Find the area of the hyperbolic (geodesic) triangle with vertices (0, 0),(0, 1) and (1, 0).
0
I'm not really sure where to start for this question. I'm aware of the Gauss-Bonnet theorem $Area = pi -(A+B+C)$ where $A,B,C$ are the interior angles of the triangle. However I am not sure how to go about calculating these angles given the vertices.
hyperbolic-geometry geodesic noneuclidean-geometry
asked Dec 9 '18 at 22:54
martinhynesone
367
|
show 10 more comments
0
I'm not really sure where to start for this question. I'm aware of the Gauss-Bonnet theorem $Area = pi -(A+B+C)$ where $A,B,C$ are the interior angles of the triangle. However I am not sure how to go about calculating these angles given the vertices.
hyperbolic-geometry geodesic noneuclidean-geometry
asked Dec 9 '18 at 22:54
martinhynesone
367
1
Where do these points live? Apparently in neither the hyperbolic half-plane nor the hyperbolic disk. So what's your model for the geometry?
– Ted Shifrin
Dec 9 '18 at 22:59
@TedShifrin could be a doubly ideal triangle in the disc
– Will Jagy
Dec 9 '18 at 23:03
@WillJagy: Indeed, could be ... Could be doubly ideal in the half-plane, too, for that matter.
– Ted Shifrin
Dec 9 '18 at 23:12
1
LOL, @Will, I think a piece of paper and pencil will suffice. And knowledge of what curves are the geodesics.
– Ted Shifrin
Dec 9 '18 at 23:29
1
Yes, you need a semicircle perpendicular to the $x$-axis passing through $(0,1)$ and $(1,0)$. I believe $pi/2$ is correct.
– Ted Shifrin
Dec 10 '18 at 0:57
|
show 10 more comments
0
0
0
I'm not really sure where to start for this question. I'm aware of the Gauss-Bonnet theorem $Area = pi -(A+B+C)$ where $A,B,C$ are the interior angles of the triangle. However I am not sure how to go about calculating these angles given the vertices.
hyperbolic-geometry geodesic noneuclidean-geometry
asked Dec 9 '18 at 22:54
martinhynesone
367
I'm not really sure where to start for this question. I'm aware of the Gauss-Bonnet theorem $Area = pi -(A+B+C)$ where $A,B,C$ are the interior angles of the triangle. However I am not sure how to go about calculating these angles given the vertices.
hyperbolic-geometry geodesic noneuclidean-geometry
hyperbolic-geometry geodesic noneuclidean-geometry
asked Dec 9 '18 at 22:54
martinhynesone
367
asked Dec 9 '18 at 22:54
martinhynesone
367
asked Dec 9 '18 at 22:54
martinhynesone
367
asked Dec 9 '18 at 22:54
martinhynesone
367
asked Dec 9 '18 at 22:54
martinhynesone
367
367
1
Where do these points live? Apparently in neither the hyperbolic half-plane nor the hyperbolic disk. So what's your model for the geometry?
– Ted Shifrin
Dec 9 '18 at 22:59
@TedShifrin could be a doubly ideal triangle in the disc
– Will Jagy
Dec 9 '18 at 23:03
@WillJagy: Indeed, could be ... Could be doubly ideal in the half-plane, too, for that matter.
– Ted Shifrin
Dec 9 '18 at 23:12
1
LOL, @Will, I think a piece of paper and pencil will suffice. And knowledge of what curves are the geodesics.
– Ted Shifrin
Dec 9 '18 at 23:29
1
Yes, you need a semicircle perpendicular to the $x$-axis passing through $(0,1)$ and $(1,0)$. I believe $pi/2$ is correct.
– Ted Shifrin
Dec 10 '18 at 0:57
|
show 10 more comments
1
Where do these points live? Apparently in neither the hyperbolic half-plane nor the hyperbolic disk. So what's your model for the geometry?
– Ted Shifrin
Dec 9 '18 at 22:59
@TedShifrin could be a doubly ideal triangle in the disc
– Will Jagy
Dec 9 '18 at 23:03
@WillJagy: Indeed, could be ... Could be doubly ideal in the half-plane, too, for that matter.
– Ted Shifrin
Dec 9 '18 at 23:12
1
LOL, @Will, I think a piece of paper and pencil will suffice. And knowledge of what curves are the geodesics.
– Ted Shifrin
Dec 9 '18 at 23:29
1
Yes, you need a semicircle perpendicular to the $x$-axis passing through $(0,1)$ and $(1,0)$. I believe $pi/2$ is correct.
– Ted Shifrin
Dec 10 '18 at 0:57
1
1
Where do these points live? Apparently in neither the hyperbolic half-plane nor the hyperbolic disk. So what's your model for the geometry?
– Ted Shifrin
Dec 9 '18 at 22:59
Where do these points live? Apparently in neither the hyperbolic half-plane nor the hyperbolic disk. So what's your model for the geometry?
– Ted Shifrin
Dec 9 '18 at 22:59
@TedShifrin could be a doubly ideal triangle in the disc
– Will Jagy
Dec 9 '18 at 23:03
@TedShifrin could be a doubly ideal triangle in the disc
– Will Jagy
Dec 9 '18 at 23:03
@WillJagy: Indeed, could be ... Could be doubly ideal in the half-plane, too, for that matter.
– Ted Shifrin
Dec 9 '18 at 23:12
@WillJagy: Indeed, could be ... Could be doubly ideal in the half-plane, too, for that matter.
– Ted Shifrin
Dec 9 '18 at 23:12
1
1
LOL, @Will, I think a piece of paper and pencil will suffice. And knowledge of what curves are the geodesics.
– Ted Shifrin
Dec 9 '18 at 23:29
LOL, @Will, I think a piece of paper and pencil will suffice. And knowledge of what curves are the geodesics.
– Ted Shifrin
Dec 9 '18 at 23:29
1
1
Yes, you need a semicircle perpendicular to the $x$-axis passing through $(0,1)$ and $(1,0)$. I believe $pi/2$ is correct.
– Ted Shifrin
Dec 10 '18 at 0:57
Yes, you need a semicircle perpendicular to the $x$-axis passing through $(0,1)$ and $(1,0)$. I believe $pi/2$ is correct.
– Ted Shifrin
Dec 10 '18 at 0:57
|
show 10 more comments
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
});
}
});
draft saved
draft discarded
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%2f3033153%2ffind-the-area-of-the-hyperbolic-geodesic-triangle-with-vertices-0-0-0-1%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
draft saved
draft discarded
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
To learn more, see our tips on writing great answers.
draft saved
draft discarded
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%2f3033153%2ffind-the-area-of-the-hyperbolic-geodesic-triangle-with-vertices-0-0-0-1%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
Where do these points live? Apparently in neither the hyperbolic half-plane nor the hyperbolic disk. So what's your model for the geometry?
– Ted Shifrin
Dec 9 '18 at 22:59
@TedShifrin could be a doubly ideal triangle in the disc
– Will Jagy
Dec 9 '18 at 23:03
@WillJagy: Indeed, could be ... Could be doubly ideal in the half-plane, too, for that matter.
– Ted Shifrin
Dec 9 '18 at 23:12
1
LOL, @Will, I think a piece of paper and pencil will suffice. And knowledge of what curves are the geodesics.
– Ted Shifrin
Dec 9 '18 at 23:29
1
Yes, you need a semicircle perpendicular to the $x$-axis passing through $(0,1)$ and $(1,0)$. I believe $pi/2$ is correct.
– Ted Shifrin
Dec 10 '18 at 0:57