Intersection points between a circle and a straight line on a sphere












0












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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
















0












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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














0












0








0





$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 21 '18 at 17:25







Neel Basu

















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


















  • $begingroup$
    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










0






active

oldest

votes












Your Answer








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


















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
















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.




draft saved


draft discarded














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





















































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







-

Popular posts from this blog

Plaza Victoria

Image
Plaza Victoria Plaza Victoria en 2017 Otros nombres Plaza de La Victoria Localización El Almendral, Valparaíso, Chile Vías adyacentes Chacabuco, Molina, Plaza Victoria y Edwards Creación 1841 Metro Bellavista Ubicación 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 Coordenadas: 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 [editar datos en Wikidata] La Plaza Victoria , también llamada Plaza de la Victoria , [ 1 ] ​ es una plaza ubicada en el sector El Almendral del plan de la ciudad de Valparaíso, Chile, de importancia patrimonial, que incluye valiosas estatuas y especies vegetales. [ 2 ] ​ La plaza se creó como tal en la primera mitad del siglo XIX, tomando previamente los nombres de Plaza Nueva y Plaza de Orrego . Si bien desde sus inicios tuvo relevancia histórica, no fue sino a partir de fines de los años 1860 cuando comenzó a cobrar mayor importancia y se convirtió en uno de los principales centros sociales de l...
Read more

Puebla de Zaragoza

Image
Para otros usos de este término, véase Puebla (desambiguación). Heroica Puebla de Zaragoza Localidad De arriba a abajo de izquierda a derecha: Estrella de Puebla, Volcán Popocatépetl, Catedral de Puebla, Monumento a Ignacio Zaragoza, Estadio Cuauhtémoc, Angelópolis y Vista panorámica de la ciudad. Escudo Otros nombres: El relicario de América/ La ciudad de los Ángeles/ La Angelópolis Lema: Muy noble y muy leal ciudad de Puebla de los Ángeles Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en México Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en Puebla Mapa interactivo Coordenadas 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Coordenadas: 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Idioma oficial Español Entidad Localidad  • País   México  • Estado Puebla  • Municipio Puebla Presidente municipal Claudia Rivera Vivanc...
Read more

Musa

Image
Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo...
Read more