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) equidistant 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 (x_{0},y_{0}) 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 x_{0} = y_{0} = 0, and consider the set of complex numbers x = x_{r} + ix_{i} and y = y_{r} + iy_{i} that satisfy the equation 

_{} 

where R, x_{r}, x_{i}, y_{r}, y_{i} are all purely realvalued 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 x_{i} and y_{i} to have nonzero values, we have solutions for all sets of values of x_{r},y_{r} outside the original circle, i.e., such that the sum of the squares of x_{r} and y_{r} exceeds R^{2}. 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 y_{i}, insert this into the right hand condition, and solve for x_{r} to give 

_{} 

If x_{i} = 0 this equation simply gives x as a function of y for an ordinary real circle. For any other given value of x_{i} this equation gives x_{r} as a function of y_{r} for lines of constant x_{i}. Since x_{r} must be realvalued, we know the values of y_{r} must lie within the range 

_{} 

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


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 x_{i}) viewed from the side is shown below. 


In the same way that we found the loci of constant x_{i} as a function of x_{r} and y_{r}, we can also solve for the loci of constant values of y_{i}, 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 threedimensional space representing the coordinate components x_{r}, y_{r}, x_{i}, we see that an algebraically complete circle can be described as a doublecovering surface that extends from the real circle out to infinity. Likewise in the space of x_{r}, y_{r}, y_{i} the circle is a doublecovering surface. However, these two surfaces are not independent, because the relation x_{r}x_{i} + y_{r}y_{i} = 0 implies that the signs of x_{i} and y_{i} are linked. At any given point on the x_{r},y_{r} plane, there are two possible values of x_{i} (of equal magnitude and opposite signs), and two possible values of y_{i}, but this doesn’t represent four possible combinations, because we have y_{i} = (x_{r}/y_{r})x_{i}. For example, if x_{r}/y_{r} is positive, then y_{i} and x_{i} must have opposite signs. Hence there are only two complex “points on the circle” for any given values of x_{r} and y_{r}. (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 x_{i} (and also y_{i}) would be imaginary, which means the value of x = x_{r} + ix_{i} and y = y_{r} + iy_{i} 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 (x_{r},y_{r}) inside the circle to a point on the circle. Specifically, it maps the point (x_{r},y_{r}) to the point with the coordinates 

_{} 

where r^{2} = x_{r}^{2} + y_{r}^{2}. 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 r_{1} and r_{2} denote the radii of the two circles, and let (x_{1},y_{1}) and (x_{2},y_{2}) 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 r_{1}, r_{2}, 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 r_{1}^{2} – r_{2}^{2} = L^{2}, the first two terms of the equations for the x and y coordinates of the intersection points are simply x_{2} and y_{2} respectively, so the equations reduce to 

_{} 

where we have put Dx = x_{2} – x_{1} and Dy = y_{2} – y_{1}. Also, the area of the triangle with edge lengths L, r_{1}, r_{2} in this case is simply Lr_{2}/2 (which can be confirmed by Heron’s formula), so we have 

_{} 

By similar triangles we have r_{2}/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 r_{1} = r_{2} = 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 r_{1} = r_{2}, 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 x_{i} and y_{i} rotates, but they are 90 degrees out of phase, since x_{i} is proportional to the y component of the vector between the centers, and y_{i} 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 midpoint 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 = (L^{2}/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 d_{1} and d_{2} denote the distances from the origin to the centers of the (vanishingly small) circles, and if we let d_{1} and d_{2} denote the distances from the origin to the points of intersection of those circles, we have d_{1}d_{2} = d_{1}d_{2}. 
