Intersection of 3 circles

Intersection of 3 circles calculator

Circles of the form: (x-a)² + (y-b)² = r²

Circle 1:	( x - )² + ( y - )² = ²
	x² + y² + x + y + = 0

Circle 2:	( x - )² + ( y - )² = ²
	x² + y² + x + y + = 0

Circle 3:	( x - )² + ( y - )² = ²
	x² + y² + x + y + = 0

Circles of the form: x² + y² + Ax + By + C = 0

Distance circles centers 1-2
Distance circles centers 1-3
Distance circles centers 2-3
Triangle ABC area
Lapping area

Input Limit:

Circles 1-2 intersection points
Circles 1-3 intersection points
Circles 2-3 intersection points
Lapping segments points
Total intersection points

Detailed solution of fig. - 11

Finding the lapping area between 3 circles

While finding the intersection points between the three circles are relatively an easy task, finding the lapping area of all 3 circles is a bit more difficult as the circles can be placed in each of the different arrangement as shown bellow. It is necessary to determine the relative position of the circles because each arrangements have its own method to calculate the lapping area.

Note: the circles can be tangent to each other, in this case only one intersection point exists.

Three circles arranging possibilities (the circles can be also tangent)

Fig. 1	Fig. 2	Fig. 3
Fig. 4	Fig. 5	Fig. 6
Fig. 7	Fig. 8	Fig. 9
Fig. 10	Fig. 11	Fig. 12
Fig. 13

To write a subroutine to calculate the common area we have first to recognize the relationship between the circles, for this purpose we can use the following table.

Table - 1

Fig Num	Number of intersection points			Over lapping areas			Internal points (1)			Overlap areas	Number of lapping segments
Fig Num	Circles 1,2	Circles 1,3	Circles 2,3	Circles 1,2	Circles 1,3	Circles 2,3	Circles 1,2	Circles 1,3	Circles 2,3	Overlap areas	Number of lapping segments
1	0	0	0	No	No	No	0	0	0	No	-
2	2	0	0	Yes	No	No	0	0	0	No	-
3	2	2	0	Yes	Yes	No	0	0	0	No	-
4	0	0	0	Yes	Yes	No	0	0	0	No	-
5	2	0	0	Yes	Yes	No	0	0	0	No	-
6	0	0	0	Yes	Yes	Yes	0	0	0	Yes	1
7	2	0	0	Yes	Yes	Yes	2	0	0	Yes	2
8	2	0	0	Yes	Yes	Yes	0	0	0	Yes	1
9	2	2	0	Yes	Yes	Yes	2	0	0	Yes	2
10	2	2	2	Yes	Yes	Yes	2	0	0	Yes	2
11	2	2	2	Yes	Yes	Yes	1	1	1	Yes	3
12	2	2	2	Yes	Yes	Yes	2	2	0	Yes	4
13	2	2	2	Yes	Yes	Yes	0	0	0	No	-

(1) Number of intersection points of two circles that lies inside the third circle.

A basic check to verify if lapping area exists is to check if at least one intersection point between two circles lies inside the third circle, if this check is positive for any pair of circles (total 3 cases to check) then lapping area exists.
This check is good to verify figs. 7, 9, 10, 11, 12, 13.

The condition of a point (x_p, y_p) to be inside a circle of radius r and center at (a, b) is:

This condition alone is not enough, for example fig. 8 contains common lapping area even though condition 1. is not satisfy. Another condition we have to check is if one circle is completely inside the other circle, this condition is stated by the formula:

| r₁ - r₂ | > D

(D is the distance between circles centers and is equal to:
With these criteria we can verify cases 6 and 8.

Another condition that we can check is if any of the circles are not intersecting at all, in this case no lapping area exists, the condition for separate circles is by the formula: r₁ + r₂ < D With these criteria we can verify cases, 1, 2, 3, 4, 5. (see table - 1)

Table - 2 Calculating lapping area

The lapping area of 3 circles can be either one of the following forms.

Calculating
cases:
7, 9, 10

The lapping area contains two circular segments area.

Cord a can be calculated according to the two intersection points (x_i1 y_i1) and (x_i2 y_i2).

The area of segment with radius r₁ and cord a and angle θ₁ is:

In order to find angle θ₁ We define:

Note: We used both circles radii in order to calculate the angle θ₁ this method enables us

to get angles greater then 180 degrees. See solved example

The same way we find the other segment area:

Total lapping area is:

A = A_seg1 + A_seg2

Calculating
case:
11

See note (2)

The lapping area contains the area of three segments and an inner triangle whose sides are the 3 circles segments cords (Circular triangles).

The area of the triangle with sides a, b and c is according to Heron's formula:

Where s is equal to:

The sides a, b and c of the inner triangle can be calculated according to the intersection points between each pair of the circles for example.

In order to calculate the segment area, we can't use formula (2) because we have 2 radii involved in each cord, instead we use the formula:

The combined lapping area is the sum:

Calculating
case:
12

See note (3)

This case is the most complicated shape of lapping area and it contains 4 circular segments and an area of an irregular convex quadrilateral.

The area of an irregular convex quadrilateral whose sides are a, b, c, d and diagonals p and q is by the following formula (we chose this formula among many others, also see note 3).

is the angle between diagonals p and q.

The lengths of the diagonals p and q can be calculated same way as the sides a, b, c and d:

The combined area of the segments is:

Notes

(2) Formula (4) obtained by subtracting the area of the triangle from the sector area.

Sector area is:

Using the formulas above are not working in all cases for example the set of circles:

The result will be wrong that is because the segment of circle with the radius of 5.5 should be the bigger one and not the small one (in this case the segment area is more then half circle area and the
angle θ₁ > 180 degree).

A way to deal with this problem is to add a check that will verify which area should be used, this is done by the following method (see the drawing at the right):

First find the mid point of the segment cord

The equation of the line connecting mid point and circle center is of the form y=mx+n:
Now we have to find the intersection points of this line and the circle, this is done by defining a new constant:
And the circle line intersection points will be:

We define:

Circle 1 equation:

Now we have to find which point of the two points found lies inside the other two circles (circle 2 and 3), this will be done by equation (1) performed on both points (x_ic1 y_ic1) and (x_ic2 y_ic2).

Once we found the correct point all we have to do is to find if the segment area is the small or the big portion of the circle that will be done by measuring the distance of the point to the mid point and compare it to the radius of the circle. If the distance is less or more then the radius then apply equation (5) as follows; we assumed in the next formula that the correct point is (x_ic2 y_ic2).

And finally, the correct area of the segment is:

(3) We can use the following formula which uses the sides and the diagonals lengths only to calculate the
quadrilateral area.

Note: circle 1 contain all 4 points of the lapping area (see also fig 12 in table 1).
The problem in this case is how we determine the correct sequence of the intersection points (not to get a crossed quadrilateral).
One way to deal with this problem is to calculate inner angles α, β, γ, and δ. The sum of the angles should be 360 degree for a convex quadrilateral.
The angles can be calculated according to the cosine law for example for angle α we have:

Second way to deal with this problem is to find the intersection points of sides a - c and b - d. If the intersection points of both pairs lie outside of circle 1 then the sequence are correct.

Intersection point between 2 sides a and c is:

And the intersect point of sides b and d is at:

Detailed solution of case 11

Find the lapping area between the three circles given by the equations:

① (x + 3)² + (y + 1)² = 8²

② (x + 1)² + (y − 1)² = 7²

③ (x − 1)² + y² = 5.5²

Step 1

Find all the intersection points between the three circles, this is performed according to the process described at the two circles intersection, for the circles ① - ② for example we get:

Distance between circles 1 and 2 centers

The two intersection points are (4.75, -3) and (-5, 6.75) it can be verified that the other two possible points are not lying on both circles.

Step 2

Find which point or points lying inside the third circle, these points will determine the boundaries of the lapping area.

The center of circle ③ is at (1, 0) and the distance of intersection points to this center are:

Because the radius of the third circle is 5.5 then only d₁ (4.75, − 3) is inside the circle.

Step 3

Repeat steps 1 and 2 for the other two pairs of circles ① ③ then ② ③ to get the intersection points that defines the boundary of the lapping area (1.73, 5.45) and (1.71, -5.45).

Step 4

Now we are ready to start calculating the lapping area, first we will find the area of the triangle defined by 3 point that is:

A_T=1/2 [1.73(-5.45+3)+1.71(-3-5.45)+4.75(-5.45+3)]=15.16

The segments area will be found by:

A_s=r^2 sin^(-1)⁡(c/2r)-c/4 √(4r^2-c^2 )

Seg. 1

Seg. 2

Seg. 3

	Notice that the cord c₃ is to the right of the center of circle ③ while we need the area at the left side so:
	A_s3 = π r₃² − A_s33 = π 5.5² − 39.4 = 55.63

Step 5

The lapping area is: A = A_T + A_s1 + A_s2 + A_s3 = 16.54 + 8.38 + 0.74 + 55.63 = 81.21