Given $n$ points, what is the locus of points $X$ such that the sum of the squares of the distances from $X$...

0

$begingroup$

Given $n$ points $P_{1},P_{2},dots P_{n}$ and a real number $c,$ find the locus of points $X$ such that $$sum_{i=1}^{n}XP_{1}^{2}=c.$$

Actually, I'm also interested in a more general case: Given $n$ points $P_{1},P_{2},dots ,P_{n},$ and reals $r_{1},r_{2},dots r_{n},$ and $c,$ find the locus of points $X$ such that $$sum_{i=1}^{n}r_{1}XP_{1}^{2}=c.$$

Is the locus anything significant? Is there a way of constructing it?

Also, is there a way of finding this locus without resorting to analytical techniques?

Thanks.

geometry locus

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edited Jan 21 at 1:08

P-addict

asked Jan 20 at 1:56

P-addictP-addict

304

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The answer can be easily found once you expand out the square of distances: it is a circle, centred at the centroid of the points. The second case is similar.
  $endgroup$
  – Aretino
  Jan 20 at 15:54
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The word "sum" is lacking in your title.
  $endgroup$
  – Jean Marie
  Jan 20 at 19:30
  

  
  
  
  

add a comment | 

0

$begingroup$

Given $n$ points $P_{1},P_{2},dots P_{n}$ and a real number $c,$ find the locus of points $X$ such that $$sum_{i=1}^{n}XP_{1}^{2}=c.$$

Actually, I'm also interested in a more general case: Given $n$ points $P_{1},P_{2},dots ,P_{n},$ and reals $r_{1},r_{2},dots r_{n},$ and $c,$ find the locus of points $X$ such that $$sum_{i=1}^{n}r_{1}XP_{1}^{2}=c.$$

Is the locus anything significant? Is there a way of constructing it?

Also, is there a way of finding this locus without resorting to analytical techniques?

Thanks.

geometry locus

share|cite|improve this question

edited Jan 21 at 1:08

P-addict

asked Jan 20 at 1:56

P-addictP-addict

304

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The answer can be easily found once you expand out the square of distances: it is a circle, centred at the centroid of the points. The second case is similar.
  $endgroup$
  – Aretino
  Jan 20 at 15:54
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The word "sum" is lacking in your title.
  $endgroup$
  – Jean Marie
  Jan 20 at 19:30
  

  
  
  
  

add a comment | 

0

0

0

0

$begingroup$

Given $n$ points $P_{1},P_{2},dots P_{n}$ and a real number $c,$ find the locus of points $X$ such that $$sum_{i=1}^{n}XP_{1}^{2}=c.$$

Actually, I'm also interested in a more general case: Given $n$ points $P_{1},P_{2},dots ,P_{n},$ and reals $r_{1},r_{2},dots r_{n},$ and $c,$ find the locus of points $X$ such that $$sum_{i=1}^{n}r_{1}XP_{1}^{2}=c.$$

Is the locus anything significant? Is there a way of constructing it?

Also, is there a way of finding this locus without resorting to analytical techniques?

Thanks.

geometry locus

share|cite|improve this question

edited Jan 21 at 1:08

P-addict

asked Jan 20 at 1:56

P-addictP-addict

304

$endgroup$

Given $n$ points $P_{1},P_{2},dots P_{n}$ and a real number $c,$ find the locus of points $X$ such that $$sum_{i=1}^{n}XP_{1}^{2}=c.$$

Actually, I'm also interested in a more general case: Given $n$ points $P_{1},P_{2},dots ,P_{n},$ and reals $r_{1},r_{2},dots r_{n},$ and $c,$ find the locus of points $X$ such that $$sum_{i=1}^{n}r_{1}XP_{1}^{2}=c.$$

Is the locus anything significant? Is there a way of constructing it?

Also, is there a way of finding this locus without resorting to analytical techniques?

Thanks.

geometry locus

geometry locus

share|cite|improve this question

edited Jan 21 at 1:08

P-addict

asked Jan 20 at 1:56

P-addictP-addict

304

share|cite|improve this question

edited Jan 21 at 1:08

P-addict

asked Jan 20 at 1:56

P-addictP-addict

304

share|cite|improve this question

share|cite|improve this question

edited Jan 21 at 1:08

P-addict

edited Jan 21 at 1:08

P-addict

edited Jan 21 at 1:08

P-addict

asked Jan 20 at 1:56

P-addictP-addict

304

asked Jan 20 at 1:56

P-addictP-addict

304

asked Jan 20 at 1:56

P-addictP-addict

304

304

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The answer can be easily found once you expand out the square of distances: it is a circle, centred at the centroid of the points. The second case is similar.
  $endgroup$
  – Aretino
  Jan 20 at 15:54
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The word "sum" is lacking in your title.
  $endgroup$
  – Jean Marie
  Jan 20 at 19:30
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The answer can be easily found once you expand out the square of distances: it is a circle, centred at the centroid of the points. The second case is similar.
  $endgroup$
  – Aretino
  Jan 20 at 15:54
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The word "sum" is lacking in your title.
  $endgroup$
  – Jean Marie
  Jan 20 at 19:30
  

  
  
  
  

1

1

$begingroup$
The answer can be easily found once you expand out the square of distances: it is a circle, centred at the centroid of the points. The second case is similar.
$endgroup$
– Aretino
Jan 20 at 15:54

$begingroup$
The answer can be easily found once you expand out the square of distances: it is a circle, centred at the centroid of the points. The second case is similar.
$endgroup$
– Aretino
Jan 20 at 15:54

$begingroup$
The word "sum" is lacking in your title.
$endgroup$
– Jean Marie
Jan 20 at 19:30

$begingroup$
The word "sum" is lacking in your title.
$endgroup$
– Jean Marie
Jan 20 at 19:30

add a comment | 

1 Answer
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$begingroup$

@Aretino has answered you concerning an analytical technique.

Here is a (classical) vector technique for this issue.

Consider the case of three points $P_1,P_2,P_3$ with centroid $G$ (the general case with $n$ points and hopefully weights for these points is completely similar).

We are looking for the locus of points $X$ such that

$$(XP_1)^2 + (XP_2)^2 + (XP_3)^2=c, text{a given constant}$$

But we can write :

$$(XP_1)^2 + (XP_2)^2 + (XP_3)^2=overrightarrow{XP_1}^2+overrightarrow{XP_2}^2+overrightarrow{XP_3}^2=$$

$$=(overrightarrow{XG}+overrightarrow{GP_1})^2+(overrightarrow{XG}+overrightarrow{GP_2})^2+(overrightarrow{XG}+overrightarrow{GP_3})^2$$

Expanding, we get :

$$3 overrightarrow{XG}^2+2overrightarrow{XG} . underbrace{sum_i overrightarrow{GP_i}}_{= 0}+underbrace{sum_i overrightarrow{GP_i}^2}_{text{another constant} k} = c.$$

Can you conclude from here that the locus of points $X$ is either a circle centered in $G$ or the void set ?

share|cite|improve this answer

edited Jan 21 at 1:13

answered Jan 20 at 19:48

Jean MarieJean Marie

30.4k42153

$endgroup$

add a comment | 

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1 Answer
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1 Answer
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oldest

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active

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2

$begingroup$

@Aretino has answered you concerning an analytical technique.

Here is a (classical) vector technique for this issue.

Consider the case of three points $P_1,P_2,P_3$ with centroid $G$ (the general case with $n$ points and hopefully weights for these points is completely similar).

We are looking for the locus of points $X$ such that

$$(XP_1)^2 + (XP_2)^2 + (XP_3)^2=c, text{a given constant}$$

But we can write :

$$(XP_1)^2 + (XP_2)^2 + (XP_3)^2=overrightarrow{XP_1}^2+overrightarrow{XP_2}^2+overrightarrow{XP_3}^2=$$

$$=(overrightarrow{XG}+overrightarrow{GP_1})^2+(overrightarrow{XG}+overrightarrow{GP_2})^2+(overrightarrow{XG}+overrightarrow{GP_3})^2$$

Expanding, we get :

$$3 overrightarrow{XG}^2+2overrightarrow{XG} . underbrace{sum_i overrightarrow{GP_i}}_{= 0}+underbrace{sum_i overrightarrow{GP_i}^2}_{text{another constant} k} = c.$$

Can you conclude from here that the locus of points $X$ is either a circle centered in $G$ or the void set ?

share|cite|improve this answer

edited Jan 21 at 1:13

answered Jan 20 at 19:48

Jean MarieJean Marie

30.4k42153

$endgroup$

add a comment | 

2

$begingroup$

@Aretino has answered you concerning an analytical technique.

Here is a (classical) vector technique for this issue.

Consider the case of three points $P_1,P_2,P_3$ with centroid $G$ (the general case with $n$ points and hopefully weights for these points is completely similar).

We are looking for the locus of points $X$ such that

$$(XP_1)^2 + (XP_2)^2 + (XP_3)^2=c, text{a given constant}$$

But we can write :

$$(XP_1)^2 + (XP_2)^2 + (XP_3)^2=overrightarrow{XP_1}^2+overrightarrow{XP_2}^2+overrightarrow{XP_3}^2=$$

$$=(overrightarrow{XG}+overrightarrow{GP_1})^2+(overrightarrow{XG}+overrightarrow{GP_2})^2+(overrightarrow{XG}+overrightarrow{GP_3})^2$$

Expanding, we get :

$$3 overrightarrow{XG}^2+2overrightarrow{XG} . underbrace{sum_i overrightarrow{GP_i}}_{= 0}+underbrace{sum_i overrightarrow{GP_i}^2}_{text{another constant} k} = c.$$

Can you conclude from here that the locus of points $X$ is either a circle centered in $G$ or the void set ?

share|cite|improve this answer

edited Jan 21 at 1:13

answered Jan 20 at 19:48

Jean MarieJean Marie

30.4k42153

$endgroup$

add a comment | 

2

2

2

$begingroup$

@Aretino has answered you concerning an analytical technique.

Here is a (classical) vector technique for this issue.

Consider the case of three points $P_1,P_2,P_3$ with centroid $G$ (the general case with $n$ points and hopefully weights for these points is completely similar).

We are looking for the locus of points $X$ such that

$$(XP_1)^2 + (XP_2)^2 + (XP_3)^2=c, text{a given constant}$$

But we can write :

$$(XP_1)^2 + (XP_2)^2 + (XP_3)^2=overrightarrow{XP_1}^2+overrightarrow{XP_2}^2+overrightarrow{XP_3}^2=$$

$$=(overrightarrow{XG}+overrightarrow{GP_1})^2+(overrightarrow{XG}+overrightarrow{GP_2})^2+(overrightarrow{XG}+overrightarrow{GP_3})^2$$

Expanding, we get :

$$3 overrightarrow{XG}^2+2overrightarrow{XG} . underbrace{sum_i overrightarrow{GP_i}}_{= 0}+underbrace{sum_i overrightarrow{GP_i}^2}_{text{another constant} k} = c.$$

Can you conclude from here that the locus of points $X$ is either a circle centered in $G$ or the void set ?

share|cite|improve this answer

edited Jan 21 at 1:13

answered Jan 20 at 19:48

Jean MarieJean Marie

30.4k42153

$endgroup$

@Aretino has answered you concerning an analytical technique.

Here is a (classical) vector technique for this issue.

Consider the case of three points $P_1,P_2,P_3$ with centroid $G$ (the general case with $n$ points and hopefully weights for these points is completely similar).

We are looking for the locus of points $X$ such that

$$(XP_1)^2 + (XP_2)^2 + (XP_3)^2=c, text{a given constant}$$

But we can write :

$$(XP_1)^2 + (XP_2)^2 + (XP_3)^2=overrightarrow{XP_1}^2+overrightarrow{XP_2}^2+overrightarrow{XP_3}^2=$$

$$=(overrightarrow{XG}+overrightarrow{GP_1})^2+(overrightarrow{XG}+overrightarrow{GP_2})^2+(overrightarrow{XG}+overrightarrow{GP_3})^2$$

Expanding, we get :

$$3 overrightarrow{XG}^2+2overrightarrow{XG} . underbrace{sum_i overrightarrow{GP_i}}_{= 0}+underbrace{sum_i overrightarrow{GP_i}^2}_{text{another constant} k} = c.$$

Can you conclude from here that the locus of points $X$ is either a circle centered in $G$ or the void set ?

share|cite|improve this answer

edited Jan 21 at 1:13

answered Jan 20 at 19:48

Jean MarieJean Marie

30.4k42153

share|cite|improve this answer

share|cite|improve this answer

edited Jan 21 at 1:13

edited Jan 21 at 1:13

edited Jan 21 at 1:13

answered Jan 20 at 19:48

Jean MarieJean Marie

30.4k42153

answered Jan 20 at 19:48

Jean MarieJean Marie

30.4k42153

answered Jan 20 at 19:48

Jean MarieJean Marie

30.4k42153

30.4k42153

add a comment | 

add a comment | 

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