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#1 2013-06-19 23:14:04

Herc11
Member
Registered: 2013-06-19
Posts: 169

Define the intersection points of polynomials

I am facing the following problem.

Let’s consider 2 points that are not known

I know that from these 2 unknown points

For each of these polynomials I know one point


Is it possible to find the intersection points (i.e. the 2 unknown points) of the aforementioned polynomials?

Last edited by Herc11 (2013-06-19 23:22:12)

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#2 2013-06-19 23:33:57

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 90,818

Re: Define the intersection points of polynomials

Hi Herc11;

Sounds like you have 3 points for each quadratic. That is enough to determine the two of them uniquely.

Wait you do not have 3 points. You have one point and the leading coefficient of each one. That may not be enough.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#3 2013-06-19 23:39:35

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,538

Re: Define the intersection points of polynomials

hi Herc11

Welcome to the forum.

Someone will probably be able to answer this.  There are members who are far brainier than me.  smile

I'll edit your post into Latex to make the powers and subscripts easier to read.  Please check in 5 minutes that I've got it right.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#4 2013-06-19 23:45:05

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

Hmmm.Thats the problem. I know the leading coefficient and one point  from each of the polynomials.

But I m thinking that if I know the leading coef. and one point  from four polynomials

I might be able to find a set of 4 equations to define the four variables x0 x1 y0 y1.

Is ti possible?

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#5 2013-06-19 23:49:29

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 90,818

Re: Define the intersection points of polynomials

Hi;

First of all taking one quadratic at at a time I cannot uniquely determine the other two coefficients just because I know the first one and a point.

If I told you that the the coefficient of x^2 was 2 and the point it passed through was (5,28) you would end up with this.


2x^2 + a*x + b = 28

50 + 5a + b = -22

5a + b = - 22

I am stuck. This has an infinite number of solutions for a and b.

The next quadratic will yield another 2 variables c and d. Coupled with the 4 other variables that define the unknown points we have maybe 4 equations and 8 unknowns. An overdetermined set. This spells doom for a unique solution.

I think I need more relationships between the points and the quadratics.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#6 2013-06-20 00:03:03

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

Ok, That's right.

Let’ s use Newton Interpolation method to construct the known leading coefficient of the quadratic.

a_(2_known )=(y_known-(y_0+(y_1-y_(0  ))/(x_1-x_0 )(x_known-x_1))/((x_known-x_1)(x_known-x_0))

If I use the previous Equation for 4 quadratics, is it possible to define x0 x1 y0  y1  ?

P.S. How can I write efficiently the formulas??

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#7 2013-06-20 00:10:56

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,818

Re: Define the intersection points of polynomials

Hi;

Hold on. There are many interpolation formulas, as a matter of fact in the old days you were not considered a mathematician unless you discovered one.

The fact is you are going to end up with variables a,b,c,d,x0,y0,x1,y1 and we know few facts about them. What relationships we do know will amount to 4 equations in 8 unknowns. Even if these were linear which they are not we would not be able to come up with a unique solution. Interpolation will not help us out. I can get the solutions using mathematica. They will be large because they will have a lot of variables left in them.

Is there anymore that you can provide me, no matter how tiny?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#8 2013-06-20 00:23:32

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

Consider only coefficient a2.

The only unknowns are x0 x1 y0 y1.

If I have the coef.  a2 of four other polynomials could I determine the x0 x1 y0 y1?

I might be cinfused...

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#9 2013-06-20 00:31:14

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 90,818

Re: Define the intersection points of polynomials

Let me give you a concrete example here are two polynomials with the the leading coefficient known. Also each has one point known. Can you determine point A and B?

View Image: 2013-06-19_210627.gif

In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#10 2013-06-20 00:40:48

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

Ok...What I am thinking is:

a2=2=   (7+    (y0-  ( (y1-y0) / (x1-x0))*(1-x1)))/ ( (1-x1)*(1-x0)
similarly for the second polynomial

a2=-3=  (1+ (y0- (( y1-y0)/(x1-x0))*(2-x1))/ (2-x0)*(2-x1)

If i had more polynomials could i find the x0 x1 y0 y1?

Last edited by Herc11 (2013-06-20 00:41:59)

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#11 2013-06-20 00:43:56

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 90,818

Re: Define the intersection points of polynomials

If i had more polynomials could i find the x0 x1 y0 y1

That is what I am saying. You need more relationships between the variables. We call these equations. You have to make sure they are unique equations and not just the same as the others.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#12 2013-06-20 00:44:01

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

It seems that I used wrong a2 but the idea is still the same.

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#13 2013-06-20 00:46:32

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

So, you mean that If I had 4 equations a2=.... I wouldn t be able to define x0,,y1?

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#14 2013-06-20 00:50:34

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 90,818

Re: Define the intersection points of polynomials

You want A, B in my drawing. Therefore you (x0,y0) and (x1,y1) that is 4 variables. To have a chance at getting those 4 as a number I need 4 equations. Then to get the 2 polynomials I need 4 more equations.

I need 8 equations to determine the uniquely. To give you answers that are numbers.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#15 2013-06-20 00:55:03

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

To have a chance at getting those 4 as a number I need 4 equations.

So If you have at your disposal two more polynomials passing from the same intersection points you were able to define x0,y1?

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#16 2013-06-20 00:58:56

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 90,818

Re: Define the intersection points of polynomials

Only if you knew the equations of those polynomials. If you did not you would be just adding more variables!


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#17 2013-06-20 00:58:58

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,673

Re: Define the intersection points of polynomials

I have a feeling that the intersections can be determined. You can set up three equations in x0,y0,x1,y1,a12, and a13. Then each three new equations add just 2 new variables, ai2 and ai3. So, with 12 equations and 12 variables, there is aa chance that the system will be determined.


“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 2013-06-20 00:59:51

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,818

Re: Define the intersection points of polynomials

With 8 and 8 it could be determined. He has more variables than relationships that is the problem.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#19 2013-06-20 01:06:40

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

a2=2=   (7+    (y0-  ( (y1-y0) / (x1-x0))*(1-x1)))/ ( (1-x1)*(1-x0)


a2=-3=  (1+ (y0- (( y1-y0)/(x1-x0))*(2-x1))/ (2-x0)*(2-x1)

a2=8=  (6+ (y0- (( y1-y0)/(x1-x0))*(2-x1))/ (2-x0)*(2-x1)                          polynomial with a2=8 and a point 2,6

a2=11=  (12+ (y0- (( y1-y0)/(x1-x0))*(8-x1))/ (8-x0)*(8-x1)                     polynomial with a2=11 and a point 8,2

I can not solve this system of equations?

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#20 2013-06-20 01:09:10

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 90,818

Re: Define the intersection points of polynomials

Where is (2,6) and (8,2) coming from?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#21 2013-06-20 01:12:06

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

bobbym wrote:

Where is (2,6) and (8,2) coming from?

Consider that there are two more polynomials passing from the intersection points illustrated in your figure and I know their leading coefficient and one point form each of them i.e. 2,6 8,2.

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#22 2013-06-20 01:14:32

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,818

Re: Define the intersection points of polynomials

That will add more variables because you will not know two terms from each of them. You are adding 2 equations and 4 more unknowns. See post #4.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#23 2013-06-20 01:16:25

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

Which are the unknowns?

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#24 2013-06-20 01:18:52

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,818

Re: Define the intersection points of polynomials

You will have the x^2 coefficient but not the x and the constant term for both. You will need to know what the equation is for each.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Offline

#25 2013-06-20 01:20:40

Herc11
Member
Registered: 2013-06-19
Posts: 169

Re: Define the intersection points of polynomials

a2=2=   (7+    (y0-  ( (y1-y0) / (x1-x0))*(1-x1)))/ ( (1-x1)*(1-x0)

If i use this equation for a2, which unknons does it introduce?

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