The Joy of Mirror Making

Mel Bartels

Rough Grinding

The first milestone is putting a curve into the mirror face.  The curve's depth dictates the mirror's focal length.  The curve should be spherical.

Creating the curve can be done by several methods:
The first method is by far the most popular.  Grinding the curve into the mirror face can be done by one of the following approaches:
The first approach is almost universally used.  I recommend the ring tool method as it rough grinds in half the time as the tiled tool.  The other approaches have drawbacks.  The diamond wheel in particular can be deadly because the airborne glass dust causes siliconsis.  Also there's the difficulty of creating a spherical curve in the mirror face.  The grinding wheel on a boom approach is difficult because it is hard to anchor the mirror and the boom's pivot so that they do not move in relation to each other.  Glass tools can stick and cause scratches in larger sizes.  More importantly, the glass tool can be saved and used later as a mirror blank for a second project.

Unglazed ceramic tiles, 1 inch [2.5cm] square can be attached to a several substrates:
I've successfully used plywood tools up to 8 inches.  I glue plywood layers together for a total thickness of 1 inch [2.5cm], then embed the tiles in polyester resin that also coats the rim and backside.  Plywood tools are very lightweight.  By contrast, stone and concrete tools are very heavy.  A granite stone tool can be used without tiles, if channels are cut into the face.  Concrete tools take weeks to cure and are hard to handle gently because of their weight, cracking when dropped.

Ring Tools

A metal ring of half the mirror diameter is ground against the glass, stroking across the center and overhanging a bit at each end. Half sized ring tools are the best compromise between grinding in a curve quickly and overall grinding action.  Take a half dozen strokes, then move a step in one direction. Repeat, repeat, repeat.  This rapidly tears into the glass, creating a curve quickly.  I generated a 16 inch [41cm] f/6 plate glass mirror in about 4 hours with 80 silicon carbide grit.  When the sound dies down, recharge with another sprinkle of grit and a spritzing of water.  Many use too much grit and not enough water.  As the mud or discarded grit and glass builds up, wash it or wipe it away.  It is best to clean the mirror by dunking it in a bucket.  Don't pour the mud down the drain as it can cake up.

Extreme pressure can be applied to the ring tool.  Because the ring tool is metal, it changes shape very little while grinding on the glass.  A tiled plaster tool grinds both tiles and plaster, with both starting flat, meaning that the tile must be ground convex and the glass ground concave.  Ring tools reach desired depth  about twice as fast as tiled ceramic tools.  Ring tools for smaller mirrors should be no more than half the mirror diameter size; otherwise grinding time is increased.  Don't forget to apply pressure with smaller ring tools; they can be hard to grip.  

Stop just before desired depth: about 5/6 is good.  The center will deepen during the first stage of fine grinding as contact is achieved across the mirror face and fine grinding tool.  On larger mirrors, the ring tool may leave the zones part way to the edge a little underground.  For these mirrors, as you near the depth, you can alter the strokes to be off-center.  This will prevent the mirror zones partway to the edge not being ground deep enough.

I use a discarded pulley.  Ring tools can be pipe floor flanges - anything round metal object that will touch in a ring on its perimeter.

16 inch [41cm], 10 inch [25cm], 6 inch [15cm], 16 inch [41cm] mirrors being rough ground with ring tools.  Images by Jerry Oltion


Geometrically, two surfaces when rubbed against each other, must create a concave sphere on one surface, and a convex sphere on the other surface. Flat surfaces are special conditions where the curve is infinite.  Any high spots are ground off, and any low spots are not ground.  A ring also generates a concave sphere because a circle touching a two dimensional curve acts in both dimensions, wearing down the high spots and avoiding the low spots.  

Tiled Plaster Tools

Tiles allow grit and water to flow across the mirror blank.  This channeling effect helps prevent sticking and scratching.

Plaster thicknesses of 1 to 1.5 inches [2.5 to 3.8cm] are a good compromise between rigidity and heaviness.  The plaster is poured onto the mirror face that has a paper dam taped around the edge.  Smearing the glass with a release agent like butter or grease or oil helps, but is not necessary in smaller sizes.  After the plaster sets in a few minutes, twist off the tool from the glass.  Let the plaster finish drying overnight.

Leaving the tiles attached to their ribbing, I place dobs of JBWeld on the tile faces, placing the tile matt over the plaster face.  The ribbing will quickly grind off.  If necessary for larger deeper work where the tiles might grind through before finishing, the tiles can be glued tightly together and placed on their edge on the plaster tool.

Grindwith the tool on top in a to and fro pattern, overhanging each end by about 1/6 the mirror diameter.  After a half dozen strokes, take a step to one side and repeat. Then spin the tool on top in the opposite direction.  Repeat ad naseum.  When the sound dies down, recharge with another sprinkle of grit and a spritzing of water.  Many use too much grit and not enough water.  As the mud or discarded grit and glass builds up, wash it or wipe it away.  It is best to clean the mirror by dunking it in a bucket.  Don't pour the mud down the drain as it can cake up.

20 inch [51cm] tile and plaster tool, used to  rough grind a 30 inch [76cm] Pyrex mirror


8 inch [20cm] tiled tool. The tile's backing is yet to be ground off.  Image by Jerry Oltion


How Deep?  The Mirror's Sagitta

The depth in the center of the mirror is the mirror's sagitta.  The sagitta determines the mirror's focal length and focal ratio.  There is plenty of time to contemplate the desired focal length while rough grinding!  Here's a table of depths across a variety of mirror diameters and mirror focal ratios:

Focal Ratio
Mirror Diameter 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10
4 0.100 0.083 0.072 0.063 0.056 0.050 0.045 0.042 0.038 0.036 0.033 0.031 0.029 0.028 0.026 0.025
6 0.150 0.125 0.107 0.094 0.083 0.075 0.068 0.063 0.058 0.054 0.050 0.047 0.044 0.042 0.039 0.038
8 0.201 0.167 0.143 0.125 0.111 0.100 0.091 0.083 0.077 0.071 0.067 0.063 0.059 0.056 0.053 0.050
10 0.251 0.209 0.179 0.156 0.139 0.125 0.114 0.104 0.096 0.089 0.083 0.078 0.074 0.069 0.066 0.063
12 0.301 0.250 0.215 0.188 0.167 0.150 0.136 0.125 0.115 0.107 0.100 0.094 0.088 0.083 0.079 0.075
14 0.351 0.292 0.250 0.219 0.195 0.175 0.159 0.146 0.135 0.125 0.117 0.109 0.103 0.097 0.092 0.088
16 0.401 0.334 0.286 0.250 0.222 0.200 0.182 0.167 0.154 0.143 0.133 0.125 0.118 0.111 0.105 0.100
18 0.451 0.376 0.322 0.282 0.250 0.225 0.205 0.188 0.173 0.161 0.150 0.141 0.132 0.125 0.118 0.113
20 0.501 0.417 0.358 0.313 0.278 0.250 0.227 0.208 0.192 0.179 0.167 0.156 0.147 0.139 0.132 0.125
22 0.551 0.459 0.393 0.344 0.306 0.275 0.250 0.229 0.212 0.196 0.183 0.172 0.162 0.153 0.145 0.138
24 0.602 0.501 0.429 0.375 0.334 0.300 0.273 0.250 0.231 0.214 0.200 0.188 0.177 0.167 0.158 0.150
26 0.652 0.543 0.465 0.407 0.361 0.325 0.296 0.271 0.250 0.232 0.217 0.203 0.191 0.181 0.171 0.163
28 0.702 0.584 0.501 0.438 0.389 0.350 0.318 0.292 0.269 0.250 0.233 0.219 0.206 0.194 0.184 0.175
30 0.752 0.626 0.536 0.469 0.417 0.375 0.341 0.313 0.289 0.268 0.250 0.234 0.221 0.208 0.197 0.188
32 0.802 0.668 0.572 0.500 0.445 0.400 0.364 0.333 0.308 0.286 0.267 0.250 0.235 0.222 0.211 0.200
34 0.852 0.710 0.608 0.532 0.473 0.425 0.387 0.354 0.327 0.304 0.283 0.266 0.250 0.236 0.224 0.213
36 0.902 0.751 0.644 0.563 0.500 0.450 0.409 0.375 0.346 0.322 0.300 0.281 0.265 0.250 0.237 0.225
38 0.952 0.793 0.679 0.594 0.528 0.475 0.432 0.396 0.366 0.339 0.317 0.297 0.279 0.264 0.250 0.238
40 1.003 0.835 0.715 0.626 0.556 0.500 0.455 0.417 0.385 0.357 0.333 0.313 0.294 0.278 0.263 0.250
42 1.053 0.877 0.751 0.657 0.584 0.525 0.478 0.438 0.404 0.375 0.350 0.328 0.309 0.292 0.276 0.263
44 1.103 0.918 0.787 0.688 0.612 0.550 0.500 0.459 0.423 0.393 0.367 0.344 0.324 0.306 0.290 0.275
46 1.153 0.960 0.822 0.719 0.639 0.575 0.523 0.479 0.442 0.411 0.383 0.359 0.338 0.320 0.303 0.288
48 1.203 1.002 0.858 0.751 0.667 0.600 0.546 0.500 0.462 0.429 0.400 0.375 0.353 0.333 0.316 0.300
50 1.253 1.043 0.894 0.782 0.695 0.625 0.568 0.521 0.481 0.447 0.417 0.391 0.368 0.347 0.329 0.313
52 1.303 1.085 0.930 0.813 0.723 0.650 0.591 0.542 0.500 0.464 0.433 0.406 0.382 0.361 0.342 0.325
54 1.353 1.127 0.966 0.845 0.751 0.675 0.614 0.563 0.519 0.482 0.450 0.422 0.397 0.375 0.355 0.338
56 1.404 1.169 1.001 0.876 0.778 0.700 0.637 0.584 0.539 0.500 0.467 0.438 0.412 0.389 0.368 0.350
58 1.454 1.210 1.037 0.907 0.806 0.725 0.659 0.604 0.558 0.518 0.483 0.453 0.427 0.403 0.382 0.363
60 1.504 1.252 1.073 0.938 0.834 0.750 0.682 0.625 0.577 0.536 0.500 0.469 0.441 0.417 0.395 0.375

The precise formula for calculating sagitta is:
sagitta = RadiusCurvature - square root (RadiusCurvature squared - MirrorRadius squared).  See http://liutaiomottola.com/formulae/sag.htm

There's a simpler formula floating around that easy to calculate using pencil and paper.  It gives somewhat incorrect numbers for fast low focal ratio large mirrors.  There is no reason in today's computer aged not to use the precise formula.  Here are two calculators using the precise formula that will calculate the sagitta or the focal ratio:

Sagitta Calculator
Mirror diameter
Focal ratio
Resulting Sagitta

Focal Ratio Calculator
Mirror diameter
Sagitta
Resulting focal ratio


The Oltion Relationship

Jerry Oltion points out that if the sagitta is 1/16 of the unit of measure, the focal ratio is the mirror diameter.   For instance:

Measuring the Sagitta

I use a level with a micrometer whose caliper I cut off.  Drill bits can also be inserted under a straight edge that stretches from mirror side to mirror side.  Make sure you measure at the center of the mirror.  In many cases, less precision is called for.  Inserting American coins work well.  Here is a table of coins and their thicknesses in inches:

coin rounded precise
thickness thickness
penny 0.06 0.061024
nickel 0.08 0.076772
dime 0.05 0.05315
quarter 0.07 0.068898
half-dollar 0.08 0.084646
dollar 0.08 0.07874

Radius of Curvature and Focal Length, Spheroidal and Paraboloidal

The radius of curvature is the distance from the mirror to a light source where the focus is directed back onto the light source.  This is the distance that tests such as Foucault, Caustic and Ronchi are positioned from the mirror.  The mirror must be spheroidal in shape for it to return all the light directed to its surface back exactly where it originated.

The focal length is the distance from the mirror to the focus when light is received from distant astronomical objects.  This is the distance that eyepieces and imagers are placed from the mirror when focused on the planets, stars, and galaxies.  The mirror must be paraboloidal in shape for it to reflect all the light hitting its surface from a distant star to a single focus point.

Why the radius of curvature is twice the focal length

The radius of curvature is twice the focal length.  Why this is is a very interesting question, having to do with the fundamental geometric properties of our universe and of a special characteristic called symmetry.  Symmetry is the ability to remain invariant under certain transformations.  For instance, rotating a cube 90 or 180 or 270 or 360 degrees is indistinguishable from the original cube.  We say the cube has a rotational symmetry of 4.  Rotating a cube some other angle such as 30 degrees does not look like the original cube, hence the cube is not symmetric under a 30 degree rotation.  A sphere by contrast is infinitely rotationally symmetrical.  The telescope mirror can reflect light to a single focal point thanks to the directions in space being symmetrically invariant.  Symmetry also has additional meanings relating to the balance or resolution of forces and can be applied aesthetically (the painting's foreground and background display a perfect symmetry).  For more on symmetry, start with http://en.wikipedia.org/wiki/Rotational_symmetry.

The paraboloid is the special curved surface that reflects light coming from distant astronomical objects to a single focal point.  The sphere is the special curve that reflects light coming from a single point back onto itself.  The two are related because they represent the symmetry breaking of a cone in space.  Look at http://en.wikipedia.org/wiki/Conic_section.  You can see that the circle/ellipse, the parabola and the hyperbola are generated by how the cone is sliced.  The parabola is the special case when the intersecting plane is parallel to the cone's side. See http://en.wikipedia.org/wiki/Parabola.  You can see how the parallel light from a distance object is reflected to a single point.

The parabola is defined as the curve that results from the series of points where each point is equidistance from the focus and from a line or plane called the directrix.  Look what happens with the point that is exactly between the focus and the directrix.  The distance to the directrix is exactly twice that of the distance to the parabola.  The curve that reflects the light emanating from the parabola's focus back from the directrix is the sphere that is centered on the parabola's focus.  The sphere's radius is twice that of the parabola's focus.  This is one way to understand why the radius of curvature of the sphere is twice the focal length of the parabola.  See http://www.mathwarehouse.com/quadratic/parabola/interactive-parabola.php for an interactive graphing of the parabola.  Notice that no matter where you move the parabola or otherwise define it, the focal length of the parabola is half that to the directrix, or the radius of curvature.

In a deep way, the radius of curvature and focal length are related just like ice and steam are related.  Both represent symmetry breaking.  For water, it's the phase.  For a reflecting surface, it is the original of the light rays: from a point reflecting back through the point, or from a distant point reflecting back to a focus.  Back to the conic, if we rotate the plane that intersects the cone that generates the parabola about the parabola's focus, when the plane reaches horizontal, it will generate a circle whose radius is twice that of the parabola's focus.

Another way to understand that the radius of curvature is twice the focal length is to consider the rate of change of the change in slope of the sphere compared to the parabola.  See  http://www.atmpage.org/contrib/Brown//MirrGeom/MirrGeom.html#RadiusOfCurvature.  Note that the rate of change of the change in slope of the parabola is the second derivative, y" = 2 / focal length.  The rate of change of the change in slope for a sphere is 1 / focal length.  The difference is our 2x factor.  Stated another way, the parabolic curve at the closest point to focus changes at twice the sphere's changes, so is half the distance.

Finally, consider the inverse transform of the parabola: the cardioid.  Executing the cardioid results in a curve that is twice the distance of the rolling circle: another instance of the special 2:1 relationship between sphere and parabola. See http://en.wikipedia.org/wiki/Cardioid.

Why we need both sphere and parabola

When rubbing tool against glass, a spheroid is produced: concave for the mirror and convex for the tool.  But the shape that will focus light coming from distant astronomical objects to a single point is a paraboloid. Because the mirror's focal length is large in relation to the mirror's diameter, in other words, the focal ratio is large, the difference between sphere and parabola is very little: a few millionths of an inch in the mirror's surface for a typical amateur telescope.  We grind and polish to a sphere, then figure to a parabola.  We test at the sphere's focus (the distance being the radius of curvature), calculating and looking for the slight deformity caused by the parabola.  It is this slight difference between parabola and sphere that we can calculate exactly and test for to a fraction of the wavelength of light that allows us to figure paraboloidal mirrors that perfectly focus light from distant astronomical objects.

Three ways to parabolize

Finally, we can use the symmetry breaking of the cone, where a property called eccentricity is defined.  For ellipses (circles being a special case), e < 1, for a parabola, e = 1, and for hyperbolas, e > 1.  There are three groups of eccentricities.  Knowing that figuring consists of transitioning from a sphere to a parabola and since there are three eccentricities, we can deduce that there will be three ways to turn a sphere into a parabola.  Indeed, that's exactly the case.  This is a very important realization because we can select the easiest of the three parabolization approaches, depending on the mirror's exact shape.

The Bevel

Beveling a mirror is critical to avoid small chips and flaking at the mirror's edge.  I like to maintain a 1/10 inch [2.5mm] bevel.  A diamond belt or cloth will round the mirror edge in minutes; a wetstone will take longer.  Stroke down and across to avoid lifting flakes off the mirror's face.  Use water if necessary.  Renew the bevel periodically if it becomes too small.

When the curve reaches the desired depth, the mirror will be ready for fine grinding, which will repair the damage done to the mirror's face.

(end of rough grinding)