Spherometer Calculator / Calibration Tool

Entry fields for the geometry of this specific spherometer A B C Dial Indicator
Enter A to B measurement:
Your spherometer will have 3 fixed legs (call them A, B, and C) on the perimeter and one adjustable leg (the dial indicator) in the middle. In these 3 boxes you would enter the center-to-center distances between each pair of legs. It is best to use a caliper for this since the measurements need to be as precise as possible (garbage in, garbage out, and all that...)
Your browser (if it supports HTML5 local storage) will store these values for you (each time you hit submit button -- and reload previously saved values when you open or refresh the page) so you won't have to keep re-entering them each time you use this (until you clear your browser cache, at least). If you have multiple spherometers you'd like to automatically save the values for, just use a different browser for each, e.g. Chrome for spherometer #1, Firefox for spherometer #2, IE for spherometer #3, etc.
.......... Enter A to Dial Indicator measurement:
In these 3 boxes you enter the distances (center-to-center) between the 3 legs on the outer perimeter and the central inner leg. We'll refer to this center leg as the dial indicator even though it might be just a simple screw or a digital indicator. Try to measure the distance with the indicator leg level with the tips of the other legs, such as when on a flat surface with the indicator leg all the way down to the surface, rather than on a diagonal. These values are used to determine how far (offset error) from the ideal center of the spherometer the central dial indicator is actually positioned.

Enter B to C measurement: .......... Enter B to Dial Indicator measurement:
Enter C to A measurement: .......... Enter C to Dial Indicator measurement:

Select Calculator Output Precision: decimal digits
Project entry fields Enter Dial Indicator Reading:
Enter the dial indicator reading here.

Enter Target Focal Length:
(optional)
If you want to determine what your target indicator reading should be for a desired focal length, enter the desired focal length in this box. The target indicator reading will appear in the "Target Dial Indicator Reading at Targeted Focal Length" box below. For example, if you want to know what indicator reading your spherometer will give when measuring a mirror with a 60 inch focal length, enter 60 in this box and look for the answer in the above-mentioned box.
(optional)
Spherometer-specific information (calculated for you) Spherometer Radius:
The value that will appear in this box after you hit submit is the radius of your spherometer. In particular, it's the radius of the circle defined by where the 3 legs on the perimeter of your spherometer come into contact with the surface of the mirror. The legs form a triangle whose vertices are all on the circumference of the unique circle they define in this manner. The radius of that circle (called the circumcircle) is the radius of your spherometer.

Notice how the center of the circle in the above image doesn't quite match up with the intuitive center of the triangle. If it were an equilateral triangle (one with the 3 side lengths all equal to each other) the center of the circle would also be what we'd consider to be the center of the triangle. That's why it's important to try to get the legs of your spherometer the same distance apart from each other and the same distance apart from the center dial indicator, but this calculator is designed to account for errors of this type and compensate for them.

Crosscheck Error Value:
The closer this number is to zero the more accurate your measurements you entered in above were to the actual dimensions of the spherometer. Remember, the measurements are all center-to-center. If you used inner-to-inner or outer-to-outer instead of center-to-center it could lead to non-zero values in this box. You can expect to have some measurement error due to precision issues.
The calculator graphs positions for legs A, B, and C, and for the Dial Indicator, all based on the measurements entered in the 6 boxes (A to B, B to C, C to A, A to Dial, B to Dial, and C to Dial), assigning all 4 positions cartesian coordinates in the form of (x,y). A is assigned (0,0), B is assigned (A to B, 0), and C is derived mathematically based on the measurements provided (B to C and C to A). The (x,y) coordinates for the Dial Indicator are calculated similarly, but using A to Dial and B to Dial entries. At this point, the calculator has (x,y) coordinates for A, B, C, and the Dial Indicator. It calculates the distance from C to the Dial Indicator and compares this calculated value to the user-entered value for C to Dial. The difference between these 2 values is the crosscheck error value. If this number seems to high it could indicate an error in the measurements, which said error could be in any of the 6 boxes, not necessarily in the C to Dial measurement. The closer the crosscheck error value is to zero the more confident you can be in the calculator's results. You might be able to get it close to zero by remeasuring.

Calculated Dial Indicator Offset Error from True Center of Spherometer:
The offset error is the distance from the Dial Indicator to the true circumcenter of the spherometer. Don't confuse this with the error in the final measurement of the radius of curvature or focal length. One way to envision what the offset error is would be to consider taking a 12" ruler and placing it over a 16" diameter mirror and measuring the sagitta at 5" or 7" position rather than at the 6" (center) of the 12" ruler. In this example, your offset error would be 1". The final effect of the offset error on the measured focal length or radius of curvature will depend on the curvature of the mirror, the steeper the curve the bigger the final error. For a spherometer in the above example with a 6" spherometer radius and with an offset error 1", the sagitta error would be exactly the same as with the 12" ruler measured at the 5" or 7" position.


[image of spherometer to go here after you hit the submit button]
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Mirror-specific information (calculated for you) Radius of Curvature:
This is the calculated radius of curvature (based on the dial indicator reading, after factoring in the offset error and radius of the spherometer). The focal length is just half of this value. To visualize what is meant by radius of curvature, try to think of the curved mirror surface as a small part of the outer surface of a larger sphere. Imagine a giant perfectly round ball with a 20 foot diameter. If the ball is perfectly round you would be able to set it on top of a mirror with a 60" focal length and have the outer surface of the ball exactly conform to the surface of the mirror. This is because the focal length (60" = 5') is exactly half the radius of curvature (10') which is exactly half the diameter (20').
Focal Length:
Radius of Curvature (ignoring dial indicator offset error):
This is the radius of curvature you'd get by ignoring the offset error (in other words, by just assuming the dial indicator is exactly centered). It is being provided to the difference accounting for the offset error made in the calculations of the radius of curvature. It is expected the difference will be very negligible in most cases.
Focal Length (ignoring error):
Target Dial Indicator Reading at Targeted Focal Length:
If you entered a value in the "Enter Target Focal Length" box you'll get a number in this box ("NaN" -- Not a Number, otherwise). This number is the indicator reading your spherometer will give you once your mirror has reached the targeted radius of curvature. This does take into account the offset error. The following box is the same thing, but calculated by ignoring the offset error. It is expected these 2 values will very often be either exactly the same or very close to one another.
(ignoring error):

[image of mirror will go here after you hit the submit button]
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Telescope and eyepiece information

Telescope Calculator Lagniappe

(You only need to enter the diameter and the focal length, the other fields below are all calculated for you.)

Enter mirror (or objective lens) diameter (aperture) in inches: or in millimeters:
Enter focal length in inches: or in millimeters:
The F/number of this telescope is F/
The Maximum Resolving Power is: arcseconds
Faintest stars and other objects visible with this telescope will be of Magnitude:

Some eyepieces you might want to consider for this telescope: (As these are calculated values, some might not even be available, so just get as close as you can find.)

Optimal Eyepiece for the resolving power of the human eye (2.4mm exit pupil): mm focal length
Surface Brightness for the optimal eyepiece will be %
Magnification will be x

Maximum Magnification Eyepiece (1mm exit pupil): mm focal length
Magnification will be x
Surface Brightness for the maximum magnification eyepiece will be: %
(Dawes limit maximum resolving power for this telescope, any magnification over 200x will suffer from atmospheric interference.)

Wide Field Compromise Eyepiece for this telescope (5mm exit pupil): mm focal length
Magnification will be x
Surface Brightness for the wide field eyepiece will be: %
(Wide Field is compromise between 70% of widest possible field of view with 50% brightness.)

Maximum Brightness / Minimum Magnification Eyepiece (7mm exit pupil): mm focal length
Magnification will be x
Surface Brightness for this eyepiece will be: %
(7mm is the width of the pupil of the darkness-adjusted human eye, so going beyond 7mm exit pupil will not result in more brightness since the additional light will be lost. There are indications the human eye pupil will shrink with age -- perhaps to 5mm by age 80 -- so your mileage may vary.)

Extra-high Magnification Eyepiece (2/3mm exit pupil): mm focal length
Magnification will be x
Surface Brightness for this eyepiece will be: %
(Extra-high magnification eyepiece won't reveal more detail, but the larger size might make it easier to see things.)

(Credit to Randy Culp for the telescope and eyepiece equations in this section.)

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What is a spherometer? If you've found this page then you probably already know this, but for benefit of those who might not know, a spherometer is a device used to measure the curvature of a curved surface, such as the surface of a concave primary mirror in a newtonian reflector telescope. The spherometer gives the same measurement you'd get simply by placing a straight edge (such as a ruler) on top of the mirror and measuring the distance (called the sagitta) from the center of the ruler to the surface of the mirror using feeler gauges (though perhaps less crudely and we hope more precisely). this distance, by the way, is called the sagitta. armed with the sagitta and knowing the length of the ruler (or, analagously, the radius /> diameter of the spherometer) you can do some math to work out the radius of curvature / focal length of the mirror. This calculator does this for you, and much more. It will work out the radius of the spherometer for you, and also work out what, if anything, is the offset error of the spherometer, and, of course, the radius of curvature of the mirror and it's focal length.

"Offset Error"? What I'm referring to as offset error (my own words, not some industry standard) is the distance which the dial indicator is in a spherometer from the true center of that spherometer. Let's go back to the ruler analogy. If you place a 12-inch ruler atop a 16-inch mirror you need to measure the sagitta at the 6-inch mark (center of the ruler) of the ruler. If you measure at the 5-inch mark that would be equivalent to a 1-inch offset error. If your DIY spherometer's dial indicator position is 1 inch from the spherometer's true center and if the spherometer has a 6-inch radius, the results you get (sagitta reading) would be exactly (more or less) the same as you'd get by using the 12-inch mirror and taking your sagitta reading at the 5-inch (or 7-inch) mark instead of at the 6-inch (center) of the ruler. This calculator measures the spherometer's offset error (based on user-input data) and compensates for that offset error. Thus, even using a poorly made DIY spherometer, you can still get accurate radius of curvature measurements with this calculator (measurements as accurate as the measurements you provide for the dimensions of the spherometer).

So, what are these measurements? You need to provide the calculator with 6 measurements in order for it to determine the geometry of the spherometer. You need to provide the distances from each of the 3 legs to each of the other 2 legs, to wit: A to B, B to C, and C to A (each of the perimeter legs being labeled A, B, and C, respectively). You also need to provide the distances from each of the 3 legs to the central leg (dial indicator leg), to wit: A to dial indicator, B to dial indicator, and C to dial indicator. These are all center-to-center measurements, not inner-to-inner or outer-to-outer.

The calculator uses the first 3 measurements to determine the radius of the spherometer. Think of the 3 distances as the side lengths of a triangle. Every triangle has 2 associated circles: an incircle and a circumcircle. The incircle is the one that touches all 3 sides of the triangle (but it's not the one we're interested in here). The circumcircle is the one whose circumference (the perimeter of the circle) intersects all 3 vertices of the triangle. The radius of that circumcircle is the radius of the spherometer. The center of that circle (the circumcenter) is the true center of the spherometer.

Let's examine in some detail how the offset error and crosscheck error are calculated. The calculator assigns cartesian coordinates for legs A, B, and C, for the Dial Indicator, and for the Circumcenter of the spherometer. Let's call these A = (ax,ay), B = (bx,by), C = (cx,cy), Dial Indicator = (dx,dy) and Spherometer Circumcenter = (sx,sy). A gets assigned (ax,ay = 0,0), B gets assigned (bx,by = A to B distance user supplied, 0). C = (cx,cy) gets calculated based on distances the user supplied for B to C and C to A. D = (dx,dy) gets calculated based on the user supplied A to Dial and B to Dial measurements. The user supplied value for C to Dial Indicator is used as a crosscheck for the calculated distance between (dx,dy) and (cx,cy), the difference between them being the crosscheck error value. For example, if the calculator works out the distance from (dx,dy) to (cx,cy) is 3.257" and the user supplied a value for C to Dial of 3.287", then the crosscheck error value is 3.287 - 3.257 = 0.030. The offset error is the distance the calculator works out for the distance between (dx,dy) and (sx,sy).

Once you are satisfied with your measurements (and you're getting a nice low crosscheck error value very close to zero), you'll want to write them down somewhere (bottom of the spherometer would be a good place to write them). The calculator will remember your values for you using something called "local storage" in your browser cache. If you have multiple spherometers you should use a different browser for each (or the same browser on different computers) so you don't have to manually re-enter those measurements each time you use the calculator, which would be tedious and which would become quickly annoying. The local storage values are persistent and will remain in memory until you clear your browser cache (which the browser might do "behind your back" depending on certain browser settings, so be sure to write down your measurments just in case).

When you hit Submit the calculator will go through the calculations and create a couple images for you. (It also saves the spherometer geometry, so be sure to hit Submit before closing the browser if you want to save them.) The first image is a graph of the spherometer as seen from the bottom. It's fairly self-explanatory. The green A, B, and C represent the legs of the spherometer. The red lines from each leg will converge on the position of the Dial Indicator. The black cross is the true center (circumcenter) of the spherometer. The red circle around the black cross illustrates the offset error (which is the radius of the red circle). Notice how the Dial Indicator's position is a point on the circumference of the red circle. The closer the black cross is to the convergence point of the red lines, the lower the offset error is for this spherometer. The crosscheck error is not illustrated because it's not a physical property of the spherometer. Offset error is a physical error in the build whereas the crosscheck error represents a measurement error.

Hover your mouse over some of the entry fields in the calculator to get some popup tooltips (even with pictures included) to help you further understand what is going on with it. If you're really feeling adventurous check out the source code for this page where I go into some detail on how the formulas were derived using Mathematica. Except for the distance formula given to us by the Pythagoreans so long ago, the rest of it is pretty much my own work. I've always enjoyed puzzles, and this one (figuring out how to get the radius of curvature knowing that the sagitta being produced by the instrument wasn't exactly right, but also knowing how far from the center of the spherometer the indicator was positioned) was one that intrigued me. The "aha" moment came when I realized I had 3 known sets of coordinates where the spherometer made contact with the surface of the mirror and could use them to work out the coordinates of the center of the circle and hence the radius of curvature.

After completing the spherometer calculator I came across some interesting equations for telescopes and eyepieces, and decided to incoroprate them into some additional features for the calculator. I thought I would add this paragraph to explain some of those terms in more detail. The Maximum Resolving Power is in arcseconds. There are 60 arcseconds per arcminute, and 60 arcminutes per degree. The full moon is about 30 arcminutes (or 1/2 degree) across, for comparison purposes. The Maximum Resolving Power is how close 2 stars can be and yet still be seen (barely perhaps) as 2 separate stars when viewed through the telescope under ideal atmospheric conditions with the ideal eyepiece. The faintest star magnitude rating tells the faintest objects (in magnitude) the telescope would be able to see. It's a logarithmic scale with origins dating back to ancient times (with modern updates). The brighter the object, the lower the magnitude number. Most visible (to the naked eye) stars are Magnitude 1 through 6, with the faintest ones visible to the naked eye being around Magnitude 7. Sirius, the brightest star, is Magnitude -1.5, the Sun -26.8, the full moon -12.6, Venus -4.4 ( at its brightest), Jupiter -2 (its 4 largest moons Magnitude 5), Saturn +1, Uranus 6, Neptune 8, Pluto 14 (at its brightest). Exit Pupil refers to the diameter of the cone of light coming out of the eyepiece. The exit pupil is where the light leaving the eyepiece converges to its smallest circle -- you find the exit pupil when you bring your eye up to the eyepiece until you can see the whole image. The upper useful bound is 7mm, since that is the same diameter of the human eye pupil when it is adjusted to seeing best in the dark (pupil fully open). This may vary from person to person, depening on age, and will tend to shrink with age (but might not shrink as much for some individuals). Any exit pupil greater than 7mm will result in wasted light since the cone will be bigger than the eye can take in. Smaller exit pupils correlate to higher magnifications, but at the expense of lost brightness. The recommended eyepieces are centered on a few selected exit pupil sizes ranging from (2/3)mm to 7mm. Surface Brightness is expressed as a percentage of the maximum brightness. The larger the mirror the more light the telescope can gather, but the surface brightness percentage is a function of the geometry of the eyepiece in use. The reason the image loses brightness with magnification is really simple when you think about it. The telescope will gather an amount of light dependent upon the diameter of the mirror (and of course the amount of light coming in from the object). Viewed at size N, the object will appear to have a certain brightness, but viewed at size 2 x N, the gathered light (which remains the same as before) is now spread out to cover the larger area, and thus loses some of its brightness.

Thank you for visiting this page. If you find it of interest and/or you think you have a friend who might find it of some use or interest, let your friend know about it. Also, if you're interested in Astronomy consider a subscription to Astronomy Magazine, a complete science magazine that is out of this world.
--Mark Ganson

Send any questions/comments/flames to mwganson at hotmail dot com with spherometer in the subject line.



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