Mixing
Given two circles, arc length of the intersection
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If I have the radius of each circle, and the distance between them, how can I calculate the arc length of each arc in their intersection?
geometry
asked Jan 22 at 13:13
zane49erzane49er
31
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show 3 more comments
$begingroup$
If I have the radius of each circle, and the distance between them, how can I calculate the arc length of each arc in their intersection?
geometry
asked Jan 22 at 13:13
zane49erzane49er
31
$endgroup$
$begingroup$
Hint: you can compute the length of the overlap as $(r+r') - d$, where $r,r'$ are the radii and $d$ is the distance between the circle centers.
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– lightxbulb
Jan 22 at 13:17
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@lightxbulb: How does that help?
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– TonyK
Jan 22 at 13:18
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Draw an image and see for yourself. The arc length of the intersection is a function of it @TonyK.
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– lightxbulb
Jan 22 at 13:19
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Not to mention that what he's asking is available with a rudimentary google search, with derivations and everything.
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– lightxbulb
Jan 22 at 13:24
$begingroup$
@lightxbulb: But what to search for? It's like when we asked the teacher how to spell a word, and she said "Look it up in the dictionary!" Perhaps this link might help the OP (although it doesn't use your $(r+r')-d$ expression at all).
$endgroup$
– TonyK
Jan 22 at 13:30
|
show 3 more comments
$begingroup$
If I have the radius of each circle, and the distance between them, how can I calculate the arc length of each arc in their intersection?
geometry
asked Jan 22 at 13:13
zane49erzane49er
31
$endgroup$
If I have the radius of each circle, and the distance between them, how can I calculate the arc length of each arc in their intersection?
geometry
geometry
asked Jan 22 at 13:13
zane49erzane49er
31
asked Jan 22 at 13:13
zane49erzane49er
31
asked Jan 22 at 13:13
zane49erzane49er
31
asked Jan 22 at 13:13
zane49erzane49er
31
asked Jan 22 at 13:13
zane49erzane49er
31
31
$begingroup$
Hint: you can compute the length of the overlap as $(r+r') - d$, where $r,r'$ are the radii and $d$ is the distance between the circle centers.
$endgroup$
– lightxbulb
Jan 22 at 13:17
$begingroup$
@lightxbulb: How does that help?
$endgroup$
– TonyK
Jan 22 at 13:18
$begingroup$
Draw an image and see for yourself. The arc length of the intersection is a function of it @TonyK.
$endgroup$
– lightxbulb
Jan 22 at 13:19
$begingroup$
Not to mention that what he's asking is available with a rudimentary google search, with derivations and everything.
$endgroup$
– lightxbulb
Jan 22 at 13:24
$begingroup$
@lightxbulb: But what to search for? It's like when we asked the teacher how to spell a word, and she said "Look it up in the dictionary!" Perhaps this link might help the OP (although it doesn't use your $(r+r')-d$ expression at all).
$endgroup$
– TonyK
Jan 22 at 13:30
|
show 3 more comments
$begingroup$
Hint: you can compute the length of the overlap as $(r+r') - d$, where $r,r'$ are the radii and $d$ is the distance between the circle centers.
$endgroup$
– lightxbulb
Jan 22 at 13:17
$begingroup$
@lightxbulb: How does that help?
$endgroup$
– TonyK
Jan 22 at 13:18
$begingroup$
Draw an image and see for yourself. The arc length of the intersection is a function of it @TonyK.
$endgroup$
– lightxbulb
Jan 22 at 13:19
$begingroup$
Not to mention that what he's asking is available with a rudimentary google search, with derivations and everything.
$endgroup$
– lightxbulb
Jan 22 at 13:24
$begingroup$
@lightxbulb: But what to search for? It's like when we asked the teacher how to spell a word, and she said "Look it up in the dictionary!" Perhaps this link might help the OP (although it doesn't use your $(r+r')-d$ expression at all).
$endgroup$
– TonyK
Jan 22 at 13:30
$begingroup$
Hint: you can compute the length of the overlap as $(r+r') - d$, where $r,r'$ are the radii and $d$ is the distance between the circle centers.
$endgroup$
– lightxbulb
Jan 22 at 13:17
$begingroup$
Hint: you can compute the length of the overlap as $(r+r') - d$, where $r,r'$ are the radii and $d$ is the distance between the circle centers.
$endgroup$
– lightxbulb
Jan 22 at 13:17
$begingroup$
@lightxbulb: How does that help?
$endgroup$
– TonyK
Jan 22 at 13:18
$begingroup$
@lightxbulb: How does that help?
$endgroup$
– TonyK
Jan 22 at 13:18
$begingroup$
Draw an image and see for yourself. The arc length of the intersection is a function of it @TonyK.
$endgroup$
– lightxbulb
Jan 22 at 13:19
$begingroup$
Draw an image and see for yourself. The arc length of the intersection is a function of it @TonyK.
$endgroup$
– lightxbulb
Jan 22 at 13:19
$begingroup$
Not to mention that what he's asking is available with a rudimentary google search, with derivations and everything.
$endgroup$
– lightxbulb
Jan 22 at 13:24
$begingroup$
Not to mention that what he's asking is available with a rudimentary google search, with derivations and everything.
$endgroup$
– lightxbulb
Jan 22 at 13:24
$begingroup$
@lightxbulb: But what to search for? It's like when we asked the teacher how to spell a word, and she said "Look it up in the dictionary!" Perhaps this link might help the OP (although it doesn't use your $(r+r')-d$ expression at all).
$endgroup$
– TonyK
Jan 22 at 13:30
$begingroup$
@lightxbulb: But what to search for? It's like when we asked the teacher how to spell a word, and she said "Look it up in the dictionary!" Perhaps this link might help the OP (although it doesn't use your $(r+r')-d$ expression at all).
$endgroup$
– TonyK
Jan 22 at 13:30
|
show 3 more comments
1 Answer
1
active
oldest
votes
$begingroup$
Let $T$ be one of the intersection points of the circles. By Heron's formula the area of the triangle $O_1TO_2$ is
$$A_{O_1TO_2}=sqrt{P(P-R_1)(P-R_2)(P-D)},
$$
where $P=frac{R_1+D+R_2}{2}$ is the semi-perimeter of the triangle.
From the other hand the same area can be written as
$$A_{O_1TO_2}=frac{1}{2}R_iDsinphi_i,
$$
where $phi_i$ is the angle at $O_i$.
Finally we have:
$$
Phi_i=2R_iarcsinfrac{2 sqrt{P(P-R_1)(P-R_2)(P-D)}}{DR_i};quad i=1,2,
$$
where $Phi_i$ is the arc length in question.
answered Jan 22 at 13:31
useruser
5,20311030
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Let $T$ be one of the intersection points of the circles. By Heron's formula the area of the triangle $O_1TO_2$ is
$$A_{O_1TO_2}=sqrt{P(P-R_1)(P-R_2)(P-D)},
$$
where $P=frac{R_1+D+R_2}{2}$ is the semi-perimeter of the triangle.
From the other hand the same area can be written as
$$A_{O_1TO_2}=frac{1}{2}R_iDsinphi_i,
$$
where $phi_i$ is the angle at $O_i$.
Finally we have:
$$
Phi_i=2R_iarcsinfrac{2 sqrt{P(P-R_1)(P-R_2)(P-D)}}{DR_i};quad i=1,2,
$$
where $Phi_i$ is the arc length in question.
answered Jan 22 at 13:31
useruser
5,20311030
$endgroup$
add a comment |
$begingroup$
Let $T$ be one of the intersection points of the circles. By Heron's formula the area of the triangle $O_1TO_2$ is
$$A_{O_1TO_2}=sqrt{P(P-R_1)(P-R_2)(P-D)},
$$
where $P=frac{R_1+D+R_2}{2}$ is the semi-perimeter of the triangle.
From the other hand the same area can be written as
$$A_{O_1TO_2}=frac{1}{2}R_iDsinphi_i,
$$
where $phi_i$ is the angle at $O_i$.
Finally we have:
$$
Phi_i=2R_iarcsinfrac{2 sqrt{P(P-R_1)(P-R_2)(P-D)}}{DR_i};quad i=1,2,
$$
where $Phi_i$ is the arc length in question.
answered Jan 22 at 13:31
useruser
5,20311030
$endgroup$
add a comment |
$begingroup$
Let $T$ be one of the intersection points of the circles. By Heron's formula the area of the triangle $O_1TO_2$ is
$$A_{O_1TO_2}=sqrt{P(P-R_1)(P-R_2)(P-D)},
$$
where $P=frac{R_1+D+R_2}{2}$ is the semi-perimeter of the triangle.
From the other hand the same area can be written as
$$A_{O_1TO_2}=frac{1}{2}R_iDsinphi_i,
$$
where $phi_i$ is the angle at $O_i$.
Finally we have:
$$
Phi_i=2R_iarcsinfrac{2 sqrt{P(P-R_1)(P-R_2)(P-D)}}{DR_i};quad i=1,2,
$$
where $Phi_i$ is the arc length in question.
answered Jan 22 at 13:31
useruser
5,20311030
$endgroup$
Let $T$ be one of the intersection points of the circles. By Heron's formula the area of the triangle $O_1TO_2$ is
$$A_{O_1TO_2}=sqrt{P(P-R_1)(P-R_2)(P-D)},
$$
where $P=frac{R_1+D+R_2}{2}$ is the semi-perimeter of the triangle.
From the other hand the same area can be written as
$$A_{O_1TO_2}=frac{1}{2}R_iDsinphi_i,
$$
where $phi_i$ is the angle at $O_i$.
Finally we have:
$$
Phi_i=2R_iarcsinfrac{2 sqrt{P(P-R_1)(P-R_2)(P-D)}}{DR_i};quad i=1,2,
$$
where $Phi_i$ is the arc length in question.
answered Jan 22 at 13:31
useruser
5,20311030
edited Jan 22 at 18:44
answered Jan 22 at 13:31
useruser
5,20311030
answered Jan 22 at 13:31
useruser
5,20311030
answered Jan 22 at 13:31
useruser
5,20311030
5,20311030
add a comment |
add a comment |
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$begingroup$
Hint: you can compute the length of the overlap as $(r+r') - d$, where $r,r'$ are the radii and $d$ is the distance between the circle centers.
$endgroup$
– lightxbulb
Jan 22 at 13:17
$begingroup$
@lightxbulb: How does that help?
$endgroup$
– TonyK
Jan 22 at 13:18
$begingroup$
Draw an image and see for yourself. The arc length of the intersection is a function of it @TonyK.
$endgroup$
– lightxbulb
Jan 22 at 13:19
$begingroup$
Not to mention that what he's asking is available with a rudimentary google search, with derivations and everything.
$endgroup$
– lightxbulb
Jan 22 at 13:24
$begingroup$
@lightxbulb: But what to search for? It's like when we asked the teacher how to spell a word, and she said "Look it up in the dictionary!" Perhaps this link might help the OP (although it doesn't use your $(r+r')-d$ expression at all).
$endgroup$
– TonyK
Jan 22 at 13:30