Coding the Trigonometric Function

[Lydia_R]

This is what I've been working on the last few days:

 

3:38am (the next day) I just pulled up a website to do the arc tangent function because the Windows scientific calculator doesn't do that function.  I entered in the .577 number and did an atan on it and it came back with .523 which is what I was expecting.  The .577 number is the result of the opposite over the adjacent and the .523 is the circumference measure to 30 degrees and is one of those numbers I memorized in February 2004.  That's an interesting story in my book.

 

So, the lessons I learned last night are that .577 is the result of a ratio.  The idea here is how to convert that decimal back to a ratio.  Andy helped with that last night.  He said "just divide by 10!"  I was like, oh my, that makes sense!  Andy is good for a laugh like that!  I had come home from a walk to the store and was hot on the idea of division and the musical connect to Ed Sheeran and Pink Floyd's The Division Bell.  I don't know either one of them very well, but I am hot on the idea of writing a function that does division.  And that brings up the subject of how a computer does multiplication.  I talked with Andy about that and I said that my dad just beat the multiplication tables into me.  I remember not getting it.  I would tell my dad "why do I have to memorize this stuff?"  And he would say "you just do."  And ironically, it was the same way with trigonometry on the streets.  With trig, the numbers only matter so that you can grasp the concept of it all.  They don't matter, but there is the magic of the sine of 30 degrees being .5 and how that points to 12 and the overtone series / cycle of 5ths.  That's pretty amazing stuff and then Andy was trying to draw a conclusion to 24 hours a day.  I insisted that that application of 24 was man-made and he was not convinced.  Then he brought up the year being about 360 days, or perhaps I brought that up.  That is not a man-made number (365) and it's really odd having that and the connection to 12.  It's just a coincidence, buy Andy was trying to draw some conclusion from it.  And then I brought up the thing with the moon appearing to be the same size as the sun.  And that got Andy brining up more conclusions.

 

Then I abandoned the conversation and my next thought on the subject was that this whole thing is about ratios and how I was using ratios to do my collision detection before I started studying trig.  I then took a look at the calulators on my screen.  What is this .577 number?  That is not the arc tangent of 30 degrees, is it?  I swear that I'm correct about the arc tangent of 30 degrees being .523.  I've been spouting that number off consistently since Feburary 2004.  So then I got out my Plane Trigonometry book and looked up 30 degrees.  Sure enough, .523 is correct.  But then from having looked at that row of numbers, I got the idea that if you had any one of those numbers and knew which function it was under, then you should be able to figure out the rest.  They are all the result of a ratio and the idea is to convert them back to ratios and cross multiply and divide.  And I mentioned that to Andy, or did he bring up cross multiplying and dividing when I was talking about ratios?  We basically got to that at the same time and I was impressed that Andy knew about it.

 

So, I think all the keys are here and it's almost time for me to start coding a division function and then a trigonometric function.  If it can be done, I should be able to have this coded in the next couple days, although I'm stretched pretty thin with this political stuff.  We'll see.  It's an interesting project.

 

One of the key things that got me going on this the other day was how the .3333 number came up.  I had recognized that as 30 degrees expressed in grads.  And that brought up the thought of base 10.  All these hard number calculations started off with 30 degrees which is equivalent to 33.33 grads.  90 degrees = 100 grads, 360 degrees = 400 grads.  I learned that over a decade ago by entering in some of these numbers into a calculator and playing with the grads button.  Anyway.  I love triangulation.

 

[Ivy]

Okay…………

[Vidanjali]

1 hour ago, Lydia_R said:

I entered in the .577 number and did an atan on it and it came back with .523 which is what I was expecting.  The .577 number is the result of the opposite over the adjacent and the .523 is the circumference measure to 30 degrees and is one of those numbers I memorized in February 2004.  That's an interesting story in my book.

 

I may be telling you things you already know, but I enjoy the discussion.

 

The .523 is the angle measurement in radians. The .523 is actually a truncated irrational number (a number with infinitely long, nonrepeating decimal expansion). To six decimal places, it is 0.523599. Therefore, rounded to three decimal places, we actually have 0.524 as the 4th decimal place is 5 or greater. It is an irrational number because the conversion from degrees to radians entails the value pi which is irrational. The conversion is 30 X pi/180 because the angle measurement pi radians is equivalent to 180 degrees.

 

1 hour ago, Lydia_R said:

So, the lessons I learned last night are that .577 is the result of a ratio.  The idea here is how to convert that decimal back to a ratio.  Andy helped with that last night.  He said "just divide by 10!"  I was like, oh my, that makes sense! 

 

If .577 were a rational number, you can read the decimal, "five hundred seventy seven thousandths" and then write the fraction, 577/1000. From there you would try to simplify the fraction, but this fraction does not simplify because 577 is a prime number (i.e. therefore, 577 and 1000 have no common factors to cancel). But .557 is also a truncated irrational number whose origin is tan(30 degrees) = 1/sqrt(3), where "sqrt" is an abbreviation for square root. It is irrational because sqrt(3) is irrational. (In fact, the square root of any prime number is irrational). Irrational numbers are so called because they cannot be expressed as the ratio of two integers. You can, however, acquire approximate values working with truncated irrational values such as .524 and .577.

 

1 hour ago, Lydia_R said:

"why do I have to memorize this stuff?"  And he would say "you just do."  And ironically, it was the same way with trigonometry on the streets.  With trig, the numbers only matter so that you can grasp the concept of it all.  They don't matter, but there is the magic of the sine of 30 degrees being .5 and how that points to 12 and the overtone series / cycle of 5ths.  That's pretty amazing stuff

 

True that trig requires some memorization, but you can develop intuition as well. Do you understand trig in terms of the unit circle (circle with radius 1) centered at the origin of a rectangular coordinate system? And have you worked with calculations using angles measured in radians? This helps with visualization and makes memorization of values less critical. But it is critical to realize that most of the decimals which arise from trig calculations are irrational and their expressions are approximate. It is a lot of fun and indeed there are many applications in the arts.

 

1 hour ago, Lydia_R said:

and then Andy was trying to draw a conclusion to 24 hours a day.  I insisted that that application of 24 was man-made and he was not convinced.  Then he brought up the year being about 360 days, or perhaps I brought that up.  That is not a man-made number (365) and it's really odd having that and the connection to 12.  It's just a coincidence, buy Andy was trying to draw some conclusion from it.

 

It's all about revolution and rotation. So is the concept of trig from the point of view of the unit circle. Consider how sundials were developed. Some time ago I watched a documentary series on ancient American civilizations and was impressed about their advanced knowledge of astronomy. There was one ancient site which was built like a sun dial. At specific hours of the day, the sun would align in such a way as to shine through specific constructions of the site creating stunning displays of symmetrical light and shadows. There were other sites where astronomical calendars were discovered showing the position of the sun during the solstices and equinoxes. In particular, this demonstrates the relationship between rotation of radius about a unit circle and the graph of the sine function which is a waveform. 

 

 

 

1 hour ago, Lydia_R said:

What is this .577 number?  That is not the arc tangent of 30 degrees, is it?  I swear that I'm correct about the arc tangent of 30 degrees being .523. 

 

To reiterate, tan(30 degrees) = tan(pi/6 radians) = 1/sqrt(3) which is approximately equal to .577. 

And 30 degrees which is equivalent to pi/6 radians is approximately equal to 0.52359877559 which rounded to three decimal places is actually .524 (not .523). See if you can visualize this. Draw a set of axes, one vertical, the other horizontal. The point at which the axes cross is called the origin. Mark an arbitrary unit of 1 to the right, left, above, and below the origin on the axes. Now draw a circle whose circumference touches all four of those points. You have a unit circle. Now consider the radius of the circle which coincides with the right side of the horizontal axis as the base of a right triangle. Rotate the radius from its initial position counterclockwise by 30 degrees. Now drop a vertical line from that point on the circle and you have a right triangle with one angle of 30 degrees. It looks like this.

 

tan(30 degrees) = length of opposite side / length of adjacent side. Well, we know the length of the hypotenuse of this triangle because we designed it using the unit circle. The coordinates noted on the above diagram give the cosine and sine of 30 degrees, respectively. Note that sine(30 degrees) = opposite/hypotenuse = (1/2)/1 = 1/2. cosine(30 degrees) = adjacent/hypotenuse = (sqrt(3)/2)/1 = sqrt(3)/2. And tan(30 degrees) = opposite/adjacent = (1/2)/(sqrt(3)/2) = 2/(2sqrt(3)) = 1/sqrt(3) which is approximately equal to .577

 

1 hour ago, Lydia_R said:

So, I think all the keys are here and it's almost time for me to start coding a division function and then a trigonometric function.  If it can be done, I should be able to have this coded in the next couple days, 

 

Sure enough. You just need to consider that you're working with approximations of irrational numbers. I suggest using the numbers as ratios involving pi. I believe you will have better results.

 

1 hour ago, Lydia_R said:

I love triangulation.

 

It is fun stuff. 

[Vidanjali]

Oops don't know how that second diagram got in there at the end. Forgive the redundancy. 

[Lydia_R]

Thanks for responding @Vidanjali!  You rock!

 

Yes, I'm really good at trig and most of what you were writing about is old hat to me.  I studied trig while I was living on the streets in 2004.  After about two weeks of studying the Plane Trigonometry book that I had checked out from the Seattle library, I woke up at 3am on the sidewalk and realized that .5 is the sine of 30 degrees and .523 is the circumference measure in radians up to 30 degrees.  After that, it all opened up for me and a few weeks later I coded an animated webpage slinky.  I started getting into coding perspective properly.  And then after a month or two of that, I went back to applying my new trigonometry skills to the collision detection I was coding previously only with ratios to make a video game I was calling Micropede, a Centipede type of clone.

 

And then a few months after that I coded an Asteroids type clone with shields in about 4 hours that I called Bumper Bubbles.  And with a bunch of effort of getting off of the streets and more studying, I went on to have a computer programming career for the last 20 years.  I always get a kick out of getting paid to do trigonometry, but I'm certainly not doing it every day.

 

This line of thought recently has come around to the old question of how to code the trigonometric function.  What is the code inside of the multiplication, division and trig functions in a calculator?  It seems that multiplication has the 10x10 multiplication table coded in as constants and then it loops to do numbers larger than 100 (for base 10).  Division is just like how you would solve a division problem.  It involves guessing.  So a computer function to do division would have to go through loops while it guesses and uses multiplication.

 

Then comes the trigonometric function.  I've been thinking a lot lately that pi is calculated by producing a large physical wheel and rolling it out on a flat surface and measuring it.  When I was on the streets, I would take a Starbucks stir stick and put a hole in either end of it.  I'd put a push pin in one end and then my pen in the other end.  That is one way to draw a perfect circle and it doesn't matter whether the stir stick is straight because the line between two points is always a perfect line.  The other way to do it is with a string instead of the stir stick.

 

When I was on the streets, it made me wonder how humans made perfectly straight, flat objects.  My theory is that it evolved over time.  Once you can create a perfect circle or saw blade, then you can use that to make a flat surface.  But the trick is a crap in crap out deal.  You have to start with a really perfect circle.

 

But you have introduced a new concept here with the square root deal.  Andy has been mentioning square roots with a song called the square root of two that he listens to.  And that brings up irrational numbers and personally, with where I am at in math, I don't place a huge significance on "irrational" yet.  I mean, would the same number have the same infinitely repeating pattern if it were expressed in a different base?  Maybe it does.  I don't tend to find the concept of that all too interesting.  But how you mention sqrt(3)/2 is interesting and is something I haven't considered yet.

 

You know, I could have been doing drugs on the streets, but I figured that I'd have a better chance at making a good life for myself if I studied this stuff and well, I was 33yo when I wound up on the streets and it's not like I felt I was missing out on some big drug experience that I hadn't experienced before.  I got to the point where math and programming were more fun that drugs, so like that, I guess math became my drug of choice.  Or work.  Math is an inexpensive hobby.

 

Hugs,

Lydia_n

 

[Lydia_R]

And I love your new profile pic @Ivy!  Wow, looking great!

[Ivy]

24 minutes ago, Lydia_R said:

And I love your new profile pic

Thanks

But honestly, I don't understand a thing in this thread.  LOL

[Lydia_R]

Andy and I just had another discussion on this which turned to how pi is computed.  I kept on pressing for the measuring a big wheel method and he kept driving this idea home that measuring like that is inaccurate.  We agreed to disagree on that, but then he googled how pi was originally computed.  I have never looked this up because I tend to not want to get an answer out of a book like that because I enjoy the challenge of working things out on my own.

 

The answer he found said that Archimedes did it by crunching the math of a 96 side polygon.  And then I immediately thought, YES!  When I was on the streets of Seattle in January of 2004 and got interested in studying trigonometry, I went to the library and there as a little green book called Plane Trigonometry with a canvas cover and then next to it or nearby was a slightly bigger blue book with a canvas cover on Calculus.  I opened the calculus book and it said that calculus is the study of two different things.  One of them is the measurement of an irregular area through methods of exhaustion.  So that is definitely what Archimedes was doing there.  And it makes sense that you can do that and wind up with an answer that it is within the range which you can call the tolerance.

 

I got interested in coding the math of a pinball machine in 2013, so I thought about it for a few minutes and decided that I needed to know the intersection of two lines.  So I Googled that and it came back with a small linear algebra equation which I was then able to code into a computer pinball machine in about 300 hours.  The idea of this type of collision detection is that there are only points and the lines between them.  There are no true curves in the game.  I was using this process of taking an artist's drawing of a maze and using Adobe Illustrator to turn it into vector graphics (SVG) and then they had some special function that would reduce all the curves in the SVG data to a series of straight lines.  Then I would consume the SVG data in my code and display the raster image.  It worked great and of course I was using a polygon for the ball.

 

With that huge amount of vector data, I wound up having to write special code to break that vector data into buckets within an array so that the collision detection wasn't working on all the lines at the same time because that would just bog the computer down.  I had to write and inRange() function to determine what sector(s) the ball was in and then grab the vector data for what could possibly be a collision.  I would compile those buckets (array) at runtime when the maze was opened.

 

The dependency on Adobe Illustrator was the death of the project.  The conversion process was tedious and the artist wasn't up to doing that work.  I researched other code to do it and never got anything that worked.  I dated a transwoman last year who did engineering work for a machine vision company and that sparked some new ideas.  The subject of machine vision has been on my mind for 20 years, but I hadn't quite figured out how to start coding it.  Then I realized last year that the way to do it would be to make a QR code reader.  You could start out with just like a 4x4 QR code pattern.  You take an image of it.  Or, you know, I guess those registration points in the corner are important, but to start coding, you could start with a perfect image of a 4x4.  Then you would lay a theoretical grid on top of it.  Then you would iterate over the grid and find out whether the pixel dead center in each square is black or white.  Then you would look at all the pixels around it and then average them and compare them to the one pixel in the center.  If the average of the outer pixels matches the center pixel, then you have a good estimate going.  So it would be that sort of thing.

 

But then to actually extend that to interpret lines at any angle...  Well, that would be very challenging work.  It's all very interesting stuff and if I live long enough, I'd like to start working on that.  I'm 53yo and this is kind of a hobby at this point.  I did good with going from having absolutely nothing on the streets to making all this happen and buying a house 10 years after I got off the streets in 2014.  I've been able to pay for 100% of my gender out of pocket so far, so I get a kick out of the fact that writing math on paper on the streets allowed me to do that.  I've made just over $1 million in my life, so it's not like I got rich off of that at all.  Over the span of my 35 year career, I've averaged about $35,000/year.  I live an excellent life though and all I really care about is having good food and time to cook it.  And having some time to play some nice piano.

 

Anyway.  I put this in the story section.  It's an autobiographical math story.  I've written all kinds of stuff about my life, but I tend to draw it back to a math education lesson.  Perhaps that turns some people away.  I always wish that our politics would focus more on engineering and less on the social issues.  I think we really take engineering for granted in our modern societies.

[Vidanjali]

1 hour ago, Lydia_R said:

Thanks for responding @Vidanjali!  You rock!

 

Aw shucks :) I like your story and similar to you, I had dropped out of high school and was homeless before eventually applying for various aid which led me to taking classes as community college. I'd never completed a math class in high school, but turns out I'm good at it and loved it so much from the very first basic algebra class as community college that I decided I'd become a mathematics professor. And I did! I taught math at the higher ed level for 17 years before resigning due to disability.

 

1 hour ago, Lydia_R said:

I woke up at 3am on the sidewalk and realized that .5 is the sine of 30 degrees and .523 is the circumference measure in radians up to 30 degrees. 

 

True as long as the radius is 1. Else it's a multiple of .524 which is an approximation of pi/6. The entire circumference of the unit circle is 2pi. And one full rotation about the circle is 2pi radian which is equivalent to 360 degrees. 30 degrees is 1/12 of one full rotation. Divide 2pi by 12 and you get approximately .524.

 

1 hour ago, Lydia_R said:

I've been thinking a lot lately that pi is calculated by producing a large physical wheel and rolling it out on a flat surface and measuring it. 

 

You are correct. The circumference of any circle equals pi times the circle's diameter. Therefore one definition of pi is the ratio of a circle's circumference to its diameter.

 

1 hour ago, Lydia_R said:

would the same number have the same infinitely repeating pattern if it were expressed in a different base?  Maybe it does. 

 

Not totally sure what you mean. First note that irrational numbers have infinitely long nonrepeating expansion. And note that the measurements of the sides of right triangles and ratios of those numbers are in arbitrary units of length whereas angle measurements may be in degrees or radians. So, for example, if you ask a calculator what's tangent of 30, and its programmed for radian input, it'll read that as 30 radians which is about 1719 degrees or almost 5 complete rotations.

[Vidanjali]

10 minutes ago, Lydia_R said:

The answer he found said that Archimedes did it by crunching the math of a 96 side polygon.  And then I immediately thought, YES!  When I was on the streets of Seattle in January of 2004 and got interested in studying trigonometry, I went to the library and there as a little green book called Plane Trigonometry with a canvas cover and then next to it or nearby was a slightly bigger blue book with a canvas cover on Calculus.  I opened the calculus book and it said that calculus is the study of two different things.  One of them is the measurement of an irregular area through methods of exhaustion.  So that is definitely what Archimedes was doing there.  And it makes sense that you can do that and wind up with an answer that it is within the range which you can call the tolerance.

 

Yes, indeed. It's nice to think about. Calculus allows you to work with an infinitely-many-sided regular polygon which may as well be a circle. 

 

I worked with such ideas within my master's thesis which was a history of the Jordan Curve Theorem. Basically, the theorem states that if you draw a circle on a piece of paper, that the circle separates the paper into two distinct regions - the interior of the circle and the exterior of the circle. Turns out it's not so straightforward to prove rigorously, especially considering all they had to work with at the time the theorem was stated was Euclidean geometry. It took generations and the development of new fields of mathematics before any correct proof was established.

[Lydia_R]

Yes, totally.  That's a great story about your math history @Vidanjali!  Wow!

 

People tend to forget or are not aware of grads instead of degrees and radians, but that is becoming a big deal to me in the idea of coding a trigonometric function.  If you set your calculator to grads and then do the sin of 33.33, it comes up with .4999, you know, pi/6.  The significance here is that by dividing pi/2 (radians) into 100 units (grads) instead of 90 units (degrees), you are now in a base 10 space.  And when we are dealing with decimals (in base 10), and trying to convert them back to ratios, then notating our angles in the base 10 system of gradients seems like it is the key to coding the trigonometric function.

 

Then again, this has all kind of been a revelation to me the last couple days.  It's fun to combine math with code and I'm looking forward to writing some algorithms around this.  My last job was coding software for a healthcare company and there was no math involved in that and that was a depressing part of that job.  The long hours of that job kind of created a spiritual backlog of wanting to do some math work and I think that energy is busting out right now.

[Vidanjali]

1 hour ago, Ivy said:

Thanks

But honestly, I don't understand a thing in this thread.  LOL

 

Here's some inspiration. Wheels within wheels.