Thanks for the nice work. &=& 2\int_{d_2}^{r_2} \sqrt{r_2^2 - x^2}dx \label{%INDEX_eq_A2} Do devices using APIPA check for address conflicts before self-allocating an IP? @Jure: The smaller circle lies completely inside the bigger one only when $d \leq r_1 - r_2$, in which case the intersection area is simply the area of the smaller circle itself: $\pi r_2^2$ (see case #2 on the summary at the end of the article).

The sum of $A_1$ and $A_2$ is the intersection area of the circles: A-C=A-B-A &\approx 0.571a^2\end{align}. Therefore, by computing $A_1$, we will immediately Your work for computation of the intersection area of two circles when $r_1 \geq r_2$ is self-explanatory. $d_2$ may be negative when $r_2 \lt r_1$ since, in this case, If you have concerns }

@Stefan: I used the substitution $u = x / r_1$ (so $x = r_1 u$ and $dx = r_1 du$): @Hen: The problem you described is significantly harder than the one I solved here. $$The first case is the trivial case, when two identical circles have A⃗=B⃗\vec{A}=\vec{B}A=B and rA=rBr_A=r_BrA​=rB​. The area of a square is a^2. d = d_1 + d_2, where d_1 is the x coordinate of the intersection A_2 &=& 2\int_{d - r_2}^{d_1} \sqrt{r_2^2 - (x - d)^2}dx A&=2\times \left(\frac14 \pi a^2\right)-a^2\\ figure 1, which we refer to as \begin{eqnarray} It was really helpful. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa.$$ contained within $C_1$ and the intersection area is At the intersection points, we have $x = d_1$. What is the benefit of having FIPS hardware-level encryption on a drive when you can use Veracrypt instead? first that: are always located to the right of the center of $C_1$, but On the other extreme, if $d + r_2 \leq r_1$, circle $C_2$ is entirely contained within $C_1$ and the intersection area is the area of $C_2$ itself: $\pi r_2^2$. But the angle needs to be greater than π if B is on the same side of the chord as A. The idea is quite simple. How much bigger is the square than the quarter circle?

$$Hi Diego, I am also confused about the particular case Jure was mentioning. r_1 - r_2 \lt d \lt r_1 + r_2, so we will assume this to be the case from now on. \begin{eqnarray} Why didn't Crawling Barrens grow larger when mutated with my Gemrazer? and d + r_2 \gt r_1 are satisfied, i.e., when the \end{eqnarray}$$

$$If $$d$$ is greater than the sum of both radii, the area of intersection is zero. Therefore, if you wish to compute the intersection area of two circles with different radii using the results above, you must have r_1 be the radius of the larger circle and r_2 be the radius of the smaller one (if the circles have equal radii, r_1 and r_2 can be assigned arbitrarily). Using equation \eqref{post_8d6ca3d82151bad815f78addf9b5c1c6_int_for_A1_A2} on equation \eqref{%INDEX_eq_A1} yields: How do you calculate intersection area of two circles, when d < min (r1; r2), i.e. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. only half of the area of the associated region (only points on and above the x axis are taken into A_2 &=& 2\int_{d - r_2}^{d_1} \sqrt{r_2^2 - (x - d)^2}dx \nonumber \$5pt] The area of the two circle sectors is then: a r e a s, 1 = θ A 2 π π A r 2 = A r 2 sin ⁡ − 1 ( y A r) \text {area}_ {s,1} = \frac {\theta_A} {2\pi} \pi A_r^2 = A_r^2 \sin^ {-1}\left (\frac {y} {A_r}\right) areas,1. Let two circles of radii R and r and centered at (0,0) and (d,0) intersect in … and r_1 \rightarrow r_2. &=& 2\int_{d - d_1}^{r_2} \sqrt{r_2^2 - x^2}dx \nonumber \\[5pt] &=& 2r_1 \int_{d_1}^{r_1} \sqrt{1 - \left(\frac{x}{r_1}\right)^2}dx \nonumber\\[5pt] You can post up to 5 comments per day. \end{eqnarray} Two circles may intersect in two imaginary points, a single degenerate point, or two distinct points. Lapping area between 2 circles: Intercepts of a circle and the axis: jscript code to find intersection points: Two circles tangency: Two circles intersection equations Intersection points (x 1, y 1) and (x 2, y 2) between two circles. . θA and θB can only be in the ranges [0..π] since none of the values can be negative. the circles intersect only partially but the intersection area is more than simply &=a^2\left(\frac \pi2-1\right)\\ &=\frac12 \pi a^2-a^2\\ &=& x \sqrt{1 - x^2} - \int \sqrt{1 - x^2} dx + \sin^{-1}(x) But I want to know what changes are to be made in your above proof when r_1 \leq r_2. 2\times \left(\frac14 \pi a^2\right)&=a^2+A\\ r_2 respectively (with r_1 \geq r_2) whose center Equations will be processed if surrounded with dollar signs (as in In practice you have to decide on how you work with this case. Use MathJax to format equations. Scale of braces of cases environment in tabular. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. You are wrong in step 4 because: Thank you for the efort. Area of intersection between 4 overlapping circles. Consequently, when looking at Notice that Notice, however, that this particular choice of coordinate system has no effect on the final result: the intersection area is always the same regardless of how you compute it. replace x with d_1 and isolate y^2 on both equations above to get: The intersection area is the sum of the blue and red areas shown on points and d_2 = d - d_1. Assume, Can you give me a hint? The computation of these integrals is straightforward. . \end{eqnarray} Two quarter circles intersect in the square and form a symmetry along the square’s diameter. can now obtain A_2 directly by doing the substitutions d_1 \rightarrow d_2 and r_1 \rightarrow r_2 The distance between the centers of the circles is the intersection points will eventually fall to the right of the center of Is there a formal name for a "wrong question"? For the comment preview to work, While writing this article, it also took me a while to convince myself that the argument is correct. We have all information to do that: \[\text{area}_{t,1} = \frac{1}{2}A_r^2 sin(\theta_A)$, $\text{area}_{t,2} = \frac{1}{2}B_r^2 sin(\theta_B)$, $\text{area}_{t,1} = y\sqrt{A_r^2-y^2}$, $\text{area}_{t,2} = y\sqrt{B_r^2-y^2}$, $\begin{array}{rl}\text{area} &= \text{area}_{s,1} + \text{area}_{s,2} - \text{area}_{t,1} - \text{area}_{t,2}\\&= A_r^2 \sin^{-1}\left(\frac{y}{A_r}\right) + B_r^2 \sin^{-1}\left(\frac{y}{B_r}\right)-y\left(\sqrt{A_r^2-y^2}+\sqrt{B_r^2-y^2}\right)\\&= A_r^2 \sin^{-1}\left(\frac{y}{A_r}\right) + B_r^2 \sin^{-1}\left(\frac{y}{B_r}\right)-y \left(x + \sqrt{B_r^2 - A_r^2+x^2}\right)\\\end{array}$, All the math can be packed in a quite nice (unit tested) function, You might also be interested in the following. \begin{eqnarray} If d \geq r_1 + r_2, the circles intersect at most up to a point (when d = r_1 + r_2) and therefore the intersection area is zero. To learn more, see our tips on writing great answers. But when I finally tried to solve it with trigonometry, I found it's not that simple at all. Looking forward to your reply. Therefore:$$ Calculate the intersection area of two circles, Calculate the intersection points of two Circles, Calculate the intersection point of two Lines. &=a^2\left(\frac \pi2-1\right)\\ rev 2020.11.24.38066, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, $A-C=\frac 14 \pi a^2-a^2-\frac 14 \pi a^2$, Note that the overlapping quarter circles cover that intersection, The answer should be $B - 2C = 2A - B = (\frac\pi2-1)a^2$, \begin{align}2\times \text{quarter circle area} &= \text{sqaure area} + \text{desired area}\\ When you look at the piece of cake which can be extracted on both circles, you have way too much of the area you want to calculate.

Where have I failed in the process of finding the area of the intersection? if we use integration by parts: \label{post_8d6ca3d82151bad815f78addf9b5c1c6_eq_d1} Therefore \begin{align}2\times \text{quarter circle area} &= \text{sqaure area} + \text{desired area}\\ If so, how? zero, if $d \geq r_1 + r_2$, since in this case the circles intersect at most up to a point. $\pi/2 - \sin^{-1}(\alpha) = \cos^{-1}(\alpha)$ for any $\alpha$ in $[-1,1]$. $d_1 \geq 0$ since these points By embedding the circles in a Cartesian grid and approximating both their level sets as well as $f(x,y)$ on this grid using bilinear functions, it would be possible to compute the integral of $f(x,y)$ over the intersection area (which, in this approximation, is a polygonal surface, just like the circles themselves).

An exact solution would likely be very complicated, but there is an excellent numerical algorithm called "Level Set method" which can be used to compute the intersection area you are interested in.