When looking at the basic 10–20 system one can create a coordinate plane and then find the equations of the two lines intersecting at the F3 location. Without loss of generality we will orient this coordinate plane with the nasion on 270° (the negative y-axis) and the inion on 90° (the positive y-axis) and the vertex in the center of the plane, as shown in .

First, we find the polar coordinates of the four desired points, which will allow us to find the two equations of the lines intersecting at the F3 location. We will let R1 be the distance from the vertex to the point F_{pz}. Likewise, we will let R_{2} be the distance from the vertex to the point T_{3}.

The coordinates for the points F_{z}, F_{7}, F_{p1}, and C_{3} are now intersected by an imaginary circle with radius R_{1}. Note that a point in the polar coordinate system is expressed as two coordinates: the radial coordinate and the angular coordinate. The radial coordinate is the distance from the center of the plane and the point and the angular coordinate is the angle of the ray beginning at the center and containing the point. The angular coordinate is measured counter clockwise from the 0°ray (which is equivalent to the ray making up the positive half of the x-axis on the Cartesian plane)

The four points are expressed first as polar coordinates and then as Cartesian coordinates.

The equation to the line containing the points F

_{7} and F

_{z} is expressed in y intercept form as:

The equation of the line containing the points C

_{3} and F

_{p1} is expressed in y intercept form as:

In order to find F3 we will set these two equations equal to one another, and then solve for x. Plug x into either of the first two equations and then solve for y. This will give you the following coordinates on the cartestian plane.

This pair of Cartesian coordinates must then be translated back polar coordinates

Now that we know the angle from the line from tragus to tragus, we will use its complementary angle in order to find the angle off of the midline.

Let this new angle be Φ. Then to find the distance(u) along the circumference(c) beginning at the midline we will use the following equation

Clearly, the head is not a flat plane nor is it a sphere. The distance from the vertex to the nasion or any other points around the circumference will be much longer if measured along the scalp than the shortest distance between the two points. This is because of the curvature of the head.

The radius coordinate of the polar coordinates for F3 must now go through a small correction in order to account for this. When measuring from vertex to the desired location we have assumed that the head is a sphere; however, upon looking at this measurement, the line we are measuring is actually closer to being on a plane than a sphere. The following diagram illustrates the shortest distance between a and b is a straight line as shown. The distance of the arc along a circle going through a and b with its center at the origin is shown.

Unlike our measurements from the vertex to the inion, nasion, or trachus, the measurement from the vertex to the F3 location close to a straight line. Therefore, we must find the correction for this distance. Assuming the distance from a to the center and be to the center are the same, say r, the distance from a to b is given by r(2)^{1/2}. Whereas arc length from a to b will be given by (0.25)2π r = π r(0.5). Now to find the correction factor q we will solve for q in the following equation.

Thus we multiply the radius coordinate of the polar coordinates for F3 by 0.9 in order to account for the head not being a sphere.

Evaluation of the Beam F3 System

Ten healthy adults were enrolled in a preliminary pilot study to investigate the accuracy of the Beam F3 system. Participants were randomly assigned either to undergo the Beam F3 measurement followed by a standard 10–20 measurement, or to undergo the 10–20 measurement followed by the Beam F3 measurement. The 10–20 measurement system was implemented by a trained neurophysiology research assistant. All participants wore a form-fitting vinyl head cap that was held in place with a chin strap. All measurements were marked on the caps with a felt-tipped marker. After both system measurements were complete, the F3 locations determined by each system were compared and any difference between the two locations was measured with a fabric ruler. The locations matched exactly (i.e., distance of 0 mm between the points identified by each method) in 80% of participants, and were within 1mm of each other in the remainder of the sample (note, however that the observed differences could have been due to error in the administration of either the 10–20 or the Beam F3 systems). The Beam F3 system required only 20% of the time to conduct compared to the 10–20 system. These preliminary results support the reliability and utility of the Beam F3 System, however, future investigations might be warranted to conduct more in-depth analyses of its utility and potential limitations. Future studies should also be conducted comparing this system with not only the 10–20 system, but also the 5cm method as well as neuroimaging/co-registration systems.