Intersection points between a circle and a straight line on a sphere
$begingroup$
I have a circle on the surface of a sphere. I need to check whether the circle intersects with a given straight line or not. The center of the circle $c$ is given in terms of latitude and longitude $(phi, lambda)$, and the radius $r$ is given in meters. The latitude and longitudes of two endpoints of the straight line are given $(phi_{p}, lambda_{p}), (phi_{q}, lambda_{q})$.
I assume $t$ is a point on $pq$ such that $ct$ is the shortest distance from $c$ to $pq$. In the following scenarios intersections are possible. In all other cases no intersection will happen.
$$Vert cp Vert geq r, Vert cq Vert geq r, Vert ct Vert leq rRightarrow 2;text{intersections}$$
$$Vert cp Vert < r, Vert cq Vert geq r, Vert ct Vert leq rRightarrow 1;text{intersections}$$
$$Vert cp Vert geq r, Vert cq Vert < r, Vert ct Vert leq rRightarrow 1;text{intersections}$$
In the first case I assume the two intersection points are $x, y$. The point near $p$ is $x$, and the point near $q$ is $y$. We already know $Vert pq Vert$. Using the Law of Cosines we can derive $Vert xt Vert$, thus $Vert pq Vert = Vert pt Vert - Vert xt Vert$.
$$Vert xt Vert = cos^{-1}Big(frac{cos(r)}{cos(Vert ct Vert)}Big)$$
But after this I am loosing track. I understand that both $x, y$ are in the great circle constructed by $p, q$. But I don't know how to determine the latitude and longitudes of $x$ and $y$ from this information. I targeted the first case assuming solving this will solve the other two cases. My actual problem is to know all the intersection points.
geometry spherical-coordinates spherical-geometry spherical-trigonometry
asked Dec 21 '18 at 17:11
Neel BasuNeel Basu
201110
$endgroup$
add a comment |
$begingroup$
I have a circle on the surface of a sphere. I need to check whether the circle intersects with a given straight line or not. The center of the circle $c$ is given in terms of latitude and longitude $(phi, lambda)$, and the radius $r$ is given in meters. The latitude and longitudes of two endpoints of the straight line are given $(phi_{p}, lambda_{p}), (phi_{q}, lambda_{q})$.
I assume $t$ is a point on $pq$ such that $ct$ is the shortest distance from $c$ to $pq$. In the following scenarios intersections are possible. In all other cases no intersection will happen.
$$Vert cp Vert geq r, Vert cq Vert geq r, Vert ct Vert leq rRightarrow 2;text{intersections}$$
$$Vert cp Vert < r, Vert cq Vert geq r, Vert ct Vert leq rRightarrow 1;text{intersections}$$
$$Vert cp Vert geq r, Vert cq Vert < r, Vert ct Vert leq rRightarrow 1;text{intersections}$$
In the first case I assume the two intersection points are $x, y$. The point near $p$ is $x$, and the point near $q$ is $y$. We already know $Vert pq Vert$. Using the Law of Cosines we can derive $Vert xt Vert$, thus $Vert pq Vert = Vert pt Vert - Vert xt Vert$.
$$Vert xt Vert = cos^{-1}Big(frac{cos(r)}{cos(Vert ct Vert)}Big)$$
But after this I am loosing track. I understand that both $x, y$ are in the great circle constructed by $p, q$. But I don't know how to determine the latitude and longitudes of $x$ and $y$ from this information. I targeted the first case assuming solving this will solve the other two cases. My actual problem is to know all the intersection points.
geometry spherical-coordinates spherical-geometry spherical-trigonometry
asked Dec 21 '18 at 17:11
Neel BasuNeel Basu
201110
$endgroup$
$begingroup$
Your title says ON a sphere. Am I to assume the "on" refers only to the intersection points and the circle and not the "straight line" or is the "straight line" a great circle?
$endgroup$
– fleablood
Dec 21 '18 at 17:27
$begingroup$
The straight line is on a great circle. Like the straight line and the circle are drawn on the ground with a chalk.
$endgroup$
– Neel Basu
Dec 21 '18 at 17:28
$begingroup$
HINT: A stereo-graphic projection is useful as it maps circles to circles. A great circle and and a segment of another great circle $ pq $ are given. When the latter segment is extended around the sphere there will be two points of intersection. There would be $(0,1,2 ) $ number of projections depending on where the projected arc of circle starts and where mapped length of segment $ p^{'} q^{'}$ compared to $2 pi r^{'}$ ends..
$endgroup$
– Narasimham
Dec 21 '18 at 19:01
$begingroup$
Sorry, I am not accustomed to these type of math. I need to plot those intersection points on a map based User Interface.
$endgroup$
– Neel Basu
Dec 22 '18 at 6:43
add a comment |
$begingroup$
I have a circle on the surface of a sphere. I need to check whether the circle intersects with a given straight line or not. The center of the circle $c$ is given in terms of latitude and longitude $(phi, lambda)$, and the radius $r$ is given in meters. The latitude and longitudes of two endpoints of the straight line are given $(phi_{p}, lambda_{p}), (phi_{q}, lambda_{q})$.
I assume $t$ is a point on $pq$ such that $ct$ is the shortest distance from $c$ to $pq$. In the following scenarios intersections are possible. In all other cases no intersection will happen.
$$Vert cp Vert geq r, Vert cq Vert geq r, Vert ct Vert leq rRightarrow 2;text{intersections}$$
$$Vert cp Vert < r, Vert cq Vert geq r, Vert ct Vert leq rRightarrow 1;text{intersections}$$
$$Vert cp Vert geq r, Vert cq Vert < r, Vert ct Vert leq rRightarrow 1;text{intersections}$$
In the first case I assume the two intersection points are $x, y$. The point near $p$ is $x$, and the point near $q$ is $y$. We already know $Vert pq Vert$. Using the Law of Cosines we can derive $Vert xt Vert$, thus $Vert pq Vert = Vert pt Vert - Vert xt Vert$.
$$Vert xt Vert = cos^{-1}Big(frac{cos(r)}{cos(Vert ct Vert)}Big)$$
But after this I am loosing track. I understand that both $x, y$ are in the great circle constructed by $p, q$. But I don't know how to determine the latitude and longitudes of $x$ and $y$ from this information. I targeted the first case assuming solving this will solve the other two cases. My actual problem is to know all the intersection points.
geometry spherical-coordinates spherical-geometry spherical-trigonometry
asked Dec 21 '18 at 17:11
Neel BasuNeel Basu
201110
$endgroup$
I have a circle on the surface of a sphere. I need to check whether the circle intersects with a given straight line or not. The center of the circle $c$ is given in terms of latitude and longitude $(phi, lambda)$, and the radius $r$ is given in meters. The latitude and longitudes of two endpoints of the straight line are given $(phi_{p}, lambda_{p}), (phi_{q}, lambda_{q})$.
I assume $t$ is a point on $pq$ such that $ct$ is the shortest distance from $c$ to $pq$. In the following scenarios intersections are possible. In all other cases no intersection will happen.
$$Vert cp Vert geq r, Vert cq Vert geq r, Vert ct Vert leq rRightarrow 2;text{intersections}$$
$$Vert cp Vert < r, Vert cq Vert geq r, Vert ct Vert leq rRightarrow 1;text{intersections}$$
$$Vert cp Vert geq r, Vert cq Vert < r, Vert ct Vert leq rRightarrow 1;text{intersections}$$
In the first case I assume the two intersection points are $x, y$. The point near $p$ is $x$, and the point near $q$ is $y$. We already know $Vert pq Vert$. Using the Law of Cosines we can derive $Vert xt Vert$, thus $Vert pq Vert = Vert pt Vert - Vert xt Vert$.
$$Vert xt Vert = cos^{-1}Big(frac{cos(r)}{cos(Vert ct Vert)}Big)$$
But after this I am loosing track. I understand that both $x, y$ are in the great circle constructed by $p, q$. But I don't know how to determine the latitude and longitudes of $x$ and $y$ from this information. I targeted the first case assuming solving this will solve the other two cases. My actual problem is to know all the intersection points.
geometry spherical-coordinates spherical-geometry spherical-trigonometry
geometry spherical-coordinates spherical-geometry spherical-trigonometry
asked Dec 21 '18 at 17:11
Neel BasuNeel Basu
201110
asked Dec 21 '18 at 17:11
Neel BasuNeel Basu
201110
edited Dec 21 '18 at 17:25
Neel Basu
asked Dec 21 '18 at 17:11
Neel BasuNeel Basu
201110
asked Dec 21 '18 at 17:11
Neel BasuNeel Basu
201110
asked Dec 21 '18 at 17:11
Neel BasuNeel Basu
201110
201110
$begingroup$
Your title says ON a sphere. Am I to assume the "on" refers only to the intersection points and the circle and not the "straight line" or is the "straight line" a great circle?
$endgroup$
– fleablood
Dec 21 '18 at 17:27
$begingroup$
The straight line is on a great circle. Like the straight line and the circle are drawn on the ground with a chalk.
$endgroup$
– Neel Basu
Dec 21 '18 at 17:28
$begingroup$
HINT: A stereo-graphic projection is useful as it maps circles to circles. A great circle and and a segment of another great circle $ pq $ are given. When the latter segment is extended around the sphere there will be two points of intersection. There would be $(0,1,2 ) $ number of projections depending on where the projected arc of circle starts and where mapped length of segment $ p^{'} q^{'}$ compared to $2 pi r^{'}$ ends..
$endgroup$
– Narasimham
Dec 21 '18 at 19:01
$begingroup$
Sorry, I am not accustomed to these type of math. I need to plot those intersection points on a map based User Interface.
$endgroup$
– Neel Basu
Dec 22 '18 at 6:43
add a comment |
$begingroup$
Your title says ON a sphere. Am I to assume the "on" refers only to the intersection points and the circle and not the "straight line" or is the "straight line" a great circle?
$endgroup$
– fleablood
Dec 21 '18 at 17:27
$begingroup$
The straight line is on a great circle. Like the straight line and the circle are drawn on the ground with a chalk.
$endgroup$
– Neel Basu
Dec 21 '18 at 17:28
$begingroup$
HINT: A stereo-graphic projection is useful as it maps circles to circles. A great circle and and a segment of another great circle $ pq $ are given. When the latter segment is extended around the sphere there will be two points of intersection. There would be $(0,1,2 ) $ number of projections depending on where the projected arc of circle starts and where mapped length of segment $ p^{'} q^{'}$ compared to $2 pi r^{'}$ ends..
$endgroup$
– Narasimham
Dec 21 '18 at 19:01
$begingroup$
Sorry, I am not accustomed to these type of math. I need to plot those intersection points on a map based User Interface.
$endgroup$
– Neel Basu
Dec 22 '18 at 6:43
$begingroup$
Your title says ON a sphere. Am I to assume the "on" refers only to the intersection points and the circle and not the "straight line" or is the "straight line" a great circle?
$endgroup$
– fleablood
Dec 21 '18 at 17:27
$begingroup$
Your title says ON a sphere. Am I to assume the "on" refers only to the intersection points and the circle and not the "straight line" or is the "straight line" a great circle?
$endgroup$
– fleablood
Dec 21 '18 at 17:27
$begingroup$
The straight line is on a great circle. Like the straight line and the circle are drawn on the ground with a chalk.
$endgroup$
– Neel Basu
Dec 21 '18 at 17:28
$begingroup$
The straight line is on a great circle. Like the straight line and the circle are drawn on the ground with a chalk.
$endgroup$
– Neel Basu
Dec 21 '18 at 17:28
$begingroup$
HINT: A stereo-graphic projection is useful as it maps circles to circles. A great circle and and a segment of another great circle $ pq $ are given. When the latter segment is extended around the sphere there will be two points of intersection. There would be $(0,1,2 ) $ number of projections depending on where the projected arc of circle starts and where mapped length of segment $ p^{'} q^{'}$ compared to $2 pi r^{'}$ ends..
$endgroup$
– Narasimham
Dec 21 '18 at 19:01
$begingroup$
HINT: A stereo-graphic projection is useful as it maps circles to circles. A great circle and and a segment of another great circle $ pq $ are given. When the latter segment is extended around the sphere there will be two points of intersection. There would be $(0,1,2 ) $ number of projections depending on where the projected arc of circle starts and where mapped length of segment $ p^{'} q^{'}$ compared to $2 pi r^{'}$ ends..
$endgroup$
– Narasimham
Dec 21 '18 at 19:01
$begingroup$
Sorry, I am not accustomed to these type of math. I need to plot those intersection points on a map based User Interface.
$endgroup$
– Neel Basu
Dec 22 '18 at 6:43
$begingroup$
Sorry, I am not accustomed to these type of math. I need to plot those intersection points on a map based User Interface.
$endgroup$
– Neel Basu
Dec 22 '18 at 6:43
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%2f3048712%2fintersection-points-between-a-circle-and-a-straight-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%2f3048712%2fintersection-points-between-a-circle-and-a-straight-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
$begingroup$
Your title says ON a sphere. Am I to assume the "on" refers only to the intersection points and the circle and not the "straight line" or is the "straight line" a great circle?
$endgroup$
– fleablood
Dec 21 '18 at 17:27
$begingroup$
The straight line is on a great circle. Like the straight line and the circle are drawn on the ground with a chalk.
$endgroup$
– Neel Basu
Dec 21 '18 at 17:28
$begingroup$
HINT: A stereo-graphic projection is useful as it maps circles to circles. A great circle and and a segment of another great circle $ pq $ are given. When the latter segment is extended around the sphere there will be two points of intersection. There would be $(0,1,2 ) $ number of projections depending on where the projected arc of circle starts and where mapped length of segment $ p^{'} q^{'}$ compared to $2 pi r^{'}$ ends..
$endgroup$
– Narasimham
Dec 21 '18 at 19:01
$begingroup$
Sorry, I am not accustomed to these type of math. I need to plot those intersection points on a map based User Interface.
$endgroup$
– Neel Basu
Dec 22 '18 at 6:43