Intersecting Circles

 

The bonds that unite another person to ourself exist only in our mind.

                                                                            Marcel Proust

 

A “circle” is ordinarily defined as the locus of points (on a plane) equi-distant from a given point called the “center”. Algebraically, in terms of a system of Cartesian coordinates “x” and “y”, we typically express this by defining a circle as the locus of points with coordinates (x,y) such that

 

 

where (x0,y0) are the real coordinates of the center, and R is the real radius of the circle, i.e., the metrical distance from the center to each point of the circle. Usually we tacitly restrict our attention to points with purely real coordinates, but without this restriction the algebraic definition actually implies a more elaborate structure, consisting of all the “points” with complex coordinates that satisfy the above equation.

 

As an example, suppose the center of the circle is at the origin of the coordinates, so x0 = y0 = 0, and consider the set of complex numbers x = xr + ixi and y = yr + iyi that satisfy the equation

 

 

where R, xr, xi, yr, yi are all purely real-valued numbers. This implies

 

 

Obviously if the imaginary components of x and y are both zero, the left hand condition is automatically satisfied and the right hand condition reduces to the usual equation for a real circle centered on the origin. However, if we allow xi and yi to have non-zero values, we have solutions for all sets of values of xr,yr outside the original circle, i.e., such that the sum of the squares of xr and yr exceeds R2. The locus of these solutions could be regarded as the algebraically complete “circle”.

 

To visualize a complete circle, we can solve the left hand condition for yi, insert this into the right hand condition, and solve for xr to give

 

 

If xi = 0 this equation simply gives x as a function of y for an ordinary real circle. For any other given value of xi this equation gives xr as a function of yr for lines of constant xi. Since xr must be real-valued, we know the values of yr must lie within the range

 

 

A plot of the lines of constant xi for a circle of radius R = 1 is shown below. Note that since xi appears only squared in the equation for these loci, they correspond to both positive and negative values of xi.

 

 

As an aside, these contours are remarkably similar to the stream lines in potential fluid flow around a sphere or cylinder. A contour plot of the upper surface (i.e., the positive values of xi) viewed from the side is shown below.

 

 

In the same way that we found the loci of constant xi as a function of xr and yr, we can also solve for the loci of constant values of yi, and by symmetry it is the same as those shown above, except rotated 90 degrees. Thus a complete depiction of an algebraically complete circle is as shown below.

 

 

In the three-dimensional space representing the coordinate components xr, yr, xi, we see that an algebraically complete circle can be described as a double-covering surface that extends from the real circle out to infinity. Likewise in the space of xr, yr, yi the circle is a double-covering surface. However, these two surfaces are not independent, because the relation xrxi + yryi = 0 implies that the signs of xi and yi are linked. At any given point on the xr,yr plane, there are two possible values of xi (of equal magnitude and opposite signs), and two possible values of yi, but this doesn’t represent four possible combinations, because we have yi = -(xr/yr)xi. For example, if xr/yr is positive, then yi and xi must have opposite signs. Hence there are only two complex “points on the circle” for any given values of xr and yr. (Of course, these two “points” merge into one on the real circle, where the two parts of each surface meet.)

 

Incidentally, the flat “basin” inside the real circle (as depicted in the countour plot above) is the region where the computed value of xi (and also yi) would be imaginary, which means the value of x = xr + ixi and y = yr + iyi would be purely real. This implies that the resulting values of x and y are on the real circle, so the values of xi and yi are just such as to map each point (xr,yr) inside the circle to a point on the circle. Specifically, it maps the point (xr,yr) to the point with the coordinates

 

 

where r2 = xr2 + yr2. It’s easy to verify that these points do indeed lie on the circle of radius R, and the second terms are perpendicular to the first terms, so it follows that each point inside the circle is mapped to the circle along the line perpendicular to the line from the center of the circle, as shown below.

 

 

Now we consider the intersections between two circles, given the (purely real) radii and the coordinates of the centers. Let r1 and r2 denote the radii of the two circles, and let (x1,y1) and (x2,y2) denote the coordinates of their centers (in terms of a Cartesian coordinate system). For convenience we also let L denote the distance between the centers. It is straightforward to show that the intersection of these two circles consists of two points, with the coordinates

 

 

where A is the area of the triangle with edge lengths r1, r2, L, as given by Heron’s formula

 

 

As a simple example, consider two circles of radii 5 and 3 units respectively, with centers located a distance of 4 units apart, as depicted for a typical configuration in the figure below.

 

 

Since in this example we have r12 – r22 = L2, the first two terms of the equations for the x and y coordinates of the intersection points are simply x2 and y2 respectively, so the equations reduce to

 

 

where we have put Dx = x2 – x1 and Dy = y2 – y1. Also, the area of the triangle with edge lengths L, r1, r2 in this case is simply Lr2/2 (which can be confirmed by Heron’s formula), so we have

 

 

By similar triangles we have r2/L = dx/Dy = dy/Dx, so we make these substitutions to arrive at the expected result

 

 

This merely confirms that equations (1) give the real intersection points for real circles that overlap in real space.

 

However, those equations also apply to any two circles, even if they don’t overlap in real space. For example, consider two identical circles, each of radius r1 = r2 = 2 units, with centers separated by a distance of L = 5 units.

 

 

These circles clearly have no real points of contact, but the algebraically complete circles do have two points of intersection in the full complex space. The area of the triangle with edge lengths 5,2,2 is imaginary, given by Heron’s formula as (15/4)i. Since r1 = r2, the second terms in equations (1) are zero, so the real parts of the coordinates of the intersection points are given by the first terms, which represents simply the midpoint between the centers of the circles. The imaginary parts of the intersection coordinates are proportional to the real components of the vector between the centers. For this example we have

 

 

Thus if we rotate the configuration, keeping the radii and distance between centers the same, the vector between the centers rotates and the vector consisting of xi and yi rotates, but they are 90 degrees out of phase, since xi is proportional to the y component of the vector between the centers, and yi is proportional to the x component. The magnitude of the imaginary vector is a constant (2A/L)i. For a specific numerical example, suppose the centers of the two circles are located at (12,7) and (16,10). Inserting these values into the above equations, we get the coordinates of the two points of intersection

 

 

It can easily be confirmed that each of these points is a distance from 2 units from each of the centers.

 

If the radii of the circles are unequal, the real parts of x and y are no longer at the mid-point between the centers, but it still falls on the line between the centers, shifted toward the smaller circle. For example, if the radius of the second circle in the preceding figure was reduced from 2 units to just 1, the real parts of the intersection coordinates would shift from the midpoint (14, 17/2) to (14 + 12/50, 17/2 + 9/50) as shown below.

 

 

It’s interesting to consider the case when the radii of the circles shrink toward zero, in which case the area approaches A = (L2/4)i and the coordinates of the points of intersection approach

 

 

In this limiting case the squared distances from the origin of the coordinate system to these two intersection points are

 

 

Hence if we let d1 and d2 denote the distances from the origin to the centers of the (vanishingly small) circles, and if we let d1 and d2 denote the distances from the origin to the points of intersection of those circles, we have d1d2 = d1d2.

 

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