Using Line Element to find out length of a curve on a circle
$begingroup$
Let us denote a unit sphere by $S$ and assume that $gamma: [0,1] rightarrow S$ is a continuos and differentiable function.
Let us parametrize $S$ by spherical coordinates $(a, b)$ and assume the riemannian metric on $S$ is given by $mathrm{d}s^2=mathrm{d}a^2 + sin^2(a)mathrm{d}b^2$.
To find out the length of $gamma$ we need to evaluate $int_{0}^1mathrm{d}svert_{gamma(t)},mathrm{d}t$.
Question
Circle is also a Riemannian manifold.
Can we calculate the length of a curve on a unit circle using above formula ?
What is the $ds^2$ for a unit circle that is parameterized is only by one parameter $a$?
riemannian-geometry
asked Dec 30 '18 at 16:07
Andrzej GolonkaAndrzej Golonka
488
$endgroup$
add a comment |
$begingroup$
Let us denote a unit sphere by $S$ and assume that $gamma: [0,1] rightarrow S$ is a continuos and differentiable function.
Let us parametrize $S$ by spherical coordinates $(a, b)$ and assume the riemannian metric on $S$ is given by $mathrm{d}s^2=mathrm{d}a^2 + sin^2(a)mathrm{d}b^2$.
To find out the length of $gamma$ we need to evaluate $int_{0}^1mathrm{d}svert_{gamma(t)},mathrm{d}t$.
Question
Circle is also a Riemannian manifold.
Can we calculate the length of a curve on a unit circle using above formula ?
What is the $ds^2$ for a unit circle that is parameterized is only by one parameter $a$?
riemannian-geometry
asked Dec 30 '18 at 16:07
Andrzej GolonkaAndrzej Golonka
488
$endgroup$
add a comment |
$begingroup$
Let us denote a unit sphere by $S$ and assume that $gamma: [0,1] rightarrow S$ is a continuos and differentiable function.
Let us parametrize $S$ by spherical coordinates $(a, b)$ and assume the riemannian metric on $S$ is given by $mathrm{d}s^2=mathrm{d}a^2 + sin^2(a)mathrm{d}b^2$.
To find out the length of $gamma$ we need to evaluate $int_{0}^1mathrm{d}svert_{gamma(t)},mathrm{d}t$.
Question
Circle is also a Riemannian manifold.
Can we calculate the length of a curve on a unit circle using above formula ?
What is the $ds^2$ for a unit circle that is parameterized is only by one parameter $a$?
riemannian-geometry
asked Dec 30 '18 at 16:07
Andrzej GolonkaAndrzej Golonka
488
$endgroup$
Let us denote a unit sphere by $S$ and assume that $gamma: [0,1] rightarrow S$ is a continuos and differentiable function.
Let us parametrize $S$ by spherical coordinates $(a, b)$ and assume the riemannian metric on $S$ is given by $mathrm{d}s^2=mathrm{d}a^2 + sin^2(a)mathrm{d}b^2$.
To find out the length of $gamma$ we need to evaluate $int_{0}^1mathrm{d}svert_{gamma(t)},mathrm{d}t$.
Question
Circle is also a Riemannian manifold.
Can we calculate the length of a curve on a unit circle using above formula ?
What is the $ds^2$ for a unit circle that is parameterized is only by one parameter $a$?
riemannian-geometry
riemannian-geometry
asked Dec 30 '18 at 16:07
Andrzej GolonkaAndrzej Golonka
488
asked Dec 30 '18 at 16:07
Andrzej GolonkaAndrzej Golonka
488
asked Dec 30 '18 at 16:07
Andrzej GolonkaAndrzej Golonka
488
asked Dec 30 '18 at 16:07
Andrzej GolonkaAndrzej Golonka
488
asked Dec 30 '18 at 16:07
Andrzej GolonkaAndrzej Golonka
488
488
add a comment |
add a comment |
0
active
oldest
votes
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%2f3056966%2fusing-line-element-to-find-out-length-of-a-curve-on-a-circle%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3056966%2fusing-line-element-to-find-out-length-of-a-curve-on-a-circle%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
