Level curves of $f(r) = sum_{i=1}^na_id(r,r_i)$ with $r_iin mathbb{R}^n$
Let $$f(r) = sum_{i=1}^na_id(r,r_i)$$
Where each $r_i in mathbb{R}^n$ and each $a_iin mathbb{R}$
and $d:mathbb{R}^ntimesmathbb{R}^nrightarrowmathbb{R}$ denotes the usual distance function. That is, $f$ is a linear function of the distance from $n$ points in $n$-dimensional space. What are the level surfaces (curves) of $f$? When $n=2$, I believe these are conic sections (please correct me if I'm wrong about this). Are there analogs for $n>2$? If this cannot be answered in full generality, (which I'm guessing it can't) what area of math deals with such questions?
metric-spaces conic-sections
asked Nov 30 at 1:19
BelowAverageIntelligence
5221213
add a comment |
Let $$f(r) = sum_{i=1}^na_id(r,r_i)$$
Where each $r_i in mathbb{R}^n$ and each $a_iin mathbb{R}$
and $d:mathbb{R}^ntimesmathbb{R}^nrightarrowmathbb{R}$ denotes the usual distance function. That is, $f$ is a linear function of the distance from $n$ points in $n$-dimensional space. What are the level surfaces (curves) of $f$? When $n=2$, I believe these are conic sections (please correct me if I'm wrong about this). Are there analogs for $n>2$? If this cannot be answered in full generality, (which I'm guessing it can't) what area of math deals with such questions?
metric-spaces conic-sections
asked Nov 30 at 1:19
BelowAverageIntelligence
5221213
1
They are ellipses when $n=2$ and the coefficients are equal. Interesting question. Do you need the answer for some purpose (edit to tell us if so) or are you just curious.
– Ethan Bolker
Nov 30 at 1:24
@EthanBolker I was just curious.
– BelowAverageIntelligence
Nov 30 at 1:46
The title and body of your question contradict one another. The former uses the square of the distance, the latter the distance. I think the latter is what you mean.
– Ethan Bolker
Dec 3 at 0:51
Thanks for pointing that out. I've deleted the one in the title.
– BelowAverageIntelligence
Dec 4 at 3:54
add a comment |
Let $$f(r) = sum_{i=1}^na_id(r,r_i)$$
Where each $r_i in mathbb{R}^n$ and each $a_iin mathbb{R}$
and $d:mathbb{R}^ntimesmathbb{R}^nrightarrowmathbb{R}$ denotes the usual distance function. That is, $f$ is a linear function of the distance from $n$ points in $n$-dimensional space. What are the level surfaces (curves) of $f$? When $n=2$, I believe these are conic sections (please correct me if I'm wrong about this). Are there analogs for $n>2$? If this cannot be answered in full generality, (which I'm guessing it can't) what area of math deals with such questions?
metric-spaces conic-sections
asked Nov 30 at 1:19
BelowAverageIntelligence
5221213
Let $$f(r) = sum_{i=1}^na_id(r,r_i)$$
Where each $r_i in mathbb{R}^n$ and each $a_iin mathbb{R}$
and $d:mathbb{R}^ntimesmathbb{R}^nrightarrowmathbb{R}$ denotes the usual distance function. That is, $f$ is a linear function of the distance from $n$ points in $n$-dimensional space. What are the level surfaces (curves) of $f$? When $n=2$, I believe these are conic sections (please correct me if I'm wrong about this). Are there analogs for $n>2$? If this cannot be answered in full generality, (which I'm guessing it can't) what area of math deals with such questions?
metric-spaces conic-sections
metric-spaces conic-sections
asked Nov 30 at 1:19
BelowAverageIntelligence
5221213
asked Nov 30 at 1:19
BelowAverageIntelligence
5221213
edited Dec 4 at 3:54
asked Nov 30 at 1:19
BelowAverageIntelligence
5221213
asked Nov 30 at 1:19
BelowAverageIntelligence
5221213
asked Nov 30 at 1:19
BelowAverageIntelligence
5221213
5221213
1
They are ellipses when $n=2$ and the coefficients are equal. Interesting question. Do you need the answer for some purpose (edit to tell us if so) or are you just curious.
– Ethan Bolker
Nov 30 at 1:24
@EthanBolker I was just curious.
– BelowAverageIntelligence
Nov 30 at 1:46
The title and body of your question contradict one another. The former uses the square of the distance, the latter the distance. I think the latter is what you mean.
– Ethan Bolker
Dec 3 at 0:51
Thanks for pointing that out. I've deleted the one in the title.
– BelowAverageIntelligence
Dec 4 at 3:54
add a comment |
1
They are ellipses when $n=2$ and the coefficients are equal. Interesting question. Do you need the answer for some purpose (edit to tell us if so) or are you just curious.
– Ethan Bolker
Nov 30 at 1:24
@EthanBolker I was just curious.
– BelowAverageIntelligence
Nov 30 at 1:46
The title and body of your question contradict one another. The former uses the square of the distance, the latter the distance. I think the latter is what you mean.
– Ethan Bolker
Dec 3 at 0:51
Thanks for pointing that out. I've deleted the one in the title.
– BelowAverageIntelligence
Dec 4 at 3:54
1
1
They are ellipses when $n=2$ and the coefficients are equal. Interesting question. Do you need the answer for some purpose (edit to tell us if so) or are you just curious.
– Ethan Bolker
Nov 30 at 1:24
They are ellipses when $n=2$ and the coefficients are equal. Interesting question. Do you need the answer for some purpose (edit to tell us if so) or are you just curious.
– Ethan Bolker
Nov 30 at 1:24
@EthanBolker I was just curious.
– BelowAverageIntelligence
Nov 30 at 1:46
@EthanBolker I was just curious.
– BelowAverageIntelligence
Nov 30 at 1:46
The title and body of your question contradict one another. The former uses the square of the distance, the latter the distance. I think the latter is what you mean.
– Ethan Bolker
Dec 3 at 0:51
The title and body of your question contradict one another. The former uses the square of the distance, the latter the distance. I think the latter is what you mean.
– Ethan Bolker
Dec 3 at 0:51
Thanks for pointing that out. I've deleted the one in the title.
– BelowAverageIntelligence
Dec 4 at 3:54
Thanks for pointing that out. I've deleted the one in the title.
– BelowAverageIntelligence
Dec 4 at 3:54
add a comment |
1 Answer
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Information about the two dimensional case.
Let $A$ and $B$ be points in the plane and suppose $a+b = 1$. Define
$$
f(X) = a cdot d(X,A) + b cdot d(X,B).
$$
Then the level curves of $f$ are convex ovals surrounding $A$ and $B$. They are described by fourth degree equations in the coordinates that result when you start by squaring both sides of
$$
c - a cdot d(X,A) = b cdot d(X,B).
$$
That will leave a square root on the left. Move all the other terms on the left to the right and square again to see the fourth degree expression. It's not pretty.
When $a = b = 1/2$ the level curves are ellipses with foci $A$ and $B$: the third and fourth order terms cancel.
When $a=1, b=0$ the level curves are circles centered at $A$.
When $c$ is very large $A$ and $B$ are close together relative to $c$. The level curve looks more like a circle.
edited Dec 3 at 4:18
Ben Bolker
303210
answered Dec 3 at 0:48
Ethan Bolker
40.9k546108
add a comment |
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1 Answer
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Information about the two dimensional case.
Let $A$ and $B$ be points in the plane and suppose $a+b = 1$. Define
$$
f(X) = a cdot d(X,A) + b cdot d(X,B).
$$
Then the level curves of $f$ are convex ovals surrounding $A$ and $B$. They are described by fourth degree equations in the coordinates that result when you start by squaring both sides of
$$
c - a cdot d(X,A) = b cdot d(X,B).
$$
That will leave a square root on the left. Move all the other terms on the left to the right and square again to see the fourth degree expression. It's not pretty.
When $a = b = 1/2$ the level curves are ellipses with foci $A$ and $B$: the third and fourth order terms cancel.
When $a=1, b=0$ the level curves are circles centered at $A$.
When $c$ is very large $A$ and $B$ are close together relative to $c$. The level curve looks more like a circle.
edited Dec 3 at 4:18
Ben Bolker
303210
answered Dec 3 at 0:48
Ethan Bolker
40.9k546108
add a comment |
Information about the two dimensional case.
Let $A$ and $B$ be points in the plane and suppose $a+b = 1$. Define
$$
f(X) = a cdot d(X,A) + b cdot d(X,B).
$$
Then the level curves of $f$ are convex ovals surrounding $A$ and $B$. They are described by fourth degree equations in the coordinates that result when you start by squaring both sides of
$$
c - a cdot d(X,A) = b cdot d(X,B).
$$
That will leave a square root on the left. Move all the other terms on the left to the right and square again to see the fourth degree expression. It's not pretty.
When $a = b = 1/2$ the level curves are ellipses with foci $A$ and $B$: the third and fourth order terms cancel.
When $a=1, b=0$ the level curves are circles centered at $A$.
When $c$ is very large $A$ and $B$ are close together relative to $c$. The level curve looks more like a circle.
edited Dec 3 at 4:18
Ben Bolker
303210
answered Dec 3 at 0:48
Ethan Bolker
40.9k546108
add a comment |
Information about the two dimensional case.
Let $A$ and $B$ be points in the plane and suppose $a+b = 1$. Define
$$
f(X) = a cdot d(X,A) + b cdot d(X,B).
$$
Then the level curves of $f$ are convex ovals surrounding $A$ and $B$. They are described by fourth degree equations in the coordinates that result when you start by squaring both sides of
$$
c - a cdot d(X,A) = b cdot d(X,B).
$$
That will leave a square root on the left. Move all the other terms on the left to the right and square again to see the fourth degree expression. It's not pretty.
When $a = b = 1/2$ the level curves are ellipses with foci $A$ and $B$: the third and fourth order terms cancel.
When $a=1, b=0$ the level curves are circles centered at $A$.
When $c$ is very large $A$ and $B$ are close together relative to $c$. The level curve looks more like a circle.
edited Dec 3 at 4:18
Ben Bolker
303210
answered Dec 3 at 0:48
Ethan Bolker
40.9k546108
Information about the two dimensional case.
Let $A$ and $B$ be points in the plane and suppose $a+b = 1$. Define
$$
f(X) = a cdot d(X,A) + b cdot d(X,B).
$$
Then the level curves of $f$ are convex ovals surrounding $A$ and $B$. They are described by fourth degree equations in the coordinates that result when you start by squaring both sides of
$$
c - a cdot d(X,A) = b cdot d(X,B).
$$
That will leave a square root on the left. Move all the other terms on the left to the right and square again to see the fourth degree expression. It's not pretty.
When $a = b = 1/2$ the level curves are ellipses with foci $A$ and $B$: the third and fourth order terms cancel.
When $a=1, b=0$ the level curves are circles centered at $A$.
When $c$ is very large $A$ and $B$ are close together relative to $c$. The level curve looks more like a circle.
edited Dec 3 at 4:18
Ben Bolker
303210
answered Dec 3 at 0:48
Ethan Bolker
40.9k546108
edited Dec 3 at 4:18
Ben Bolker
303210
edited Dec 3 at 4:18
Ben Bolker
303210
edited Dec 3 at 4:18
Ben Bolker
303210
303210
answered Dec 3 at 0:48
Ethan Bolker
40.9k546108
answered Dec 3 at 0:48
Ethan Bolker
40.9k546108
answered Dec 3 at 0:48
Ethan Bolker
40.9k546108
40.9k546108
add a comment |
add a comment |
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1
They are ellipses when $n=2$ and the coefficients are equal. Interesting question. Do you need the answer for some purpose (edit to tell us if so) or are you just curious.
– Ethan Bolker
Nov 30 at 1:24
@EthanBolker I was just curious.
– BelowAverageIntelligence
Nov 30 at 1:46
The title and body of your question contradict one another. The former uses the square of the distance, the latter the distance. I think the latter is what you mean.
– Ethan Bolker
Dec 3 at 0:51
Thanks for pointing that out. I've deleted the one in the title.
– BelowAverageIntelligence
Dec 4 at 3:54