Intersection points between a circle and a straight line on a sphere
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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
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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
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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?
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– fleablood
Dec 21 '18 at 17:27
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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
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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..
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– Narasimham
Dec 21 '18 at 19:01
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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 |
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$begingroup$
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