Score:3, Interesting)
Re:Wow (Score:2, Interesting)
Re:Don't worry... (Score:2, Interesting)
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.
Re:Now ... (Score:5, Interesting)
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.
Re:Don't worry... (Score:1, Interesting)
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.
Re:Wow (Score:3, Interesting)
None (well, very little) of this needed maths.
Algorithms are *not* maths. Why should they be? Anyone can derive something like a bubble sort from first principles without the use of a calculator. A binary search is intuitively obvious - people do something like it all the time in things like interviews (the game of 20 questions as it's known). I could go on... OTOH it's rare to actually work at that level these days - the STL, Java libs, etc. provide all the primitives then you just build on top of them.. there's nothing wrong with this - going back to the days when everything was written was scratch just aint fun.
Re:No sines and cosines? (Score:1, Interesting)
Re:Great for eighth grade, but ... (Score:3, Interesting)
This stuff is junk. On page 8: Square roots are to be avoided whenever possible.
Followed by page 16: To convert back to distances, take square roots.
He claims that sines and cosines are hard because the poor student can't calculate them by hand. How many here can extract a square root by hand?
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.
quack? (Score:2, Interesting)
Math is all about discarding old "axioms" and coming up with new axioms. You just have to realize that as axioms age, they often become "axioms". Get it?
Re:Wow (Score:1, Interesting)
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.
What a completely silly idea. (Score:3, Interesting)
It is not revolutionary, nor redefined. What it does it to rewrite trigonometry in similar thing as the good old "SOH-CAH-TOA" method taught in American high schools.
Wildberger's sole insight are the following:
- Instead of using the linear norm, he chooses to use the equivalent quadratic norm for distances, thus removing the squares from Pythagorean theorem. (So, for a right triangle, his version would be BASE + HEIGHT = HYPOTENUSE).
- Instead of using angles and calculating sines and cosines from it, he uses the concept of Spread, which is essentially just the sine of the angle squared!!
Well, one immediately sees a problem with the second point when trying to do something more than traditional planar Euclidean geometry: an obtuse angle will have the same spread as one other acute angle, and they share spreads with two other angles greater than pi radians!His argument is foundamentally flawed even in the first chapter when he "explains" the traditional method of measuring angles. He claims it as the following (paraphrasing):
so far so good, but he goes on to argue that That is complete bullshit. When I was in middle school in Taiwan, we were taught that 1) we start with two rays instead of two lines. The angle between rays A and B would be given by taking the arc length of on the unit circle by going counter clockwise from A to B. So, semantically, the angle between A and B, and the angle between B and A, would sum up to a nice 2pi. While colloquially, we often prefer to deal with the smaller of the two measurements.There are obvious problems with the Quadrance and Spread concept when applied to real life. For one thing, you can't just add Quadrances: say I know that points A B and C are on a line, in that order. Say I also know the distance AB and BC, then the distance AC = AB + BC. But if I measure using the Quadrance, sqrt(AC) = sqrt(AB) + sqrt(BC) would hold. He is just passing the buck in his "redefinition". While making things simpler for trigonometry, he is making things more difficult for other bits of Euclidean geometry.
Similar, while traditionally angles can just sum, his definition using a spread cannot sum angles.
In summary, his "getting rid of sines" is just replacing it with its valuation, which is hardly novel: it is something taught standard in American classrooms (remember SAHCOHTOA?).
"Decimal number plane"?? (Score:3, Interesting)
He seems to mean real plane, but with a pejorative connotation of needing decimal numbers to do ordinary trigonometry.
Google the phrase (in quotes); you get exactly one hit - this book.
Re:I don't see how this is "easier" (Score:2, Interesting)
> some bright spark suggested calculated using sines and cosines.
It would be an amazing breakthrough, because there are some very important things which are simpler and easier using sines and cosines. Read some of the other comments about the effect of rational geometry to calculus. Sines and cosines show up all over physics and more specialized descriptions of the real world (chemistry, thermodynamics, electrical engineering, etc).
Many people have been asking the question (and I haven't seen anybody posting an answer) about what is really easier to do using quadrance and spread that we don't already use some similar form for?
Distance-squared and dx/ds, dy/ds (Score:3, Interesting)
I have done programming involving coordinates and trig from time to time - originally, stuff like finding where a line is clipped by a polygon. I can remember and do SohCahToa, but that was close to the limit of my knowledge of trig.
The big problems that I found, while trying to write the code, were positive versus negative angles, infinite-angle of vertical lines, and having to calculate a lot of square-roots.
I found that two principles were a great help...
- Like the man says/implies, if distance-squared works as well as distance, use it; you avoid a square-root calculation.
- Express angles as a pair of numbers dx/ds and dy/ds (change in x and y as you move along the line).
The second point eliminated a lot of if-statements and similar but not quite identical code (if both angles are positive..., if angle A is positive and angle B is negative..., etc.)Re:No sines and cosines? (Score:2, Interesting)
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.)
Insurance claims (Score:2, Interesting)
Once I showed people the nature of their statement/position, I said, bring all the lawyers you want, my friends are engineers...
End of discussion and bs.
Worth exploring for vector graphics programs (Score:2, Interesting)
A few years ago my software house needed a subprogram to create paths offset any chosen distance from another 2D path. (Necessary for machining in the sign-making industry.) I fondly imagined this was half a day's work for a clever visiting student.
Alas, no, it turned out to be a 3-month coding nightmare. Finding the precise intersection of two nearly parallel vectors (expressed as lines, circle arcs, or Bezier curves) is surprisingly difficult, within the limits of precision and time set by computers. You end up dealing with special case after special case.
In ignorantly fumbling towards a better way of expressing the calculations, I got as far abandoning angles and using quadratures. If only Rational Trigonometry had been around at the time