Intersecting angle of point and line segment
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
Given a line segment $L$ in $mathbb R^2$ ($2D$-Space), represented by two points, and points $P_1$ and $P_2$ (also in $mathbb R^2$) not on the segment, $L$ rotates about $P_1$, its full rotation forming an annulus (doughnut-like shape). Assuming that $P_2$ is inside of the annulus, I need to find the angle that $L$ needs to rotate about $P_1$ so that it intersects $P_2$ (such that $P_2$ lies on $L$ after its rotation).
The closest to an answer I've seen is this. It uses 3 lines in $mathbb R^3$ (with one rotating about another to intersect with the last line), and I can imagine my problem being an analogue to this, the $mathbb R^3$ space being projected onto $mathbb R_2$ with two of the lines perpendicular to the plane of projection, though I struggle to understand how to execute this (and I believe there must be a simpler solution). Nor can I find a simple trigonometric answer.
I appreciate any and all help you can provide,
rbjacob
calculus linear-algebra geometry trigonometry
edited Sep 1 at 2:10
Andrei
7,9052923
asked Sep 1 at 1:56
rbjacob
82
add a comment |Â
up vote
1
down vote
favorite
Given a line segment $L$ in $mathbb R^2$ ($2D$-Space), represented by two points, and points $P_1$ and $P_2$ (also in $mathbb R^2$) not on the segment, $L$ rotates about $P_1$, its full rotation forming an annulus (doughnut-like shape). Assuming that $P_2$ is inside of the annulus, I need to find the angle that $L$ needs to rotate about $P_1$ so that it intersects $P_2$ (such that $P_2$ lies on $L$ after its rotation).
The closest to an answer I've seen is this. It uses 3 lines in $mathbb R^3$ (with one rotating about another to intersect with the last line), and I can imagine my problem being an analogue to this, the $mathbb R^3$ space being projected onto $mathbb R_2$ with two of the lines perpendicular to the plane of projection, though I struggle to understand how to execute this (and I believe there must be a simpler solution). Nor can I find a simple trigonometric answer.
I appreciate any and all help you can provide,
rbjacob
calculus linear-algebra geometry trigonometry
edited Sep 1 at 2:10
Andrei
7,9052923
asked Sep 1 at 1:56
rbjacob
82
Find the intersection of $L$ with the circle through $P_2$ with center $P_1$ and measure the resulting angle.
– amd
Sep 1 at 3:59
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Given a line segment $L$ in $mathbb R^2$ ($2D$-Space), represented by two points, and points $P_1$ and $P_2$ (also in $mathbb R^2$) not on the segment, $L$ rotates about $P_1$, its full rotation forming an annulus (doughnut-like shape). Assuming that $P_2$ is inside of the annulus, I need to find the angle that $L$ needs to rotate about $P_1$ so that it intersects $P_2$ (such that $P_2$ lies on $L$ after its rotation).
The closest to an answer I've seen is this. It uses 3 lines in $mathbb R^3$ (with one rotating about another to intersect with the last line), and I can imagine my problem being an analogue to this, the $mathbb R^3$ space being projected onto $mathbb R_2$ with two of the lines perpendicular to the plane of projection, though I struggle to understand how to execute this (and I believe there must be a simpler solution). Nor can I find a simple trigonometric answer.
I appreciate any and all help you can provide,
rbjacob
calculus linear-algebra geometry trigonometry
edited Sep 1 at 2:10
Andrei
7,9052923
asked Sep 1 at 1:56
rbjacob
82
Given a line segment $L$ in $mathbb R^2$ ($2D$-Space), represented by two points, and points $P_1$ and $P_2$ (also in $mathbb R^2$) not on the segment, $L$ rotates about $P_1$, its full rotation forming an annulus (doughnut-like shape). Assuming that $P_2$ is inside of the annulus, I need to find the angle that $L$ needs to rotate about $P_1$ so that it intersects $P_2$ (such that $P_2$ lies on $L$ after its rotation).
The closest to an answer I've seen is this. It uses 3 lines in $mathbb R^3$ (with one rotating about another to intersect with the last line), and I can imagine my problem being an analogue to this, the $mathbb R^3$ space being projected onto $mathbb R_2$ with two of the lines perpendicular to the plane of projection, though I struggle to understand how to execute this (and I believe there must be a simpler solution). Nor can I find a simple trigonometric answer.
I appreciate any and all help you can provide,
rbjacob
calculus linear-algebra geometry trigonometry
calculus linear-algebra geometry trigonometry
edited Sep 1 at 2:10
Andrei
7,9052923
asked Sep 1 at 1:56
rbjacob
82
edited Sep 1 at 2:10
Andrei
7,9052923
asked Sep 1 at 1:56
rbjacob
82
edited Sep 1 at 2:10
Andrei
7,9052923
edited Sep 1 at 2:10
Andrei
7,9052923
edited Sep 1 at 2:10
Andrei
7,9052923
7,9052923
asked Sep 1 at 1:56
rbjacob
82
asked Sep 1 at 1:56
rbjacob
82
asked Sep 1 at 1:56
rbjacob
82
82
Find the intersection of $L$ with the circle through $P_2$ with center $P_1$ and measure the resulting angle.
– amd
Sep 1 at 3:59
add a comment |Â
Find the intersection of $L$ with the circle through $P_2$ with center $P_1$ and measure the resulting angle.
– amd
Sep 1 at 3:59
Find the intersection of $L$ with the circle through $P_2$ with center $P_1$ and measure the resulting angle.
– amd
Sep 1 at 3:59
Find the intersection of $L$ with the circle through $P_2$ with center $P_1$ and measure the resulting angle.
– amd
Sep 1 at 3:59
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
accepted
Just as @amd suggested:
- Calculate $r= |P_2 - P_1|$.
- Calculate $Q$, the intersection of the circle
centered on $P_1$ of radius $r$, with $L$. - Calculate the angle $theta$ from $P_1 Q$ to $P_1 P_2$.
    Â
answered Sep 2 at 1:32
Joseph O'Rourke
17.3k248104
Elegant solution, a lot simpler than I had conceived it would be. Thanks for the help Joseph and @amd!
– rbjacob
Sep 2 at 4:03
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
Just as @amd suggested:
- Calculate $r= |P_2 - P_1|$.
- Calculate $Q$, the intersection of the circle
centered on $P_1$ of radius $r$, with $L$. - Calculate the angle $theta$ from $P_1 Q$ to $P_1 P_2$.
    Â
answered Sep 2 at 1:32
Joseph O'Rourke
17.3k248104
Elegant solution, a lot simpler than I had conceived it would be. Thanks for the help Joseph and @amd!
– rbjacob
Sep 2 at 4:03
add a comment |Â
up vote
0
down vote
accepted
Just as @amd suggested:
- Calculate $r= |P_2 - P_1|$.
- Calculate $Q$, the intersection of the circle
centered on $P_1$ of radius $r$, with $L$. - Calculate the angle $theta$ from $P_1 Q$ to $P_1 P_2$.
    Â
answered Sep 2 at 1:32
Joseph O'Rourke
17.3k248104
Elegant solution, a lot simpler than I had conceived it would be. Thanks for the help Joseph and @amd!
– rbjacob
Sep 2 at 4:03
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
Just as @amd suggested:
- Calculate $r= |P_2 - P_1|$.
- Calculate $Q$, the intersection of the circle
centered on $P_1$ of radius $r$, with $L$. - Calculate the angle $theta$ from $P_1 Q$ to $P_1 P_2$.
    Â
answered Sep 2 at 1:32
Joseph O'Rourke
17.3k248104
Just as @amd suggested:
- Calculate $r= |P_2 - P_1|$.
- Calculate $Q$, the intersection of the circle
centered on $P_1$ of radius $r$, with $L$. - Calculate the angle $theta$ from $P_1 Q$ to $P_1 P_2$.
    Â
answered Sep 2 at 1:32
Joseph O'Rourke
17.3k248104
answered Sep 2 at 1:32
Joseph O'Rourke
17.3k248104
answered Sep 2 at 1:32
Joseph O'Rourke
17.3k248104
answered Sep 2 at 1:32
Joseph O'Rourke
17.3k248104
17.3k248104
Elegant solution, a lot simpler than I had conceived it would be. Thanks for the help Joseph and @amd!
– rbjacob
Sep 2 at 4:03
add a comment |Â
Elegant solution, a lot simpler than I had conceived it would be. Thanks for the help Joseph and @amd!
– rbjacob
Sep 2 at 4:03
Elegant solution, a lot simpler than I had conceived it would be. Thanks for the help Joseph and @amd!
– rbjacob
Sep 2 at 4:03
Elegant solution, a lot simpler than I had conceived it would be. Thanks for the help Joseph and @amd!
– rbjacob
Sep 2 at 4:03
add a comment |Â
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
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2901285%2fintersecting-angle-of-point-and-line-segment%23new-answer', 'question_page');
);
Post as a guest
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
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
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
Find the intersection of $L$ with the circle through $P_2$ with center $P_1$ and measure the resulting angle.
– amd
Sep 1 at 3:59