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## #1 2012-04-29 13:17:16

Sumasoltin
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### Find an equation showing special graph

I'm a Chinese high school student who love equation and function CRAZILY ! And I've found a lot about this mathematical game.Let's look into!
Absolution,I think it useful but unpleasant,can make any project practicable.As is known to all,the equation "ab=0" is equal to"a=0 or b=0".Similarly,"g(x,y)f(x,y)=0" is equal to "g(x,y)=0 or f(x,y)=0",so now you can use "plus" to combine different plots.And another equation "f²(x,y)+g²(x,y)=0" is equal to "f(x,y)=0 and g(x,y)=0",and in this way you can cut a part of plot.How? There comes the equation "|x-a|+|x-b|-|a-b|=0",whose graph is a part of the plane.Now,let's graph "(x-y)²+(|x-1|+|x+1|-2)²=0".It's a segment.We find |x-1|+|x+1|-2>0,so this equation can be simplified as (x-y)²+|x-1|+|x+1|=2.
This is easy,but not best.Keep patient and you're due to find an equation showing special meanings.Take your aim apart and find equations describe them,and combine them together.
But I dislike absolution,so I found the "fusion".We plus "x=1" and "(x+1/2)²+y²=1",and then we got "x((x+1/2)²+y²)=1".whose plot is the fusion of a circle and straight line.Similarly,many plots can be combine in this way.
There are lots of tips but I can only convey those with my poor English.To show more about this geeks' game I will show you some my original equations.
What's more,have you found some interesting graphs of equation?Share it if it's convenient to you!

Last edited by Sumasoltin (2012-04-29 20:18:03)

## #2 2012-04-29 20:16:55

Sumasoltin
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### Re: Find an equation showing special graph

Aaaaa......the most beautiful one is larger than 100kb
The function is:
y=abs[x]-abs[x/2+3]-abs[x/2-3]+3(x-14)(x-16)/abs[2(x-14)(x-16)]+3(x-12)(x-18)/abs[(x-12)(x-18)]+(sqrt[4-(abs[x-14]-abs[x-16])^2]+sqrt[36-(abs[x-12]-abs[x-18])^2])/4-3+(18+abs[2x+24]+abs[2x-24]-abs[2x+12]-abs[2x-12]-6(x-14)(x-16)/abs[(x-14)(x-16)]-6(x-12)(x-18)/abs[(x-12)(x-18)]-18(x+14)(x+16)/abs[(x+14)(x+16)]+sqrt[4-(abs[x-14]-abs[x-16])^2]-sqrt[36-(abs[x-12]-abs[x-18])^2])sin[5pi*x]/4

Everyone knows that the heart cruve is a closed cruve so we can't find a functiom with a heart-shaped plot.Really?
y=x^(2/3)+sqrt(4-x^2)sin(5pi*x)

Last edited by Sumasoltin (2012-05-02 02:41:19)

## #3 2012-04-30 20:03:32

bobbym

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### Re: Find an equation showing special graph

Hi Sumasoltin;

Welcome to the forum.

For your heart curve, what is the interval for x?

In mathematics, you don't understand things. You just get used to them.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.

## #4 2012-04-30 21:07:08

Sumasoltin
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### Re: Find an equation showing special graph

Hi bobbym;
Thank you.I'm sorry to have forgot to say that.The first one is from x=-25 to 20,and the second is in [-5,5]
I found something interesting I set down before:Nike equation

Last edited by Sumasoltin (2012-04-30 21:19:01)

## #5 2012-04-30 21:20:18

anonimnystefy
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### Re: Find an equation showing special graph

These images are so cool! How did you derive the heart curve and Nike curve?

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón

## #6 2012-04-30 21:54:23

bobbym

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### Re: Find an equation showing special graph

Hi Sumasoltin;

What are you using to do the graphing?

In mathematics, you don't understand things. You just get used to them.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.

## #7 2012-04-30 22:04:27

Sumasoltin
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### Re: Find an equation showing special graph

Hi bobbym;
There are lots of mathematical softwares,like matlb,mathematic,and to graph equations I prefer GrafEq(w w w .xamuel.com/graphs-of-implicit-equations)

Last edited by Sumasoltin (2012-04-30 22:05:31)

## #8 2012-04-30 22:07:21

bobbym

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### Re: Find an equation showing special graph

y=x^(2/3)+sqrt(4-x^2)sin(5pi*x)

Are you sure this is the right equation?

In mathematics, you don't understand things. You just get used to them.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.

## #9 2012-04-30 22:12:09

Sumasoltin
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### Re: Find an equation showing special graph

Hi anonimnystefy;
Aaaa......they are created in different ways.Some are using the transformation in which we replace x by x±f(x).

## #10 2012-04-30 22:14:56

Sumasoltin
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### Re: Find an equation showing special graph

To bobbym;
Aaaa...what's wrong?

## #11 2012-04-30 22:19:26

bobbym

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### Re: Find an equation showing special graph

Nothing yet, I am rechecking, probably the program.

In mathematics, you don't understand things. You just get used to them.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.

## #12 2012-04-30 22:20:34

Sumasoltin
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### Re: Find an equation showing special graph

Oh,I got it ! Different softwares define "x^(2/3)" differently.It should be (x^2)^(1/3)

## #13 2012-04-30 22:22:24

anonimnystefy
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### Re: Find an equation showing special graph

Still not a heart graph!

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón

## #14 2012-04-30 22:23:56

Sumasoltin
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### Re: Find an equation showing special graph

WolframAlpha shows as this.

## #15 2012-04-30 22:27:32

anonimnystefy
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### Re: Find an equation showing special graph

Oh,yes! Got it now! Thanks for the cool graph! Got more?

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón

## #16 2012-04-30 22:28:28

bobbym

Online

### Re: Find an equation showing special graph

Yes, I got that, I did not see the heart at first.

In mathematics, you don't understand things. You just get used to them.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.

## #17 2012-04-30 22:31:01

anonimnystefy
Real Member

Online

### Re: Find an equation showing special graph

Here's another heart graph that you get by varying the constant with pi*x inside the sine function:

'Nother Heart Graph

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón

## #18 2012-04-30 22:32:36

bobbym

Online

### Re: Find an equation showing special graph

What is cbrt?

In mathematics, you don't understand things. You just get used to them.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.

## #19 2012-04-30 22:33:56

anonimnystefy
Real Member

Online

### Re: Find an equation showing special graph

Cube root function.

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón

## #20 2012-04-30 23:06:27

bobbym

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### Re: Find an equation showing special graph

Got MIF's graph to do it.

In mathematics, you don't understand things. You just get used to them.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.

## #21 2012-04-30 23:12:51

anonimnystefy
Real Member

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### Re: Find an equation showing special graph

Good! It is nice. Did you try using sin(k*pi*x) instead of sin(5*pi*x) for arbitrary values of k?

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón

## #22 2012-04-30 23:13:01

Sumasoltin
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### Re: Find an equation showing special graph

ρ=1-cosθ , an old equation shows heart cruve (bottom instead I think).But I found "(x²+y²-1)³=x²y³" on WolframAlpha.
"Who found this?",I search it on Google and finally I learned it was Siehe Beutel.But how he found this?
We chosed Archimedes spiral: x²+y²=arc tan²(y/x)
we find that "tan x" is simillar to "x³",so we got
(x²+y²)³y/x)²
we find heart isn't a symmetrical shape,so we change "y²" by "y³"
As x becomes larger,the difference between "tan x" and "x³" becomes larger,so we plus "x⁴"
Aha~just a joke ! Maybe following can be turth.
He used a simple ellipse "x²+y²-xy=1".
We transform it to "x²+y²-1x|y".
NO ABS ! We get "(x²+y²-1)²=x²y²".
But now we lose the relation of that "x²+y²-1" is the same to "y" in "±" (I can't convey in proper way)
So we try the equation "(a²x²+b²y²-1)^(2k+m)=(by)^m*(abxy)^2k"
when a、b→1,there is the equation "
(x²+y²-1)^m=y^m"
So we have "|x|^m(x²+y²-1)/y)^m=1"
Compare plots of "x²+y²-xy=1" and "x²+y²-y=1",we find only when x∈[-1/2,1/2] their difference become obvious,so m→0.
We make m=1，so k→+∞.
By trying,due to the equation's interval,when k=1 we get such a beautiful plot : (x²+y²-1)³=x²y³

## #23 2012-04-30 23:18:38

Sumasoltin
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### Re: Find an equation showing special graph

Oh, that's funny ! Without blanks "= (" was transformed into negetive face!
I did use k instead of 5,but a known number looks better.

## #24 2012-04-30 23:23:23

Sumasoltin
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### Re: Find an equation showing special graph

(x+13+sqrt((x+4)^2+4)-2sqrt((x+8)^2+4))^2*(12x*y^2+x^3-11y^2+x^2+7x-4)^3+(8y^2+6x^2-16x+4)^3*y^4*(x+13)^2=0 where x=-15 to 5, guess what's in it?
Tip:Thanksgiving Days

Last edited by Sumasoltin (2012-04-30 23:26:19)

## #25 2012-04-30 23:24:35

anonimnystefy
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### Re: Find an equation showing special graph

Check this out:Some values of k

Last edited by anonimnystefy (2012-04-30 23:24:49)

The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón