THE NATURE OF THE DISCOVERY: By means of the "Battersonian" method, it is possible to disregard the dogma which has held since the start of optical science, that optical surfaces can only be produced by the averaging out of multiple dissimilar grinding strokes. Anyone familiar with the standard method of grinding optical surfaces will see immediately how this new method goes against most of the standard rules as they have been practiced for centuries. However, it is not necessary to understand the old method to understand this new method. People can understand this new method by following carefully what we say here. The significance of the discovery is that we have uncovered new macro-physical laws by means of which we can construct a unique ellipse, or "orbit" for any pair of mirror blanks to be ground against each other, where the relative diameters of these two blanks fall within certain constraints, and where they are not equal in diameter. These are a series of "laws," not unlike Kepler's laws of planetary motion, which determine a non-arbitrary arrangement whereby an arbitrary Mirror blank of diameter "M" and an arbitrary Tool blank of diameter "T", when ground together, will produce a coherent optical surface. If these laws are violated, the result will be a zoned and useless surface. We believe that once the full implications of this discovery are fully realized, the entire current method of optical surface grinding will be rendered obsolete.
To avoid confusion, we use "M" and "T" throughout this page to signify both the diameters of the mirror blank (M) and the tool blank (T) used in a particular arrangement, but, also, in a generic way. For example, when we speak mathematically, such as in the statement "the axis = M + T," we always are using M and T to signify the diameters of the mirror blank and tool blank. However, when we speak in a generic way, as in, "grind M against T," we mean M to signify the mirror blank as an object, and T to signify the tool blank as an object.
To begin with, look at the following DOCUMENT in the ATM ARCHIVES., before continuing.
Some people have confused this method with the well-worn elliptical stroke. However, a clear reading will show quickly why this is wrong. Each pair of M and T falling within the defined constraints, will only work with one unique orbit, which orbit is a function of the relationship of M to T (, or, perhaps, vis versa). If the wrong orbit is used, or if a wrong-sized M or T is used, the method will not work. This also means that, given an orbit, we can determine what the diameters of M and T must be, as a function of the orbit.
This begs a question which will probably greatly upset some people, something which we have had on our mind since we first set out to make this discovery: If such a relationship exists on the macro-physical scale, is there some similar relationship on the astrophysical level? Is there some still undiscovered law by means of which we will be able to determine a planet's diameter by means of its orbit and vis versa?
There are three basic types:
RANGES: Following are some diagrams of the three basic types, demonstrating the simple reasons for the ranges in which the different types must fall. The geometry of the constructions produce very distinct ranges, which fall within very simple ratios. For example the range of the size of the tool in an Internal arrangement is exactly from a quarter of the diameter of the mirror blank to one half of this diameter. For sake of argument, and borrowing from astronomy, we use the term "perihelion" to mean the point on the orbit of T where T's center is closest to the center of M. We use the term "aphelion" to mean the point on T's orbit where T's center is furthest from the center of M. The ellipses delineate the orbit of an arrangement. By the term "superior" we mean a point on M or T which is closest to the perihelion of the orbit. By the term "inferior" we mean a point furthest from the perihelion of the orbit.
INTERNALS (Where T is always internal to the perimeter of M, and T is centered on M at perihelion.) (In the diagrams below, the large empty circles represent the mirror blank M, while the checkered circle represents the tool blank T.)
1. Where T < 0.25M, no point on T, when T is at the exact intermediate point on its orbit, between perihelion
and aphelion, reaches the center of the ellipse, which means that this area is left un-ground. Thus, this does not
work. Thus, we know that T must be larger than 0.25M.
2. Where T = 0.25M, one point on the edge of T exactly touches the center of the orbit; The minor-axis is
exactly twice the diameter of T. This is because the square root of MT is the minor-axis of the ellipse used.
Where T=0.25M, its square root gives a minor-axis of 0.5M, which is also 2T: thus, this point of the range is a
singularity. Therefore, we know that T > 0.25 M.
3. 1/4M < T < 1/3M: Where T is less than 1/3M but larger than 1/4M, although T, at the two extreme
positions on the major axis, no longer overlaps itself, it does overlap itself at the two extreme positions of the
minor-axis
4. T = 1/3M: At this point, the two successive positions of T, at the extreme positions of the major axis,
"perihelion" and "aphelion," just barely touch. The major axis is exactly twice the diameter of T.
5. 1/3M < T < 1/2M: For this and all earlier arrangements, where T is less than the radius of M, the superior
edge of T, when T is at aphelion, will never reach the center of M. This is a containing condition: no
arrangement which does not follow this condition will work.
6. At the point where T = 0.5M, a point on the edge of T hits the center of M, which represents another
singularity. This is the upper limit of the range. Therefore, we know that T < 0.5M.
7. Here, T > 0.5M, and part of the edge of T is superior to the center of M, at the point where T is at
"aphelion."
Thus, we end up with a range for the Internals, where T is larger than 0.25M but smaller than 0.5M.
PEREXTERNALS (Where one point on the edge of T, when T is at perihelion, is always at the center of M.)
The notes below describe the regular perexternal, unless otherwise specified. Note that the diagram on the left of each pair below is the regular perexternal, while the diagram on the right is the derivative perexternal. The only difference between the two, is that, for the derivative, the size of the tool blank is doubled. The same mirror blank M is used in both the perexternal and derivative perexternal cases.
1)Where T < 0.5M, we get a zoned surface, since T is entirely internal to M at perihelion, and,
thus,grinds a zone internal to M.
2)Where T = 0.5 M, we reach a singularity. Note that T is exactly contained within the radius of
M at perihelion.
3) Here, 0.5M < T < M, which is the only working range for the perexternal. If we double the size
of T, but use the exact same orbit constructed for the Perexternal, we get the Derivative
Perexternal, which has a slightly different range of eccentricity than the Perexternal.
4)Where T = M, we reach another singularity. The orbit becomes circular, with points on T's
edge, at both perihelion and aphelion, exactly hitting the center of M. Thus, we see why, with
the perexternal, the size of T must be more than the radius of M, but less than the diameter of
M. (The derivative perexternal has a slightly different range, but we won't deal with that here.)
Thus, we end up with a range for the Perexternals where T is larger than 0.5M but smaller than M.
APEXTERNALS ( Where one point on the edge of T, when T is at aphelion, is always at the center of M.)
1) Here T = M. It is impossible for T's superior edge to be at M's center at aphelion and for T to be smaller
than M. This orbit is circular, and happens to be the upper range of the perexternal orbits.
2) Where T < 2M, the center of T, at perihelion, is superior to the center of M.
3) Where T = 2M, T is exactly centered on M, at perihelion. This is a singularity. It is also the "orbit of origin"
(see below)
4) Where 2M < T < 3M, we have the proper, working range of the Apexternal. Here, the center of T, at
perihelion, is inferior to the center of M. If we double the size of M within this range, but keep the same orbit
constructed for the Apexternal, we produce the Derivative Apexternal, which keeps the same range of
eccentricities as the Apexternal.
5) Where T = 3M, the center of T, at perihelion, lies exactly on the inferior edge of M. Beyond this point, the
center of T will not be ground. This is a singularity. Thus, we know that T < 3M.
6) Where T > 3M, the center of T will not be ground, since it is never internal to the perimeter of M. This
particular diagram is where T=4M, where we reach another singularity: Here, the diameter of T is the minor
axis of the ellipse. Thus, T cannot even revolve around inside its orbit.
Thus, we end up with a range for the Apexternals where T is larger than 2M, but smaller than 3M.
THE ORBIT OF ORIGIN: All three of the basic types have a specific range of possible eccentricities of orbit. In fact, all
three of the basic types originate at an orbit whose eccentricity is 1/3. Thus, I call this particular orbit the "orbit of origin of
the system." Below are the different ranges of eccentricity:
INTERNAL: 1/3 to 3/5
PEREXTERNAL 1/3 to 0
APEXTERNAL 1/3 to 1/2
GRIND-SPEED RATIOS: The speed at which a point on M will grind against a point on T is determined by three separate motions, namely, the speed of rotation of M, the speed of rotation of T and the speed of revolution of T around its orbit. In addition, the rotational motion of M or T increases with the distance of a point from the center of M or T. We end up with quite a complex hyper geometry, for something originating in such a simple arrangement. The question is, what would be the optimum speeds at which to grind any arrangement, to produce the best optical surface? This is the main question which we need to solve to bring this discovery to the next level. Take the diagram below, where we show the simplest illustration of this:
DIAGRAM OF GRIND SPEED RATIOS 
The diagram shows the position of T
at "perihelion" (x), and at "aphelion" (y), in an internal arrangement. Let M = 1
and let M be rotating at 1 revolution per second.
A point on the edge of M, which is point y on the diagram (which is also the point
of aphelion), will be moving at pi per second. However, at point x, which is the
point on T, while T is at perihelion, which is the most superior to M of all possible
points on T, the speed of M is only 1.2566/second, which equals
T/M(Speed of M at y) if M is the diameter of the mirror blank and T is the
diameter of the tool.
Disregard the speed of rotation of T for the moment and look at its orbiting speed, or speed of revolution about its orbit, which we term "n" here: If we were to give T a constant speed of revolution around its orbit of, say, 2 units per second, and have it orbit in the opposite direction than that toward which M rotates, then:
a) at position x, the speed of grind at this point will be the motion of M at x plus that of T, or,
T/M(pi) + n per second, which in our case here is: 1.2566 + 2 = 3.2566 units per second.
At position y, the speed of grind at this point will be
M(pi) + n per second, which in our case here is:
pi + 2 = 5.14159 units per second.
If we were to revolve T around its orbit in the same direction as M is rotating, and keep the same speed of revolution as above (ie. 2 units per second), then:
a) at x, the speed of grinding motion would be T/M(pi) - n units per second,
or in this case, 1.2566 - 2 =0.74336
b) at y, the speed of grinding motion would be M(pi) - n units per second,
which, here, is pi - 2 = 1.14159 units per second.
What if we wanted the minimum grind speed at y, so as to maximize the grind-speed at x? This would be useful for parabolizing, where we want to take off more glass at the center, for example.
If we gave the speed of revolution of T a constant speed , equal to the speed of rotation of M at y, which, in this case, is pi per second, and revolved T in the same direction as the rotation of M, then:
a) at x, the speed of motion is pi - 1.2566 = 1.88495 units per sec.
b) at y, the speed of motion is pi - pi = 0 units per sec. So, there is no grinding motion at all at point y, which is the edge of M.
We could also do the reverse, set up an arrangement where the grind-speed at x was the least possible, such that M would be ground down more at the edge. This could be a way to reverse a hyperbolic surface back to a parabolic or spherical surface, for example. So, the difference of speeds makes a very real difference!
Since the possible types of orbit all follow Kepler's first law, do they follow his second as well? This is a question I have had for quite some time. I don't know, but I have hypothesized that they do. Take the following case: take the hypothetical least possible instant of time of T's motion of revolution, at both points x and y. Were it the case that the speed of revolution of T at perihelion (point x) were to "sweep out equal area in equal time" as the speed of revolution of T at aphelion (point y), then the following very interesting result occurs, for the least instant of time:
a) The speed of revolution of T at x converges on the value M/T(Speed of T at y) units per sec.
b) Similarly, the speed of revolution of T at y converges on T/M(Speed of T at x) units per sec.This preserves the original M:T relationship, which is the basis of the whole construction!
What if we take these two speeds as the maximum and minimum and integrate Kepler's second law into practice? What if we wanted, after integrating this in, to set up an arrangement whereby the grind-speeds at x and y were equal? Let us say that T is revolving in the opposite direction than M is rotating. Then, for our case here:
1.2566 per second + X per second = pi per second + (T/M)X per second.
So, X = Pi. Isn't that pretty?
Of course, this is the simplest illustration, and integrating in the speed of rotation of T makes it more complex; but this gives an idea of how to think about the problem. Solving this is the next step of the discovery which we must make.