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#1 2012-04-28 15:17:16

Sumasoltin
Member
Registered: 2012-04-28
Posts: 30

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!

View Image: 人脸形函数.jpg View Image: 囧方程.jpg View Image: 圣诞树方程.JPG View Image: 蜡烛图.png View Image: 蜡烛.png

Last edited by Sumasoltin (2012-04-28 22:18:03)

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#2 2012-04-28 22:16:55

Sumasoltin
Member
Registered: 2012-04-28
Posts: 30

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
n7r999.jpg
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 (2012-05-01 04:41:19)

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#3 2012-04-29 22:03:32

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 87,249

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#4 2012-04-29 23:07:08

Sumasoltin
Member
Registered: 2012-04-28
Posts: 30

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

View Image: 耐克方程图像.png View Image: 耐克方程.png

Last edited by Sumasoltin (2012-04-29 23:19:01)

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#5 2012-04-29 23:20:18

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,544

Re: Find an equation showing special graph

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


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#6 2012-04-29 23:54:23

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,249

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#7 2012-04-30 00:04:27

Sumasoltin
Member
Registered: 2012-04-28
Posts: 30

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 00:05:31)

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#8 2012-04-30 00:07:21

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,249

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#9 2012-04-30 00:12:09

Sumasoltin
Member
Registered: 2012-04-28
Posts: 30

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).

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#10 2012-04-30 00:14:56

Sumasoltin
Member
Registered: 2012-04-28
Posts: 30

Re: Find an equation showing special graph

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

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#11 2012-04-30 00:19:26

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,249

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#12 2012-04-30 00:20:34

Sumasoltin
Member
Registered: 2012-04-28
Posts: 30

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)

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#13 2012-04-30 00:22:24

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,544

Re: Find an equation showing special graph

Still not a heart graph!


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#14 2012-04-30 00:23:56

Sumasoltin
Member
Registered: 2012-04-28
Posts: 30

Re: Find an equation showing special graph

WolframAlpha shows as this.

View Image: 图MSP7421a17ie66e89h2474000025319i543hf8712f.gif View Image: 方MSP78991a17i614e20d3g7a000039b24dga4e171bi7.gif

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#15 2012-04-30 00:27:32

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,544

Re: Find an equation showing special graph

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


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#16 2012-04-30 00:28:28

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,249

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#17 2012-04-30 00:31:01

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,544

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


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#18 2012-04-30 00:32:36

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,249

Re: Find an equation showing special graph

What is cbrt?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#19 2012-04-30 00:33:56

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,544

Re: Find an equation showing special graph

Cube root function.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#20 2012-04-30 01:06:27

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 87,249

Re: Find an equation showing special graph

Got MIF's graph to do it.

View Image: plot.png

In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#21 2012-04-30 01:12:51

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,544

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?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#22 2012-04-30 01:13:01

Sumasoltin
Member
Registered: 2012-04-28
Posts: 30

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)³=(y/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+y²-1=|x|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

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#23 2012-04-30 01:18:38

Sumasoltin
Member
Registered: 2012-04-28
Posts: 30

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.

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#24 2012-04-30 01:23:23

Sumasoltin
Member
Registered: 2012-04-28
Posts: 30

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 01:26:19)

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#25 2012-04-30 01:24:35

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,544

Re: Find an equation showing special graph

Check this out:Some values of k

Last edited by anonimnystefy (2012-04-30 01:24:49)


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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