dependence of angles of a geodesic line on a sphere
$begingroup$
Is it possible to prove a unique dependence of $theta$ and $phi$ angles for a geodesic line, knowing only the definition of a geodesic line?
Otherwise, this task can be set as follows: we can find the length of a line in space, using
L= $int sqrt{(dot x(t)^2 + dot y(t)^2 + dot z(t)^2)}dt $
On the surface of the sphere, we can set a point using two coordinates (in my case these are spherical coordinates: $theta$ and $phi$). How can we prove that we can use $theta$ or $phi$ as a parameter t?
geometry differential-geometry spherical-geometry
asked Dec 21 '18 at 15:09
krys tofearkrys tofear
11
$endgroup$
add a comment |
$begingroup$
Is it possible to prove a unique dependence of $theta$ and $phi$ angles for a geodesic line, knowing only the definition of a geodesic line?
Otherwise, this task can be set as follows: we can find the length of a line in space, using
L= $int sqrt{(dot x(t)^2 + dot y(t)^2 + dot z(t)^2)}dt $
On the surface of the sphere, we can set a point using two coordinates (in my case these are spherical coordinates: $theta$ and $phi$). How can we prove that we can use $theta$ or $phi$ as a parameter t?
geometry differential-geometry spherical-geometry
asked Dec 21 '18 at 15:09
krys tofearkrys tofear
11
$endgroup$
$begingroup$
I assume that you are using spherical coordinates. What if you have two points on the equator? They have the same $theta$, so the geodesic is only $phi$ dependent. Multiple $phi$ angles correspond to the same $theta$.
$endgroup$
– Andrei
Dec 21 '18 at 15:28
add a comment |
$begingroup$
Is it possible to prove a unique dependence of $theta$ and $phi$ angles for a geodesic line, knowing only the definition of a geodesic line?
Otherwise, this task can be set as follows: we can find the length of a line in space, using
L= $int sqrt{(dot x(t)^2 + dot y(t)^2 + dot z(t)^2)}dt $
On the surface of the sphere, we can set a point using two coordinates (in my case these are spherical coordinates: $theta$ and $phi$). How can we prove that we can use $theta$ or $phi$ as a parameter t?
geometry differential-geometry spherical-geometry
asked Dec 21 '18 at 15:09
krys tofearkrys tofear
11
$endgroup$
Is it possible to prove a unique dependence of $theta$ and $phi$ angles for a geodesic line, knowing only the definition of a geodesic line?
Otherwise, this task can be set as follows: we can find the length of a line in space, using
L= $int sqrt{(dot x(t)^2 + dot y(t)^2 + dot z(t)^2)}dt $
On the surface of the sphere, we can set a point using two coordinates (in my case these are spherical coordinates: $theta$ and $phi$). How can we prove that we can use $theta$ or $phi$ as a parameter t?
geometry differential-geometry spherical-geometry
geometry differential-geometry spherical-geometry
asked Dec 21 '18 at 15:09
krys tofearkrys tofear
11
asked Dec 21 '18 at 15:09
krys tofearkrys tofear
11
edited Dec 21 '18 at 19:49
krys tofear
asked Dec 21 '18 at 15:09
krys tofearkrys tofear
11
asked Dec 21 '18 at 15:09
krys tofearkrys tofear
11
asked Dec 21 '18 at 15:09
krys tofearkrys tofear
11
11
$begingroup$
I assume that you are using spherical coordinates. What if you have two points on the equator? They have the same $theta$, so the geodesic is only $phi$ dependent. Multiple $phi$ angles correspond to the same $theta$.
$endgroup$
– Andrei
Dec 21 '18 at 15:28
add a comment |
$begingroup$
I assume that you are using spherical coordinates. What if you have two points on the equator? They have the same $theta$, so the geodesic is only $phi$ dependent. Multiple $phi$ angles correspond to the same $theta$.
$endgroup$
– Andrei
Dec 21 '18 at 15:28
$begingroup$
I assume that you are using spherical coordinates. What if you have two points on the equator? They have the same $theta$, so the geodesic is only $phi$ dependent. Multiple $phi$ angles correspond to the same $theta$.
$endgroup$
– Andrei
Dec 21 '18 at 15:28
$begingroup$
I assume that you are using spherical coordinates. What if you have two points on the equator? They have the same $theta$, so the geodesic is only $phi$ dependent. Multiple $phi$ angles correspond to the same $theta$.
$endgroup$
– Andrei
Dec 21 '18 at 15:28
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%2f3048585%2fdependence-of-angles-of-a-geodesic-line-on-a-sphere%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%2f3048585%2fdependence-of-angles-of-a-geodesic-line-on-a-sphere%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
$begingroup$
I assume that you are using spherical coordinates. What if you have two points on the equator? They have the same $theta$, so the geodesic is only $phi$ dependent. Multiple $phi$ angles correspond to the same $theta$.
$endgroup$
– Andrei
Dec 21 '18 at 15:28