Mixing
Circle rotating in circle rotating in circle
$begingroup$
There are three circles with different radii arranged such that the smallest circle is contained entirely within the next larger circle and that circle is entirely contained within the largest circle. All three circles are touching at a single point. If the inner circle is rotated, how many rotations of that circle will be required before the points on each of the circles at which they originally touched are again a single point? Assume the radii are such that the points will eventually be perfectly aligned.
The real world application of this, which may help to visualize the problem, is with a grown up version of Spirograph that has multiple rings such that you can put a gear inside a ring inside a ring. The outer ring stays fixed while the inner ring and the gear inside the inner ring rotate.
The simpler question of a gear inside a ring, or a circle inside a circle, has already been asked and answered. For the gear problem the size of the rings and gears are given in number of teeth on the gear or ring and this gives an equation of LCM(G,R)/G, where LCM is least common multiple, G is teeth on the gear and R is teeth on the ring.
The question of gear inside ring inside ring (circle in circle in circle), however, does not seem to have come up here. I know from experimentation that the answer is not a simple extension of the gear in ring equation. Any assistance on this will be greatly appreciated by me and I'm sure many others using this gear set. For extra credit and extreme gratitude, how would the answer be extended to include an additional outer ring (gear in ring in ring in ring)?
geometry circle rotations
edited Jan 18 at 21:46
Daniel Mathias
1,30518
asked Jan 18 at 16:07
Jay HeylJay Heyl
113
$endgroup$
add a comment |
$begingroup$
There are three circles with different radii arranged such that the smallest circle is contained entirely within the next larger circle and that circle is entirely contained within the largest circle. All three circles are touching at a single point. If the inner circle is rotated, how many rotations of that circle will be required before the points on each of the circles at which they originally touched are again a single point? Assume the radii are such that the points will eventually be perfectly aligned.
The real world application of this, which may help to visualize the problem, is with a grown up version of Spirograph that has multiple rings such that you can put a gear inside a ring inside a ring. The outer ring stays fixed while the inner ring and the gear inside the inner ring rotate.
The simpler question of a gear inside a ring, or a circle inside a circle, has already been asked and answered. For the gear problem the size of the rings and gears are given in number of teeth on the gear or ring and this gives an equation of LCM(G,R)/G, where LCM is least common multiple, G is teeth on the gear and R is teeth on the ring.
The question of gear inside ring inside ring (circle in circle in circle), however, does not seem to have come up here. I know from experimentation that the answer is not a simple extension of the gear in ring equation. Any assistance on this will be greatly appreciated by me and I'm sure many others using this gear set. For extra credit and extreme gratitude, how would the answer be extended to include an additional outer ring (gear in ring in ring in ring)?
geometry circle rotations
edited Jan 18 at 21:46
Daniel Mathias
1,30518
asked Jan 18 at 16:07
Jay HeylJay Heyl
113
$endgroup$
$begingroup$
The question is not clear. The described system has 2 degrees of freedom, So it can have non-periodic trajectory even if radii are integers.
$endgroup$
– Vasily Mitch
Jan 18 at 16:23
$begingroup$
Is this what you mean? Feel free to include the illustration in your question if so ().
$endgroup$
– Nominal Animal
Jan 18 at 16:43
$begingroup$
In the real world application that is the basis for the question the radii are constrained such that the trajectory is periodic. I do apologize if my command of mathematical terms is insufficient. If you can be specific about what you find unclear in the question I'll do my best to clarify.
$endgroup$
– Jay Heyl
Jan 18 at 16:48
$begingroup$
Nominal Animal -- Yes, that's it exactly. Modified to include link to your image. I don't have enough reputation to add the image directly.
$endgroup$
– Jay Heyl
Jan 18 at 16:56
add a comment |
$begingroup$
There are three circles with different radii arranged such that the smallest circle is contained entirely within the next larger circle and that circle is entirely contained within the largest circle. All three circles are touching at a single point. If the inner circle is rotated, how many rotations of that circle will be required before the points on each of the circles at which they originally touched are again a single point? Assume the radii are such that the points will eventually be perfectly aligned.
The real world application of this, which may help to visualize the problem, is with a grown up version of Spirograph that has multiple rings such that you can put a gear inside a ring inside a ring. The outer ring stays fixed while the inner ring and the gear inside the inner ring rotate.
The simpler question of a gear inside a ring, or a circle inside a circle, has already been asked and answered. For the gear problem the size of the rings and gears are given in number of teeth on the gear or ring and this gives an equation of LCM(G,R)/G, where LCM is least common multiple, G is teeth on the gear and R is teeth on the ring.
The question of gear inside ring inside ring (circle in circle in circle), however, does not seem to have come up here. I know from experimentation that the answer is not a simple extension of the gear in ring equation. Any assistance on this will be greatly appreciated by me and I'm sure many others using this gear set. For extra credit and extreme gratitude, how would the answer be extended to include an additional outer ring (gear in ring in ring in ring)?
geometry circle rotations
edited Jan 18 at 21:46
Daniel Mathias
1,30518
asked Jan 18 at 16:07
Jay HeylJay Heyl
113
$endgroup$
There are three circles with different radii arranged such that the smallest circle is contained entirely within the next larger circle and that circle is entirely contained within the largest circle. All three circles are touching at a single point. If the inner circle is rotated, how many rotations of that circle will be required before the points on each of the circles at which they originally touched are again a single point? Assume the radii are such that the points will eventually be perfectly aligned.
The real world application of this, which may help to visualize the problem, is with a grown up version of Spirograph that has multiple rings such that you can put a gear inside a ring inside a ring. The outer ring stays fixed while the inner ring and the gear inside the inner ring rotate.
The simpler question of a gear inside a ring, or a circle inside a circle, has already been asked and answered. For the gear problem the size of the rings and gears are given in number of teeth on the gear or ring and this gives an equation of LCM(G,R)/G, where LCM is least common multiple, G is teeth on the gear and R is teeth on the ring.
The question of gear inside ring inside ring (circle in circle in circle), however, does not seem to have come up here. I know from experimentation that the answer is not a simple extension of the gear in ring equation. Any assistance on this will be greatly appreciated by me and I'm sure many others using this gear set. For extra credit and extreme gratitude, how would the answer be extended to include an additional outer ring (gear in ring in ring in ring)?
geometry circle rotations
geometry circle rotations
edited Jan 18 at 21:46
Daniel Mathias
1,30518
asked Jan 18 at 16:07
Jay HeylJay Heyl
113
edited Jan 18 at 21:46
Daniel Mathias
1,30518
asked Jan 18 at 16:07
Jay HeylJay Heyl
113
edited Jan 18 at 21:46
Daniel Mathias
1,30518
edited Jan 18 at 21:46
Daniel Mathias
1,30518
edited Jan 18 at 21:46
Daniel Mathias
1,30518
1,30518
asked Jan 18 at 16:07
Jay HeylJay Heyl
113
asked Jan 18 at 16:07
Jay HeylJay Heyl
113
asked Jan 18 at 16:07
Jay HeylJay Heyl
113
113
$begingroup$
The question is not clear. The described system has 2 degrees of freedom, So it can have non-periodic trajectory even if radii are integers.
$endgroup$
– Vasily Mitch
Jan 18 at 16:23
$begingroup$
Is this what you mean? Feel free to include the illustration in your question if so ().
$endgroup$
– Nominal Animal
Jan 18 at 16:43
$begingroup$
In the real world application that is the basis for the question the radii are constrained such that the trajectory is periodic. I do apologize if my command of mathematical terms is insufficient. If you can be specific about what you find unclear in the question I'll do my best to clarify.
$endgroup$
– Jay Heyl
Jan 18 at 16:48
$begingroup$
Nominal Animal -- Yes, that's it exactly. Modified to include link to your image. I don't have enough reputation to add the image directly.
$endgroup$
– Jay Heyl
Jan 18 at 16:56
add a comment |
$begingroup$
The question is not clear. The described system has 2 degrees of freedom, So it can have non-periodic trajectory even if radii are integers.
$endgroup$
– Vasily Mitch
Jan 18 at 16:23
$begingroup$
Is this what you mean? Feel free to include the illustration in your question if so ().
$endgroup$
– Nominal Animal
Jan 18 at 16:43
$begingroup$
In the real world application that is the basis for the question the radii are constrained such that the trajectory is periodic. I do apologize if my command of mathematical terms is insufficient. If you can be specific about what you find unclear in the question I'll do my best to clarify.
$endgroup$
– Jay Heyl
Jan 18 at 16:48
$begingroup$
Nominal Animal -- Yes, that's it exactly. Modified to include link to your image. I don't have enough reputation to add the image directly.
$endgroup$
– Jay Heyl
Jan 18 at 16:56
$begingroup$
The question is not clear. The described system has 2 degrees of freedom, So it can have non-periodic trajectory even if radii are integers.
$endgroup$
– Vasily Mitch
Jan 18 at 16:23
$begingroup$
The question is not clear. The described system has 2 degrees of freedom, So it can have non-periodic trajectory even if radii are integers.
$endgroup$
– Vasily Mitch
Jan 18 at 16:23
$begingroup$
Is this what you mean? Feel free to include the illustration in your question if so (
).$endgroup$
– Nominal Animal
Jan 18 at 16:43
$begingroup$
Is this what you mean? Feel free to include the illustration in your question if so (
).$endgroup$
– Nominal Animal
Jan 18 at 16:43
$begingroup$
In the real world application that is the basis for the question the radii are constrained such that the trajectory is periodic. I do apologize if my command of mathematical terms is insufficient. If you can be specific about what you find unclear in the question I'll do my best to clarify.
$endgroup$
– Jay Heyl
Jan 18 at 16:48
$begingroup$
In the real world application that is the basis for the question the radii are constrained such that the trajectory is periodic. I do apologize if my command of mathematical terms is insufficient. If you can be specific about what you find unclear in the question I'll do my best to clarify.
$endgroup$
– Jay Heyl
Jan 18 at 16:48
$begingroup$
Nominal Animal -- Yes, that's it exactly. Modified to include link to your image. I don't have enough reputation to add the image directly.
$endgroup$
– Jay Heyl
Jan 18 at 16:56
$begingroup$
Nominal Animal -- Yes, that's it exactly. Modified to include link to your image. I don't have enough reputation to add the image directly.
$endgroup$
– Jay Heyl
Jan 18 at 16:56
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
We have a small gear with radius $G$, a gear ring (annulus) with inner radius $R_i$ and outer radius $R_o$, and containing circle with radius $C$. One orbit of the ring around the circle will bring it back to its initial position with $frac{C}{R_o}$ rotations. The small gear will have completed $frac{C}{R_o}timesfrac{R_i}{G}$ rotations. In order to have the gear and the ring return to their starting position in their starting orientation, there need to be $O$ orbits such that $Otimesfrac{C}{R_o}$ and $Otimesfrac{C}{R_o}timesfrac{R_i}{G}$ are both integers.
$$O=frac{Gtimes R_o}{text{gcd}(Gtimes R_o,Ctimes R_i)}$$
Example: $G=21,R_i=28,R_o=35,C=50implies O=21$
$21$ orbits gives $30$ rotations of the ring and $40$ rotations of the gear. I'm not certain that this will match your observations, but it's the best I can come up with.
answered Jan 18 at 17:17
Daniel MathiasDaniel Mathias
1,30518
$endgroup$
$begingroup$
Thanks for the answer. Unfortunately, it made me realize I asked the wrong question. I was looking for an answer about real world objects that don't live in the world of lines without width. In math terms I guess I have a circle inside an annulus inside a circle. From application of these formulae to my real world observations it appears the annulus significantly alters the situation. I should probably restate it as a new question.
$endgroup$
– Jay Heyl
Jan 18 at 19:13
$begingroup$
@JayHeyl Updated. Hopefully this works for you.
$endgroup$
– Daniel Mathias
Jan 18 at 22:43
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
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
});
}
});
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3078430%2fcircle-rotating-in-circle-rotating-in-circle%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
We have a small gear with radius $G$, a gear ring (annulus) with inner radius $R_i$ and outer radius $R_o$, and containing circle with radius $C$. One orbit of the ring around the circle will bring it back to its initial position with $frac{C}{R_o}$ rotations. The small gear will have completed $frac{C}{R_o}timesfrac{R_i}{G}$ rotations. In order to have the gear and the ring return to their starting position in their starting orientation, there need to be $O$ orbits such that $Otimesfrac{C}{R_o}$ and $Otimesfrac{C}{R_o}timesfrac{R_i}{G}$ are both integers.
$$O=frac{Gtimes R_o}{text{gcd}(Gtimes R_o,Ctimes R_i)}$$
Example: $G=21,R_i=28,R_o=35,C=50implies O=21$
$21$ orbits gives $30$ rotations of the ring and $40$ rotations of the gear. I'm not certain that this will match your observations, but it's the best I can come up with.
answered Jan 18 at 17:17
Daniel MathiasDaniel Mathias
1,30518
$endgroup$
$begingroup$
Thanks for the answer. Unfortunately, it made me realize I asked the wrong question. I was looking for an answer about real world objects that don't live in the world of lines without width. In math terms I guess I have a circle inside an annulus inside a circle. From application of these formulae to my real world observations it appears the annulus significantly alters the situation. I should probably restate it as a new question.
$endgroup$
– Jay Heyl
Jan 18 at 19:13
$begingroup$
@JayHeyl Updated. Hopefully this works for you.
$endgroup$
– Daniel Mathias
Jan 18 at 22:43
add a comment |
$begingroup$
We have a small gear with radius $G$, a gear ring (annulus) with inner radius $R_i$ and outer radius $R_o$, and containing circle with radius $C$. One orbit of the ring around the circle will bring it back to its initial position with $frac{C}{R_o}$ rotations. The small gear will have completed $frac{C}{R_o}timesfrac{R_i}{G}$ rotations. In order to have the gear and the ring return to their starting position in their starting orientation, there need to be $O$ orbits such that $Otimesfrac{C}{R_o}$ and $Otimesfrac{C}{R_o}timesfrac{R_i}{G}$ are both integers.
$$O=frac{Gtimes R_o}{text{gcd}(Gtimes R_o,Ctimes R_i)}$$
Example: $G=21,R_i=28,R_o=35,C=50implies O=21$
$21$ orbits gives $30$ rotations of the ring and $40$ rotations of the gear. I'm not certain that this will match your observations, but it's the best I can come up with.
answered Jan 18 at 17:17
Daniel MathiasDaniel Mathias
1,30518
$endgroup$
$begingroup$
Thanks for the answer. Unfortunately, it made me realize I asked the wrong question. I was looking for an answer about real world objects that don't live in the world of lines without width. In math terms I guess I have a circle inside an annulus inside a circle. From application of these formulae to my real world observations it appears the annulus significantly alters the situation. I should probably restate it as a new question.
$endgroup$
– Jay Heyl
Jan 18 at 19:13
$begingroup$
@JayHeyl Updated. Hopefully this works for you.
$endgroup$
– Daniel Mathias
Jan 18 at 22:43
add a comment |
$begingroup$
We have a small gear with radius $G$, a gear ring (annulus) with inner radius $R_i$ and outer radius $R_o$, and containing circle with radius $C$. One orbit of the ring around the circle will bring it back to its initial position with $frac{C}{R_o}$ rotations. The small gear will have completed $frac{C}{R_o}timesfrac{R_i}{G}$ rotations. In order to have the gear and the ring return to their starting position in their starting orientation, there need to be $O$ orbits such that $Otimesfrac{C}{R_o}$ and $Otimesfrac{C}{R_o}timesfrac{R_i}{G}$ are both integers.
$$O=frac{Gtimes R_o}{text{gcd}(Gtimes R_o,Ctimes R_i)}$$
Example: $G=21,R_i=28,R_o=35,C=50implies O=21$
$21$ orbits gives $30$ rotations of the ring and $40$ rotations of the gear. I'm not certain that this will match your observations, but it's the best I can come up with.
answered Jan 18 at 17:17
Daniel MathiasDaniel Mathias
1,30518
$endgroup$
We have a small gear with radius $G$, a gear ring (annulus) with inner radius $R_i$ and outer radius $R_o$, and containing circle with radius $C$. One orbit of the ring around the circle will bring it back to its initial position with $frac{C}{R_o}$ rotations. The small gear will have completed $frac{C}{R_o}timesfrac{R_i}{G}$ rotations. In order to have the gear and the ring return to their starting position in their starting orientation, there need to be $O$ orbits such that $Otimesfrac{C}{R_o}$ and $Otimesfrac{C}{R_o}timesfrac{R_i}{G}$ are both integers.
$$O=frac{Gtimes R_o}{text{gcd}(Gtimes R_o,Ctimes R_i)}$$
Example: $G=21,R_i=28,R_o=35,C=50implies O=21$
$21$ orbits gives $30$ rotations of the ring and $40$ rotations of the gear. I'm not certain that this will match your observations, but it's the best I can come up with.
answered Jan 18 at 17:17
Daniel MathiasDaniel Mathias
1,30518
edited Jan 18 at 22:43
answered Jan 18 at 17:17
Daniel MathiasDaniel Mathias
1,30518
answered Jan 18 at 17:17
Daniel MathiasDaniel Mathias
1,30518
answered Jan 18 at 17:17
Daniel MathiasDaniel Mathias
1,30518
1,30518
$begingroup$
Thanks for the answer. Unfortunately, it made me realize I asked the wrong question. I was looking for an answer about real world objects that don't live in the world of lines without width. In math terms I guess I have a circle inside an annulus inside a circle. From application of these formulae to my real world observations it appears the annulus significantly alters the situation. I should probably restate it as a new question.
$endgroup$
– Jay Heyl
Jan 18 at 19:13
$begingroup$
@JayHeyl Updated. Hopefully this works for you.
$endgroup$
– Daniel Mathias
Jan 18 at 22:43
add a comment |
$begingroup$
Thanks for the answer. Unfortunately, it made me realize I asked the wrong question. I was looking for an answer about real world objects that don't live in the world of lines without width. In math terms I guess I have a circle inside an annulus inside a circle. From application of these formulae to my real world observations it appears the annulus significantly alters the situation. I should probably restate it as a new question.
$endgroup$
– Jay Heyl
Jan 18 at 19:13
$begingroup$
@JayHeyl Updated. Hopefully this works for you.
$endgroup$
– Daniel Mathias
Jan 18 at 22:43
$begingroup$
Thanks for the answer. Unfortunately, it made me realize I asked the wrong question. I was looking for an answer about real world objects that don't live in the world of lines without width. In math terms I guess I have a circle inside an annulus inside a circle. From application of these formulae to my real world observations it appears the annulus significantly alters the situation. I should probably restate it as a new question.
$endgroup$
– Jay Heyl
Jan 18 at 19:13
$begingroup$
Thanks for the answer. Unfortunately, it made me realize I asked the wrong question. I was looking for an answer about real world objects that don't live in the world of lines without width. In math terms I guess I have a circle inside an annulus inside a circle. From application of these formulae to my real world observations it appears the annulus significantly alters the situation. I should probably restate it as a new question.
$endgroup$
– Jay Heyl
Jan 18 at 19:13
$begingroup$
@JayHeyl Updated. Hopefully this works for you.
$endgroup$
– Daniel Mathias
Jan 18 at 22:43
$begingroup$
@JayHeyl Updated. Hopefully this works for you.
$endgroup$
– Daniel Mathias
Jan 18 at 22:43
add a comment |
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.
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3078430%2fcircle-rotating-in-circle-rotating-in-circle%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
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
$begingroup$
The question is not clear. The described system has 2 degrees of freedom, So it can have non-periodic trajectory even if radii are integers.
$endgroup$
– Vasily Mitch
Jan 18 at 16:23
$begingroup$
Is this what you mean? Feel free to include the illustration in your question if so (
).$endgroup$
– Nominal Animal
Jan 18 at 16:43
$begingroup$
In the real world application that is the basis for the question the radii are constrained such that the trajectory is periodic. I do apologize if my command of mathematical terms is insufficient. If you can be specific about what you find unclear in the question I'll do my best to clarify.
$endgroup$
– Jay Heyl
Jan 18 at 16:48
$begingroup$
Nominal Animal -- Yes, that's it exactly. Modified to include link to your image. I don't have enough reputation to add the image directly.
$endgroup$
– Jay Heyl
Jan 18 at 16:56