Possible closed form approximation of a trigonometrical expression
I need to create a simple algorithm to draw a Venn diagram (ideally for 3-circle case, but even solving it for 2 is a good start). So given thee numbers - X & Y (sizes of two sets), and Z (size of the overlap), I need to calculate the two circle radii (r1 & r2) and the distance (d) between them. This amazing explanation has all the needed formulas, but sadly there is no closed form for the expression (the author solves it numerically). Is there an approximation I can use to solve it? I cannot solve numerically in the Vega visualization.
Quick recap of the article: calculating r1 and r2 is straightforward,
X = π*r1^2 -> r1 = sqrt(X / π)
Y = π*r2^2 -> r2 = sqrt(Y / π)
The green area equals to
Z = r1^2 * (θ1 – sin(2*θ1) / 2) + r2^2 * (θ2 – sin(2*θ2) / 2)
thus the needed distance is
d = r1 * cos(θ1) + r2 * cos(θ2)
Note that d could be less than r1 + r2 in case when more than half of one set is also part of another set. How would it be possible to approximate it in a "good enough" manner?
closed-form
asked Jan 5 at 22:30
YurikYurik
1012
New contributor
Yurik is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
I need to create a simple algorithm to draw a Venn diagram (ideally for 3-circle case, but even solving it for 2 is a good start). So given thee numbers - X & Y (sizes of two sets), and Z (size of the overlap), I need to calculate the two circle radii (r1 & r2) and the distance (d) between them. This amazing explanation has all the needed formulas, but sadly there is no closed form for the expression (the author solves it numerically). Is there an approximation I can use to solve it? I cannot solve numerically in the Vega visualization.
Quick recap of the article: calculating r1 and r2 is straightforward,
X = π*r1^2 -> r1 = sqrt(X / π)
Y = π*r2^2 -> r2 = sqrt(Y / π)
The green area equals to
Z = r1^2 * (θ1 – sin(2*θ1) / 2) + r2^2 * (θ2 – sin(2*θ2) / 2)
thus the needed distance is
d = r1 * cos(θ1) + r2 * cos(θ2)
Note that d could be less than r1 + r2 in case when more than half of one set is also part of another set. How would it be possible to approximate it in a "good enough" manner?
closed-form
asked Jan 5 at 22:30
YurikYurik
1012
New contributor
Yurik is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
I need to create a simple algorithm to draw a Venn diagram (ideally for 3-circle case, but even solving it for 2 is a good start). So given thee numbers - X & Y (sizes of two sets), and Z (size of the overlap), I need to calculate the two circle radii (r1 & r2) and the distance (d) between them. This amazing explanation has all the needed formulas, but sadly there is no closed form for the expression (the author solves it numerically). Is there an approximation I can use to solve it? I cannot solve numerically in the Vega visualization.
Quick recap of the article: calculating r1 and r2 is straightforward,
X = π*r1^2 -> r1 = sqrt(X / π)
Y = π*r2^2 -> r2 = sqrt(Y / π)
The green area equals to
Z = r1^2 * (θ1 – sin(2*θ1) / 2) + r2^2 * (θ2 – sin(2*θ2) / 2)
thus the needed distance is
d = r1 * cos(θ1) + r2 * cos(θ2)
Note that d could be less than r1 + r2 in case when more than half of one set is also part of another set. How would it be possible to approximate it in a "good enough" manner?
closed-form
asked Jan 5 at 22:30
YurikYurik
1012
New contributor
Yurik is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I need to create a simple algorithm to draw a Venn diagram (ideally for 3-circle case, but even solving it for 2 is a good start). So given thee numbers - X & Y (sizes of two sets), and Z (size of the overlap), I need to calculate the two circle radii (r1 & r2) and the distance (d) between them. This amazing explanation has all the needed formulas, but sadly there is no closed form for the expression (the author solves it numerically). Is there an approximation I can use to solve it? I cannot solve numerically in the Vega visualization.
Quick recap of the article: calculating r1 and r2 is straightforward,
X = π*r1^2 -> r1 = sqrt(X / π)
Y = π*r2^2 -> r2 = sqrt(Y / π)
The green area equals to
Z = r1^2 * (θ1 – sin(2*θ1) / 2) + r2^2 * (θ2 – sin(2*θ2) / 2)
thus the needed distance is
d = r1 * cos(θ1) + r2 * cos(θ2)
Note that d could be less than r1 + r2 in case when more than half of one set is also part of another set. How would it be possible to approximate it in a "good enough" manner?
closed-form
closed-form
asked Jan 5 at 22:30
YurikYurik
1012
New contributor
Yurik is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 5 at 22:30
YurikYurik
1012
New contributor
Yurik is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited Jan 5 at 22:54
Yurik
asked Jan 5 at 22:30
YurikYurik
1012
New contributor
Yurik is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 5 at 22:30
YurikYurik
1012
asked Jan 5 at 22:30
YurikYurik
1012
1012
New contributor
Yurik is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Yurik is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Yurik is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
0
active
oldest
votes
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
});
}
});
Yurik is a new contributor. Be nice, and check out our Code of Conduct.
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%2f3063264%2fpossible-closed-form-approximation-of-a-trigonometrical-expression%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Yurik is a new contributor. Be nice, and check out our Code of Conduct.
Yurik is a new contributor. Be nice, and check out our Code of Conduct.
Yurik is a new contributor. Be nice, and check out our Code of Conduct.
Yurik is a new contributor. Be nice, and check out our Code of Conduct.
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3063264%2fpossible-closed-form-approximation-of-a-trigonometrical-expression%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
