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 '18 at 1:19
BelowAverageIntelligenceBelowAverageIntelligence
5021213
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 '18 at 1:19
BelowAverageIntelligenceBelowAverageIntelligence
5021213
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 '18 at 1:24
@EthanBolker I was just curious.
– BelowAverageIntelligence
Nov 30 '18 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 '18 at 0:51
Thanks for pointing that out. I've deleted the one in the title.
– BelowAverageIntelligence
Dec 4 '18 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 '18 at 1:19
BelowAverageIntelligenceBelowAverageIntelligence
5021213
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 '18 at 1:19
BelowAverageIntelligenceBelowAverageIntelligence
5021213
asked Nov 30 '18 at 1:19
BelowAverageIntelligenceBelowAverageIntelligence
5021213
edited Dec 4 '18 at 3:54
BelowAverageIntelligence
asked Nov 30 '18 at 1:19
BelowAverageIntelligenceBelowAverageIntelligence
5021213
asked Nov 30 '18 at 1:19
BelowAverageIntelligenceBelowAverageIntelligence
5021213
asked Nov 30 '18 at 1:19
BelowAverageIntelligenceBelowAverageIntelligence
5021213
5021213
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 '18 at 1:24
@EthanBolker I was just curious.
– BelowAverageIntelligence
Nov 30 '18 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 '18 at 0:51
Thanks for pointing that out. I've deleted the one in the title.
– BelowAverageIntelligence
Dec 4 '18 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 '18 at 1:24
@EthanBolker I was just curious.
– BelowAverageIntelligence
Nov 30 '18 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 '18 at 0:51
Thanks for pointing that out. I've deleted the one in the title.
– BelowAverageIntelligence
Dec 4 '18 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 '18 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 '18 at 1:24
@EthanBolker I was just curious.
– BelowAverageIntelligence
Nov 30 '18 at 1:46
@EthanBolker I was just curious.
– BelowAverageIntelligence
Nov 30 '18 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 '18 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 '18 at 0:51
Thanks for pointing that out. I've deleted the one in the title.
– BelowAverageIntelligence
Dec 4 '18 at 3:54
Thanks for pointing that out. I've deleted the one in the title.
– BelowAverageIntelligence
Dec 4 '18 at 3:54
add a comment |
1 Answer
1
active
oldest
votes
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 '18 at 4:18
Ben Bolker
318210
answered Dec 3 '18 at 0:48
Ethan BolkerEthan Bolker
42.1k548111
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3019488%2flevel-curves-of-fr-sum-i-1na-idr-r-i-with-r-i-in-mathbbrn%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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 '18 at 4:18
Ben Bolker
318210
answered Dec 3 '18 at 0:48
Ethan BolkerEthan Bolker
42.1k548111
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 '18 at 4:18
Ben Bolker
318210
answered Dec 3 '18 at 0:48
Ethan BolkerEthan Bolker
42.1k548111
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 '18 at 4:18
Ben Bolker
318210
answered Dec 3 '18 at 0:48
Ethan BolkerEthan Bolker
42.1k548111
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 '18 at 4:18
Ben Bolker
318210
answered Dec 3 '18 at 0:48
Ethan BolkerEthan Bolker
42.1k548111
edited Dec 3 '18 at 4:18
Ben Bolker
318210
edited Dec 3 '18 at 4:18
Ben Bolker
318210
edited Dec 3 '18 at 4:18
Ben Bolker
318210
318210
answered Dec 3 '18 at 0:48
Ethan BolkerEthan Bolker
42.1k548111
answered Dec 3 '18 at 0:48
Ethan BolkerEthan Bolker
42.1k548111
answered Dec 3 '18 at 0:48
Ethan BolkerEthan Bolker
42.1k548111
42.1k548111
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3019488%2flevel-curves-of-fr-sum-i-1na-idr-r-i-with-r-i-in-mathbbrn%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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 '18 at 1:24
@EthanBolker I was just curious.
– BelowAverageIntelligence
Nov 30 '18 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 '18 at 0:51
Thanks for pointing that out. I've deleted the one in the title.
– BelowAverageIntelligence
Dec 4 '18 at 3:54