Calculate center coordinates of circles surrounding a larger circle
$begingroup$
I want to draw, say, 8 smaller circles that are adjacent to the big circle the edge of a big circle, similar to this picture.
I know the center coordinates of the bigger circle $(A, B)$, its radius $(R)$,radius of the smaller circles $(r)$, and the number of circles I want to draw $(n)$.
My question is very similar to the one discussed there, with one exception. I want a formula that calculates center coordinates of circles adjacent, not those on the edge of a bigger circle.
Mathematics is not my strongest side (to say the least), so I'd greatly appreciate any help. Thank you!
geometry circle
edited Jan 14 at 20:22
dantopa
6,46942243
asked Jan 14 at 20:07
Anton LeontyevAnton Leontyev
1
$endgroup$
|
show 2 more comments
$begingroup$
I want to draw, say, 8 smaller circles that are adjacent to the big circle the edge of a big circle, similar to this picture.
I know the center coordinates of the bigger circle $(A, B)$, its radius $(R)$,radius of the smaller circles $(r)$, and the number of circles I want to draw $(n)$.
My question is very similar to the one discussed there, with one exception. I want a formula that calculates center coordinates of circles adjacent, not those on the edge of a bigger circle.
Mathematics is not my strongest side (to say the least), so I'd greatly appreciate any help. Thank you!
geometry circle
edited Jan 14 at 20:22
dantopa
6,46942243
asked Jan 14 at 20:07
Anton LeontyevAnton Leontyev
1
$endgroup$
$begingroup$
I think $n$ determines $r$, so for example if you want $6$ externally tangent circles then $r$ is fixed at $r=R$ (all $7$ circles are the same size in that special case). In general, you could write $r$ as a function of $n$ and it would be strictly decreasing.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:13
$begingroup$
I mention that because you cannot presuppose $n$ and $r$ - most pairs that you choose will lead to impossible situations. Just presuppose $n$ and $R$ and those will together determine $r$.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:14
$begingroup$
Thank you, that makes sense (in lay terminology, only certain amount of circles of size will fit, correct?)
$endgroup$
– Anton Leontyev
Jan 14 at 20:16
$begingroup$
Connect the centers of the $n$ externally tangent circles and you will have a regular $n$-gon
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:16
$begingroup$
If you choose the number of small circles and the size of the big circle, the size of the small circles is fixed, invariable, set, locked.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:17
|
show 2 more comments
$begingroup$
I want to draw, say, 8 smaller circles that are adjacent to the big circle the edge of a big circle, similar to this picture.
I know the center coordinates of the bigger circle $(A, B)$, its radius $(R)$,radius of the smaller circles $(r)$, and the number of circles I want to draw $(n)$.
My question is very similar to the one discussed there, with one exception. I want a formula that calculates center coordinates of circles adjacent, not those on the edge of a bigger circle.
Mathematics is not my strongest side (to say the least), so I'd greatly appreciate any help. Thank you!
geometry circle
edited Jan 14 at 20:22
dantopa
6,46942243
asked Jan 14 at 20:07
Anton LeontyevAnton Leontyev
1
$endgroup$
I want to draw, say, 8 smaller circles that are adjacent to the big circle the edge of a big circle, similar to this picture.
I know the center coordinates of the bigger circle $(A, B)$, its radius $(R)$,radius of the smaller circles $(r)$, and the number of circles I want to draw $(n)$.
My question is very similar to the one discussed there, with one exception. I want a formula that calculates center coordinates of circles adjacent, not those on the edge of a bigger circle.
Mathematics is not my strongest side (to say the least), so I'd greatly appreciate any help. Thank you!
geometry circle
geometry circle
edited Jan 14 at 20:22
dantopa
6,46942243
asked Jan 14 at 20:07
Anton LeontyevAnton Leontyev
1
edited Jan 14 at 20:22
dantopa
6,46942243
asked Jan 14 at 20:07
Anton LeontyevAnton Leontyev
1
edited Jan 14 at 20:22
dantopa
6,46942243
edited Jan 14 at 20:22
dantopa
6,46942243
edited Jan 14 at 20:22
dantopa
6,46942243
6,46942243
asked Jan 14 at 20:07
Anton LeontyevAnton Leontyev
1
asked Jan 14 at 20:07
Anton LeontyevAnton Leontyev
1
asked Jan 14 at 20:07
Anton LeontyevAnton Leontyev
1
1
$begingroup$
I think $n$ determines $r$, so for example if you want $6$ externally tangent circles then $r$ is fixed at $r=R$ (all $7$ circles are the same size in that special case). In general, you could write $r$ as a function of $n$ and it would be strictly decreasing.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:13
$begingroup$
I mention that because you cannot presuppose $n$ and $r$ - most pairs that you choose will lead to impossible situations. Just presuppose $n$ and $R$ and those will together determine $r$.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:14
$begingroup$
Thank you, that makes sense (in lay terminology, only certain amount of circles of size will fit, correct?)
$endgroup$
– Anton Leontyev
Jan 14 at 20:16
$begingroup$
Connect the centers of the $n$ externally tangent circles and you will have a regular $n$-gon
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:16
$begingroup$
If you choose the number of small circles and the size of the big circle, the size of the small circles is fixed, invariable, set, locked.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:17
|
show 2 more comments
$begingroup$
I think $n$ determines $r$, so for example if you want $6$ externally tangent circles then $r$ is fixed at $r=R$ (all $7$ circles are the same size in that special case). In general, you could write $r$ as a function of $n$ and it would be strictly decreasing.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:13
$begingroup$
I mention that because you cannot presuppose $n$ and $r$ - most pairs that you choose will lead to impossible situations. Just presuppose $n$ and $R$ and those will together determine $r$.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:14
$begingroup$
Thank you, that makes sense (in lay terminology, only certain amount of circles of size will fit, correct?)
$endgroup$
– Anton Leontyev
Jan 14 at 20:16
$begingroup$
Connect the centers of the $n$ externally tangent circles and you will have a regular $n$-gon
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:16
$begingroup$
If you choose the number of small circles and the size of the big circle, the size of the small circles is fixed, invariable, set, locked.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:17
$begingroup$
I think $n$ determines $r$, so for example if you want $6$ externally tangent circles then $r$ is fixed at $r=R$ (all $7$ circles are the same size in that special case). In general, you could write $r$ as a function of $n$ and it would be strictly decreasing.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:13
$begingroup$
I think $n$ determines $r$, so for example if you want $6$ externally tangent circles then $r$ is fixed at $r=R$ (all $7$ circles are the same size in that special case). In general, you could write $r$ as a function of $n$ and it would be strictly decreasing.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:13
$begingroup$
I mention that because you cannot presuppose $n$ and $r$ - most pairs that you choose will lead to impossible situations. Just presuppose $n$ and $R$ and those will together determine $r$.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:14
$begingroup$
I mention that because you cannot presuppose $n$ and $r$ - most pairs that you choose will lead to impossible situations. Just presuppose $n$ and $R$ and those will together determine $r$.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:14
$begingroup$
Thank you, that makes sense (in lay terminology, only certain amount of circles of size will fit, correct?)
$endgroup$
– Anton Leontyev
Jan 14 at 20:16
$begingroup$
Thank you, that makes sense (in lay terminology, only certain amount of circles of size will fit, correct?)
$endgroup$
– Anton Leontyev
Jan 14 at 20:16
$begingroup$
Connect the centers of the $n$ externally tangent circles and you will have a regular $n$-gon
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:16
$begingroup$
Connect the centers of the $n$ externally tangent circles and you will have a regular $n$-gon
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:16
$begingroup$
If you choose the number of small circles and the size of the big circle, the size of the small circles is fixed, invariable, set, locked.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:17
$begingroup$
If you choose the number of small circles and the size of the big circle, the size of the small circles is fixed, invariable, set, locked.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:17
|
show 2 more comments
1 Answer
1
active
oldest
votes
$begingroup$
For the 'adjacent' circles use : $delta = 360/n$ where $n$ is the number of circles you want. Then the centers are $c_i = ((R+r)cos idelta + phi, (R+r)sin idelta + phi)$, where $i=0,1,...,n-1$ and $phi$ is some offset rotation. Note that the small circles will not necessarily touch, but they will touch the large circle.
Edit:
Here's a shadertoy example:
https://www.shadertoy.com/view/wsfGWj
answered Jan 14 at 20:15
lightxbulblightxbulb
75619
$endgroup$
$begingroup$
thank you, your comment is of immense help
$endgroup$
– Anton Leontyev
Jan 14 at 20:22
$begingroup$
I just edited it, realized I had already divided by $n$, so check the corrected version. @AntonLeontyev
$endgroup$
– lightxbulb
Jan 14 at 20:25
$begingroup$
Could you please clarify what do you mean by "offset rotation"? Thank you!
$endgroup$
– Anton Leontyev
Jan 14 at 20:29
$begingroup$
@AntonLeontyev check the shadertoy example I gave, the offset is literally what rotates the circles around.
$endgroup$
– lightxbulb
Jan 14 at 20:41
$begingroup$
Thank you so much, now I understand.
$endgroup$
– Anton Leontyev
Jan 14 at 20:50
add a comment |
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1 Answer
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1 Answer
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votes
$begingroup$
For the 'adjacent' circles use : $delta = 360/n$ where $n$ is the number of circles you want. Then the centers are $c_i = ((R+r)cos idelta + phi, (R+r)sin idelta + phi)$, where $i=0,1,...,n-1$ and $phi$ is some offset rotation. Note that the small circles will not necessarily touch, but they will touch the large circle.
Edit:
Here's a shadertoy example:
https://www.shadertoy.com/view/wsfGWj
answered Jan 14 at 20:15
lightxbulblightxbulb
75619
$endgroup$
$begingroup$
thank you, your comment is of immense help
$endgroup$
– Anton Leontyev
Jan 14 at 20:22
$begingroup$
I just edited it, realized I had already divided by $n$, so check the corrected version. @AntonLeontyev
$endgroup$
– lightxbulb
Jan 14 at 20:25
$begingroup$
Could you please clarify what do you mean by "offset rotation"? Thank you!
$endgroup$
– Anton Leontyev
Jan 14 at 20:29
$begingroup$
@AntonLeontyev check the shadertoy example I gave, the offset is literally what rotates the circles around.
$endgroup$
– lightxbulb
Jan 14 at 20:41
$begingroup$
Thank you so much, now I understand.
$endgroup$
– Anton Leontyev
Jan 14 at 20:50
add a comment |
$begingroup$
For the 'adjacent' circles use : $delta = 360/n$ where $n$ is the number of circles you want. Then the centers are $c_i = ((R+r)cos idelta + phi, (R+r)sin idelta + phi)$, where $i=0,1,...,n-1$ and $phi$ is some offset rotation. Note that the small circles will not necessarily touch, but they will touch the large circle.
Edit:
Here's a shadertoy example:
https://www.shadertoy.com/view/wsfGWj
answered Jan 14 at 20:15
lightxbulblightxbulb
75619
$endgroup$
$begingroup$
thank you, your comment is of immense help
$endgroup$
– Anton Leontyev
Jan 14 at 20:22
$begingroup$
I just edited it, realized I had already divided by $n$, so check the corrected version. @AntonLeontyev
$endgroup$
– lightxbulb
Jan 14 at 20:25
$begingroup$
Could you please clarify what do you mean by "offset rotation"? Thank you!
$endgroup$
– Anton Leontyev
Jan 14 at 20:29
$begingroup$
@AntonLeontyev check the shadertoy example I gave, the offset is literally what rotates the circles around.
$endgroup$
– lightxbulb
Jan 14 at 20:41
$begingroup$
Thank you so much, now I understand.
$endgroup$
– Anton Leontyev
Jan 14 at 20:50
add a comment |
$begingroup$
For the 'adjacent' circles use : $delta = 360/n$ where $n$ is the number of circles you want. Then the centers are $c_i = ((R+r)cos idelta + phi, (R+r)sin idelta + phi)$, where $i=0,1,...,n-1$ and $phi$ is some offset rotation. Note that the small circles will not necessarily touch, but they will touch the large circle.
Edit:
Here's a shadertoy example:
https://www.shadertoy.com/view/wsfGWj
answered Jan 14 at 20:15
lightxbulblightxbulb
75619
$endgroup$
For the 'adjacent' circles use : $delta = 360/n$ where $n$ is the number of circles you want. Then the centers are $c_i = ((R+r)cos idelta + phi, (R+r)sin idelta + phi)$, where $i=0,1,...,n-1$ and $phi$ is some offset rotation. Note that the small circles will not necessarily touch, but they will touch the large circle.
Edit:
Here's a shadertoy example:
https://www.shadertoy.com/view/wsfGWj
answered Jan 14 at 20:15
lightxbulblightxbulb
75619
edited Jan 14 at 20:40
answered Jan 14 at 20:15
lightxbulblightxbulb
75619
answered Jan 14 at 20:15
lightxbulblightxbulb
75619
answered Jan 14 at 20:15
lightxbulblightxbulb
75619
75619
$begingroup$
thank you, your comment is of immense help
$endgroup$
– Anton Leontyev
Jan 14 at 20:22
$begingroup$
I just edited it, realized I had already divided by $n$, so check the corrected version. @AntonLeontyev
$endgroup$
– lightxbulb
Jan 14 at 20:25
$begingroup$
Could you please clarify what do you mean by "offset rotation"? Thank you!
$endgroup$
– Anton Leontyev
Jan 14 at 20:29
$begingroup$
@AntonLeontyev check the shadertoy example I gave, the offset is literally what rotates the circles around.
$endgroup$
– lightxbulb
Jan 14 at 20:41
$begingroup$
Thank you so much, now I understand.
$endgroup$
– Anton Leontyev
Jan 14 at 20:50
add a comment |
$begingroup$
thank you, your comment is of immense help
$endgroup$
– Anton Leontyev
Jan 14 at 20:22
$begingroup$
I just edited it, realized I had already divided by $n$, so check the corrected version. @AntonLeontyev
$endgroup$
– lightxbulb
Jan 14 at 20:25
$begingroup$
Could you please clarify what do you mean by "offset rotation"? Thank you!
$endgroup$
– Anton Leontyev
Jan 14 at 20:29
$begingroup$
@AntonLeontyev check the shadertoy example I gave, the offset is literally what rotates the circles around.
$endgroup$
– lightxbulb
Jan 14 at 20:41
$begingroup$
Thank you so much, now I understand.
$endgroup$
– Anton Leontyev
Jan 14 at 20:50
$begingroup$
thank you, your comment is of immense help
$endgroup$
– Anton Leontyev
Jan 14 at 20:22
$begingroup$
thank you, your comment is of immense help
$endgroup$
– Anton Leontyev
Jan 14 at 20:22
$begingroup$
I just edited it, realized I had already divided by $n$, so check the corrected version. @AntonLeontyev
$endgroup$
– lightxbulb
Jan 14 at 20:25
$begingroup$
I just edited it, realized I had already divided by $n$, so check the corrected version. @AntonLeontyev
$endgroup$
– lightxbulb
Jan 14 at 20:25
$begingroup$
Could you please clarify what do you mean by "offset rotation"? Thank you!
$endgroup$
– Anton Leontyev
Jan 14 at 20:29
$begingroup$
Could you please clarify what do you mean by "offset rotation"? Thank you!
$endgroup$
– Anton Leontyev
Jan 14 at 20:29
$begingroup$
@AntonLeontyev check the shadertoy example I gave, the offset is literally what rotates the circles around.
$endgroup$
– lightxbulb
Jan 14 at 20:41
$begingroup$
@AntonLeontyev check the shadertoy example I gave, the offset is literally what rotates the circles around.
$endgroup$
– lightxbulb
Jan 14 at 20:41
$begingroup$
Thank you so much, now I understand.
$endgroup$
– Anton Leontyev
Jan 14 at 20:50
$begingroup$
Thank you so much, now I understand.
$endgroup$
– Anton Leontyev
Jan 14 at 20:50
add a comment |
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$begingroup$
I think $n$ determines $r$, so for example if you want $6$ externally tangent circles then $r$ is fixed at $r=R$ (all $7$ circles are the same size in that special case). In general, you could write $r$ as a function of $n$ and it would be strictly decreasing.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:13
$begingroup$
I mention that because you cannot presuppose $n$ and $r$ - most pairs that you choose will lead to impossible situations. Just presuppose $n$ and $R$ and those will together determine $r$.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:14
$begingroup$
Thank you, that makes sense (in lay terminology, only certain amount of circles of size will fit, correct?)
$endgroup$
– Anton Leontyev
Jan 14 at 20:16
$begingroup$
Connect the centers of the $n$ externally tangent circles and you will have a regular $n$-gon
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:16
$begingroup$
If you choose the number of small circles and the size of the big circle, the size of the small circles is fixed, invariable, set, locked.
$endgroup$
– Zubin Mukerjee
Jan 14 at 20:17