Implicit Graphs

ryos · 2006-01-14 07:04:21

Sometimes I graph the weird functions we get in my Calculus class, and surprising things come out. These were cool enough to share. (They wouldn't reduce down to <100K without unacceptable quality losses. And, I'm too lazy to link to them all. )

The consecutive numbers are me zooming out from a frame of like -2.5 - 2.5 to a frame in the tens of thousands. Some of them aren't that great. If you're on a slow connection, #3 is my favorite. Oh, and "Out" was the product of zooming back in randomly with the marquee zoomer. I would have gone farther too, but after that one the program froze.

The red line is cos(xy) = 0, the green is sin(xy) = 0, and the blue is tan(xy) = 0.

While I'm not entirely sure I want to know , there are some really brilliant people here who enjoy a challenge, so I pose it: can anyone explain what's happening with these graphs?

Last edited by ryos (2006-01-14 07:09:08)

Ricky · 2006-01-14 10:18:02

Red:

cos(xy) = 0
cos(θ) = 0 when θ = pi/2 or 3pi/2. So xy = pi/2, y = pi/(2x) or xy = 3pi/2, y = 3pi/(2x).

Green:

sin(xy) = 0
sin(θ) = 0 when θ = 0 or pi. So xy = 0, x = 0 or y = 0, or xy = pi so y = pi/x.

Blue:

tan(xy) = 0
tan has periodic 0's at n*Pi. This means that xy = n*Pi, so y = nPi / x, where n is any integer.
I believe your graph of this is wrong, it should just be curves of y = pi/x, 2pi/x, 3pi/x, etc.

And remember that each one of these is periodic, which means it repeats.

What software did you use to make them?

Last edited by Ricky (2006-01-14 10:21:05)

MathsIsFun · 2006-01-14 10:27:01

Very nice patterns those functions make.

The squiggly blue line *may* be affected by the plotting program. That is the tan(xy)=0 line isn't it?

ryos · 2006-01-14 11:40:44

I used the "Grapher" program that comes with Mac OS X 10.4. Yes, the blue line is tanxy = 0.

It's possible that the squiqqlies are an artifact of the way the program calculates implicit graphs. Or of the resolution reduction - I had the resolution down to half (more and it got too cluttered and didn't look as cool).

Here are some graphs of just tanxy = 0 so we can see more clearly what's going on. I also turned the resolution all the way up this time.

The first one is zoomed out a bit, because it just looks cool.

2 is the default view. I turned off the axes so it's easier to see that the function itself has "axes." If you look really closely, you can see that they don't quite meet in the middle - they're 1/x-shaped bow thingies.

Note also that every other line in the graph is smooth, and the others are jagged. Not that they're random - they definitely have a pattern to them.

In the third, I showed the region that I would focus my zooming in on.

The fourth is zoomed waaayyyy in (note the labels on the axes; I exceeded their precision, so they all look the same). These graphs seem to possess an almost fractal ability to provide detail, even when zoomed infinitely in. Note also that a definite pattern has developed, and this pattern is unaltered no matter how far I zoomed in.

#5 is just an intermediate zoomed-out view.

I love to play with these sorts of things. I could spend hours just zooming and scrolling around, looking for interesting things.

Last edited by ryos (2006-01-14 11:47:48)

MathsIsFun · 2006-01-14 17:35:02

The first one could be used as wallpaper:

"Hey, dude, love your freaky wallpaper"

"Yeah, man, its tan(xy)=0"

mikau · 2006-01-15 06:43:59

Totally "rad"! ;-)

Last edited by mikau (2006-01-15 13:13:07)

John E. Franklin · 2006-01-18 18:30:44

I tried graphing cos(xy), sin(xy), and tan(xy), but
my curves were very predictable curves with no
zig zagging. Maybe by zooming way in, you are
showing graphs with rounding errors?
What are fractals?

MathsIsFun · 2006-01-18 18:50:32

Fractals are most wonderful things ...

In Geometry we are used to things having 1 dimension (lines), 2 dimensions (planes), 3 dimensions (solids), and so on ...

... but what if something had 1.3 dimensions? or 2.1? is that possible?

Enter the wonderful world of Fractals!

It happens when something has more and more detail the closer you look.

A real world example would be a coastline - you could measure an island's coastline using a map and get a length of 100km. But if you walked along the shores you might walk 110km, and if you measured more accurately, going around rocks, etc, you might get 120km. But to be more accurate you should curve around grains of sand, and get 130km, etc ...

So how long is a coastline - it is more than 1D and less than 2D !

OK, there may be a limit to this in the real world, but you can imagine a "bumpy line" that has bumps on bumps into infinity. It's length will completely depend on what level of detail you go to!

Ohh ... and Fractals have really cool graphics.

Start here, and enjoy!

mikau · 2006-01-19 06:39:43

Neat! Thanks for the explanation, MathsIsFun! Fractals were mentioned in num3rs the other week but I'd never heard of them. Now I have!

ryos · 2006-01-19 07:28:57

I'm surprised you didn't mention this.

Fractal math makes my head hurt, but I love the images. One assignment for an intro to computer programming course I took years ago was to write a java applet to calculate and graphically display the mandelbrot set. Fascinated, after the course I extended the program be a real java application. It did custom color gradients, had a marquee zoomer as well as numerical inputs, supported multiple windows with multithreaded rendering, etc. I was quite proud of it. I never did get to finish it, though, and now, years later, I don't have the time.

One of the coolest graphical effects in the program was the result of a moment of insanity. In Java, you represent colors as 4 byte integers, with the ARGB channels in subsequent runs of 8 bits. Of course, in Java all integers are 4 bytes, which means that any integer at all can be interpreted as a color. The result of the mandelbrot calculations in this particular calculator was an integer, one per pixel. You can see where this is heading...

I thought it would be cool to see what happened when I plugged the mandelbrot result directly into the color representation. The result looked a lot like an illustration in a Dr. Suess book, so I called it "Suess Mode." Some of the zoomed-in pictures in Suess mode were really fantastic. I wish I had saved some.

MathsIsFun · 2006-01-19 09:53:52

Could that java app be made available on the web? You could share your creation! You could have it here somewhere if you wanted.

ryos · 2006-01-19 17:01:09

Umm...Let's see. I think the source is on a CD somewhere, if I even have it at all.

As far as the app goes, I'd have to do a fair amount of work on it before it were usable by the general public (incomplete interface--that's the biggest pain in Java programming). I never did finish it, see. And, it's a Swing app, not an applet.

I could look at taking the applet I wrote for the original assignment (if I still have that code) and adding Suess mode in, plus a marquee zoomer. I don't know how well it would run inside a web browser. Plus, I haven't written Java in about 4 years...

You've piqued my interest; I'll look for it. If something does come of it, I'd happily release the code as open source. But, no promises.

ryos · 2006-01-19 17:52:46

Heh. I just found it (the applet) and tried to run it. It didn't quite work (the applet size was too small). Funny, since it worked fine when they graded it.

Now to see what sort of state the program is in...

John E. Franklin · 2006-01-21 04:06:26

What is the simplest fractal someone has ever heard of that might help me understand? Is it an equation or a complete algorithm to describe the 2-D picture? Is it still a fractal if rounding errors occur within the numerical processes?

kimrei · 2006-04-08 07:31:05

A true fractal is a repeating pattern that repeats itself and looks sort of the same no matter how far you zoom in or out on it, there are a couple formulas to make them, some alogorithms, some equations,

the simplest fractal I know of is an equilateral triangle with an upside down triangle inside of it and triangles inside of each triangle formed by the placing of the second one, I'll upload an image of it soon.

krassi_holmz · 2006-04-08 22:23:13

Here's a simple example of fractal:
Strt with string:"0".
Use this rules:
"0"->"1";
"1"->"01";
and you get:
0
1
01
101
01101
10101101
and so on.
this is a fractal structure

mathsyperson · 2006-04-09 05:28:42

Does that make the Fibonacci sequence a fractal structure?

George,Y · 2006-04-09 12:43:17

uh, that may explain the Fibonacci sequence's DYNAMIC!

Math Is Fun Forum

#1 2006-01-14 07:04:21

Implicit Graphs

#2 2006-01-14 10:18:02

Re: Implicit Graphs

#3 2006-01-14 10:27:01

Re: Implicit Graphs

#4 2006-01-14 11:40:44

Re: Implicit Graphs

#5 2006-01-14 17:35:02

Re: Implicit Graphs

#6 2006-01-15 06:43:59

Re: Implicit Graphs

#7 2006-01-18 18:30:44

Re: Implicit Graphs

#8 2006-01-18 18:50:32

Re: Implicit Graphs

#9 2006-01-19 06:39:43

Re: Implicit Graphs

#10 2006-01-19 07:28:57

Re: Implicit Graphs

#11 2006-01-19 09:53:52

Re: Implicit Graphs

#12 2006-01-19 17:01:09

Re: Implicit Graphs

#13 2006-01-19 17:52:46

Re: Implicit Graphs

#14 2006-01-21 04:06:26

Re: Implicit Graphs

#15 2006-04-08 07:31:05

Re: Implicit Graphs

#16 2006-04-08 22:23:13

Re: Implicit Graphs

#17 2006-04-09 05:28:42

Re: Implicit Graphs

#18 2006-04-09 12:43:17

Re: Implicit Graphs

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