Possible closed form approximation of a trigonometrical expression
$begingroup$
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
1013
$endgroup$
add a comment |
$begingroup$
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
1013
$endgroup$
add a comment |
$begingroup$
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
1013
$endgroup$
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
1013
asked Jan 5 at 22:30
YurikYurik
1013
edited Jan 5 at 22:54
Yurik
asked Jan 5 at 22:30
YurikYurik
1013
asked Jan 5 at 22:30
YurikYurik
1013
asked Jan 5 at 22:30
YurikYurik
1013
1013
add a comment |
add a comment |
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