O.K. I admit it. Only a few of these questions have actually been asked of me, and only one has been asked more than once. Some have come up on the ATM e-mail list. Others have been implied in ATM list discussions, if not explicitly asked. Some are things I wondered about myself, but have never heard or read from anyone else. I think most of them may be useful to potential users of Plop or Plop designed mirror cells.

Have Plop's cell designs been tested?

What do P-V and RMS mean?

How do I convert Plop's PV and RMS results to wavelengths?

How do I interpret Plop's color deformation plot?

Are the old cell designs always bad?

What about David Chandler's program Cell?

I never see 4, 6 or 12 point cells in the ATM books. Are they really any good?

I never see commercial cells with Plop derived dimensions. Don't the pro's know the best way?

Do the collimation screws have to be under the mirror support points?

What about Richard Schwartz's 5 and 7 point cells with a point in the center?

What is the equal area rule?

What is finite element analysis, and why is it better than the equal area rule?

What is refocusing?

If refocusing allows parabolic deformation, is it bad for mirrors with other shapes?

How about flats?

Where can I find setup files (.gr files) for cell designs that do not come with Plop?

Why does the word Plop appear in a different typeface than the rest of this web site?

Have Plop's cell designs been tested?

I am not aware of any systematic, quantitative tests of mirror cells in the sizes used by ATM's. Many ATM's have built telescopes using Plop designed cells. Many of these telescopes have been evaluated by experienced and critical opticians and observers. So far as I know, Plop cells have performed well. This is especially significant given that many ATM built telescopes in the last decade have used mirrors much thinner than in previous times. Thin mirrors are more susceptible to cell induced deformation. Poorly performing cells should show up quickly with these mirrors. To date, I think Plop has performed pretty well.

What do P-V and RMS mean?

P-V means peak to valley. RMS means root mean square. These are two different ways of specifying the magnitude of mirror shape errors.

P-V is pretty simple. First, you have to specify the shape that you want the mirror to be. For the majority of amateur telescope mirrors, this is a paraboloid, but other shapes are used. The height of the perfect desired shape is subtracted from the height of the real surface. The place on the mirror with the lowest value is the valley. The place on the mirror with the highest value is the peak. Peak to valley is the vertical distance between them. This is exactly anaogous to most maps of the Earth. Most maps ignore the Earth's spherical shape. It is "subtracted" out. The height of mountains and the depth of valleys is compared to a mean value for the sphere. The P-V surface error for the Earth would be the height of Mount Everest minus the height of the Marianas Trench. (The height of the Marianas Trench is a negative number, so minus a negative is a plus. The P-V value is the sum of the absolute values.)

RMS is more complex, but turns out to be a more useful value. Consider the Earth example again. Although Mount Everest and the Marianas Trench are very real, using their elevations to characterize the Earth as a whole would be pretty misleading. Only a tiny fraction of the Earth's surface is as high or low as those locations. Most of the Earth is a lot smoother. The RMS calculation takes into account the amount of area at each height and finds an average value based on the area at each height. If you know basic statistics, you will see that the RMS calculation uses exactly the same math as finding the standard deviation of a population. It is correct to think of the RMS error as the standard deviation of the surface error.

The RMS error value is useful because prediction of image quality depends on the amount of light coming from each part of the mirror, as well as the height of each part of the mirror. The amount of light, for a uniformly reflective mirror, depends on the area. An RMS value can be directly used to calculate a prediction of image quality called the Strehl ratio.

How do I convert Plop's PV and RMS results to wavelengths?

Plop reports PV and RMS residual deformations in millimeters using scientific notation. A typical value might be 1.8125e-6 millimeter. This is the same as writing 1.8125/1,000,000 millimeter. It is also equal to 1.8125 nanometers. (1 millionth of a millimeter is equal to 1 billionth of a meter, one nanometer.)

The formula for converting is:
Wavelengths = Deformation (in millimeters) / Reference Wavelength (in millimeters)

If you want a fractional result, such as 1/4 wave, or 1/8 wave, invert the equation:
Wavelength Fraction Denominator = Reference Wavelength (in millimeters) / Deformation (in millimeters)
This equation gives the denominator of the fraction, the bottom number.

What reference wavelength value should I use? Here is a table with common values.

Example: Using the deformation value above, 1.8125e-6 millimeter and a reference wavelength of 550 nanometers.
Wavelengths = 0.0000018125 / 0.00055 = 0.0033 Wavelengths = 1/303 Wavelengths

How do I interpret Plop's color deformation plot?

Blue areas on the color plot represent deformations above the "perfect" mirror figure. Red represents deformations below. Green is the happy medium. The default setting scales whatever deformation results for a particular cell so that it's high point is always blue and its low point always red. Thus, even a cell producing very little deformation will still show the full spectrum of color. In order to use the color plot to compare different designs, one has to set a fixed color scale. Plop allows this under the options menu item of the color plot screen.

Are the old cell designs always bad?

Not necessarily. Goodness or badness in a cell is a quantitative question. One decides on a level of deformation that one is willing to accept, and then compares the results from different cells. If one of the older designs gives acceptable results for your mirror, then it is OK. I recently analyzed the Kriege & Berry 18-point design for an ATM with a 17.5 x 2.1 inch mirror. In that case, the K&B design turned out to be plenty good enough.

Allowing Plop to optimize the design produced somewhat smaller residual deformation without making the cell any more difficult to construct or more sensitive to error, but the K&B deformation was already so small that there was no practical reason to prefer one over the other. It is also true that Plop shows a 9-point cell should produce good results for this mirror. The 9-point cell would probably be more sensitive to construction tolerances because it's triangles are so narrow.

Does this mean that the K&B 18-point design is always good? No. A thinner, or larger diameter mirror would accentuate the difference between the K&B and Plop designs. Plop optimization will allow some 18-point design to be used for a larger or thinner mirror than would be suitable for the K&B design.

What about David Chandler's program Cell?

I have never seen or used David Chandler's program Cell. The description I have read indicates that Cell uses the equal area rule for determining the spacing of mirror cell supports. Since Plop's finite element analysis is superior to the equal area rule, one should no longer use Cell.

I never see 4, 6 or 12 point cells in the ATM books. Are they really any good?

You bet!

I never see commercial cells with Plop derived dimensions. Don't the pro's know the best way?

Nope! A lot of the commercial cell suppliers are still behind the curve, big time, on this issue. A few, especially those that ”went professional“ after being atm's, are using Plop. Ask before you buy. Ask if they have done a Plop analysis of their cell and if it produced less than 2.5 nm RMS deformation.

Do the collimation screws have to be under the mirror support points?

Nope! Many cells are built this way, but there is no absolute reason for it. Often, one sees it in simple three point cells for small mirrors. If the mirror support points are located at about 0.4r as Plop indicates, and the screws are also, it makes a very narrow base for the collimation action. The collimation adjustment is overly sensitive, and the cell is easier to knock out of adjustment. There is no reason at all that the screws can not be moved much farther out, even outside the diameter of the mirror. If the builder is clever, he may be able to use this freedom to mount the mirror a little lower, often good for balance.

In the Kriege and Berry, Obsession style tailgate, the collimation screws form the pivot points for the first layer of supports in a flotation system. If you use this design, the collimation screws have to be where the first layer pivots are, because the rest of the cell has no opportunity for collimation adjustment.

What about Richard Schwartz's 5 and 7 point cells with a point in the center?

With respect to Richard, I have analyzed a couple of these cells using Plop and they have not turned out too well. Richard did his analyses using Nastran, a finite element analysis program designed for more general mechanical analysis. Richard has said that Nastran does not perform the refocusing that Plop uses. Instead, Nastran minimizes deformation all across the mirror surface. I think the cells with a center support look good when minimizing total deformation, but not when using refocusing. On the other hand, Richard has pointed out that Nastran uses a more sophisticated finite element analysis model than Plop. There may be instances where this extra sophistication may be important. These instances probably occur chiefly with rather large (by ATM standards) and thick mirrors.

There is one situation where Richard's cells might be better than Plop's: Flats. Typically, with a flat mirror, one does not wish to allow any deformation at all. One would not want to use the refocusing feature of Plop for a cell designed for a flat and Richard's cells might look better in that case.

What is the equal area rule?

The equal area rule refers to a mirror cell design strategy invented or at least popularized by John Hindle. This approach is described in the books Amateur Telescope Making, Vols. 1-3. The equal area rule divides a mirror into symmetrical segments of equal area. Assuming that the mirror is of equal thickness and density throughout, this also divides it into segments of equal weight. The idea is that, if each point of the support system is centered under a segment of equal weight, then the load from the mirror will be evenly distributed, and the mirror will be optimally supported. If the mirror were broken into individual segments, so that there were no mechanical linkage between segments, the equal area rule would be closer to optimum. ATM mirrors are one piece, and usually solid. Stresses in one part of the mirror are transmitted in some degree to all parts of the mirror. The equal area rule ignores this. This turns out to be a major flaw.

What is finite element analysis, and why is it better than the equal area rule?

Finite element analysis (FEA) is a computational mechanical engineering method for predicting the stresses and deformations in a structure. It is impractical to break a solid down to the molecular level to calculate stresses. FEA breaks a structure into small, but still macroscopic, spatial elements and calculates the stresses between these. Each element deforms according to the stresses transmitted to it and according to it's physical properties. In Plop, the mechanical properties modulus, Poisson ratio and density are used to calculate the deformation of each element. Finite element analysis is used widely in modern engineering to predict the behavior of a wide range of structures. It is an accepted, reliable part of modern structural engineering.

Despite being accepted and reliable, FEA is by no means an easy technique. The mathematical methods involved are arcane and difficult. Commercial FEA programs are very expensive and hard to learn. ATM's are fortunate that Toshimi Taki, David Lewis, Marc Arnold and others have contributed their knowledge and effort so that we now have Plop.

What is refocusing?

Refocusing is a subtle and clever feature of Plop. David Lewis recognized that not all mirror deformations are necessarily bad. A deformation that changes only a mirror's focal length, but does not change it's parabolic shape, may be quite acceptable. Accordingly, Plop has a refocusing option. When refocusing is turned on, Plop ignores deformations that change only the focal length of a parabola. Plop works to minimize deformations that cause the mirror surface to deviate from a parabola, but the parabola's focal length is free to change.

This seemingly small change in approach turns out to lead to large benefits. Many mirror cell designs can work for larger and thinner mirrors than anyone thought possible primarily because Plop adjusts the support points to allow a fair amount of parabolic deformation.

In most cases, the focal length change due to refocusing is rather small. The value is displayed in Plop's Run screen.

All of the cell designs tabulated at this web site have been calculated with refocusing turned on.

If refocusing allows parabolic deformation, is it bad for mirrors with other shapes?

At first I thought this might be so. The primary mirrors of a Ritchy-Chretien telescopes, for example are hyperbolic rather than parabolic. Schmidt camera's have spherical primary mirrors, etc. I thought that the parabolic deformation accepted by Plop's refocusing might be bad for these mirrors. David Lewis set my mind at ease. David has analyzed the deformations resulting from refocusing. He modeled them with a polynomial equation. (I think he used the Zernicke polynomials commonly used in optical mathematics.) Almost all of the deformation occurs in the first term of the polynomial. This is the term that changes only the focal length, without changing the form of the surface. The form of the surface, spherical, paraboloid, hyperbolic, etc. is contained in the higher terms of the polynomial and those higher terms are not significantly affected by the refocusing deformation.

How about flats?

There is one situation where refocusing might not be a good idea: Flats. Typically, with a flat mirror, one does not wish to allow any deformation at all. One would not want to use the refocusing feature of Plop for a cell designed for a flat.

I have not yet tabulated any cells for flats at this web site. If you need a cell for a flat, such as for a heliostat, and do not wish to do the Plop work yourself, please e-mail me.

Where can I find setup files (.gr files) for cell designs that do not come with Plop??

Plop setup files have a .gr extension, but, inside, they are simple ASCII text. A free, downloadable collection of Plop setup files is at http://www.atmsite.org/contrib/Holm/cells/plopsetup.html

Why does the word Plop appear in a different typeface than the rest of this web site?

Plop is the name of a computer program. HTML has the <code> tag designed to use for setting off computer code. Perhaps it is silly of me, but I usually use the <code> tag to set off the names of programs as well. Most browser's render <code> tagged text in a different font from other text. Usually it is a fixed spacing font similar to the output of old fashioned computer terminals and printers. I think most programmers still code with fixed spacing fonts. There are times where the number of characters on a line is important and that is much easier to see with fixed spacing. Fixed spacing also makes indenting much easier and more predictable. Indenting is an important feature of well written code.

© 2005 Mark D. Holm