The spherometer

The spherometer is a measuring instrument used in the machining of an astronomical mirror, to verify the depth of the sagitta, and then the radius of curvature of the surface under examination.
The instrument itself is not complicated, It's consists of a rigid supporting structure on which are placed 3 support feet arranged so as to be located at the vertices of an equilateral triangle, and in the center is positioned a screw device or a dial indicator to measure the depth of the excavation.


Knowing the radial distance of the legs from the Central rod and depth measured, You can, by means of appropriate mathematical formulas, go up to the value of the radius of curvature of the surface.

Mobile measuring rod, If screw type system, is generally fromed by a screw whit lead equal to 1 mm integral with a graduated disk with 100 divisions, which then allows a reading of the measure of the order of hundredths of a millimeter.
The three legs instead, can be formed by sharp rods, or three spheres of known diameter. The main difference between the two solutions are in the different formulas that you must use to calculate the final result.

In particular the formulas to be used are:

formule 2

R = radius of curvature of the surface (2 times the focal length)
h = measured value of the sagitta
r = radial distance from the central srew-legs
d = diameter of spheres used

± Present in the formula for the calculation of the radius of curvature for the spherometer with spheres should be understood as a sign + if it is analyzing a concave surface, While as a sign – if it is analyzing a convex surface.

Another variation of the instrument is shown below, where in place of the three supports is a cylindrical tubular section. In this variant, if you analyze a concave surface to rest on it will be the outer circular ring, while if one analyzes a convex surface to rest on it will be the inner ring. It will be important to use the correct value of the RADIUS (r) in the formula, Depending on the case.



  • The first operation to perform is to verify the zeroing of the instrument going to making a measurement on a flat surface.


  • Proceed with the measuring surface to be analyzed.
    If you have a spherometer dial indicator measurement will take place immediately and and without fatigue, while if you have one of srew spherometer, you should turn the screw to get it down until its tip comes in contact with the surface.
    Techniques to understand when contact occurred are mainly 2:
    In the first, Once you get in contact and you continue to turn the screw, the entire spherometer will tend to turn on itself, then return back slightly until it reaches the ideal location (sensitivity 3-5 hundredths of mm); in the second, if also just slightly exceeds,, the value of the sagitta, one of the legs will detach from the surface of the glass. If pressure is applied alternately over the feet you should hear a clicking sound (as with a wobbly table), then go back up to the disappearance of the effect (sensitivity 1-2 hundredths of mm).


  • Now read the measurement and using the correct formula to calculate the radius of curvature of the surface (or the focal length by dividing by 2 the result).
    The following example sets the value of the arrow is just over 0.81 mm (being the concave surface of the dipstick is decreased with respect to the zero position, then the pointer is moved in a counterclockwise direction).
    By applying the above formula, the result of the radius of curvature (R) amounted to 920.6 mm (having the spherometer concerned r = 39 mm and d = 7.95 mm).


For the construction of this spherometer I used as a base of support in a fairly sturdy plastic gear and of appropriate diameter recovered from an old copier..

I have used this support because as you can see from the image below has a circular area on the outside of delimited by two edges that allow me to position accurately (at least radial level) the three spheres of steel support. it is 12 radial ribs that allow me to place the spheres at 120° to each other.


To lock in position the comparator instead, it's drill a hole of adequate size to allow passage of the outer Rod cursor that has been clamped into place by a transversal screw.


And this is the spherometer full placed on a flat surface for zeroing:


2 comments on "The spherometer”

  1. deabis deabis

    In the first stage of lavodazione of a parabolic mirror, When you are working with a coarse grain for scooping the mirror and reach the desired arrow, the ball is not perfect. At this stage I think it's appropriate to use a spherometer that rests on the edge of the mirror or as close as possible to it, in order to clear any measurement error due to a ball still imperfect. Obviously this problem does not exist or is minimal in the later stages of processing, When, i.e., you are working with fine grits and the ball is now formed.
    On the other hand it is also true that a spherometer with three support points (or stop plate) with small diameter, or less than the diameter of the mirror, allows us to make a preliminary check of the regularity of the sphere: It will simply make several measurements at different points of the mirror and make sure that the size returned by the tool is consistent. In contrast, the ball will still be inaccurate.
    I conclude that, having the opportunity to establish itself exclusively a spherometer, It would be interesting to produce at least two, one with sufficiently broad to be positioned on the edge of the mirror and one with multiple supports “tight”: in this way I will have two useful tools in two different phases, that will enable us also to compare, and then check the measurements in the advanced stages of processing.

  2. Giulio TiberinI

    Keep in mind that the spherometer is useful only when you have some assurance that the curved surface to be measured is objectively at least one uniform spheroid, otherwise there is no guarantee that that measured is actually a spherical surface.

    In fact, the detection of radius of curvature from the unknown sphere that connects the Central measuring point of the spherometer, to the diameter of the plane identified by its three peripheral feets, is accurate only if the surface between the measurement points is spherical.

    From the perspective of "do it yourself" the amateur astronomer who creates his own mirror, the spherometer is not necessary, because the only guarantee that the curve produced by manual labor is a good spheroid, is given by the disappearance of the air bubble lingering after blocking because trapped between the mirror and the tool.
    In fact until the air bubble is present, It is a sign that the two surfaces mirror and tool does not have the same radius of curvature.

    According to the recommendations of the same Foucault (who was the advocate of the feasibility by hand of glass mirrors , in a time when it seemed possible to realize them only in metal), to get surely to a good sphere, It must be put into practice only 5 following rules:
    – Use a full tool diameter;
    – Having the foresight to apply of amplitude in the way forward-backward 1/3 Diameter, "center over center" (then with overflow forwards and backwards to 1/6 mirror diameter);
    – While at the same time to remain centered with horseracing in left-right direction, i.e. without overflow side (Because the lateral overflow would lead to flare too early the sphere toward the parable, When it still isn't perfectly polished; Or to produce a surface flared much over the parable).
    – Finally it is necessary to work turning small steps around the table,
    – by turning a little at a time into the opposite direction, the glass in your hand.
    And it's all.

    Given the large amount of strokes that unitaryly remove very little glass, is the statistics of large numbers that applying similar behavior, produces good spherical surfaces reaching huge machining precision even Freehand, as small errors of opposite sign ELIDE each other.

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