There are many ways to curve generate a telescope mirror. Each method is performed to make the mirror concave and the tool convex. You are finished when the mirror is the correct depth and the mirror and tool move over each other in spherical contact. Also, when you are in spherical contact the bubbles and material will move uniformly between the two without spots where they seem to not move on either the tool or the mirror. The correct depth will be when where s is the sagitta, or depth of the curve, r is the radius of the mirror, and R is the radius of curvature of the mirror. R is twice the focal length. To measure the sagitta, place a straightedge across the mirror passing over the center and measure the gap between the center of the mirror and the straightedge. You can also use a spherometer.
For example, for a 12” mirror with a 60” focal length, R = 120”, r = 6”, so , or 0.150”. While you are working you can compare your glass with a template. To make a template, pivot a pencil using a tape measure at the correct radius of curvature to make a curve that will match your final mirror result. Grind until the mirror matches the template.
Here are a few methods that we at the SAS ATM shop have used.
1] Full-Sized Tool on Top Method. Grit and water are applied to the mirror, which is on the barrel face up. The tool is placed over the mirror and pressure is applied over edge of tool as it passes over center of mirror. Use 60 grit if it is a large mirror, 80 grit if it is a smaller mirror. Walk around the barrel as you do this. As you walk around the barrel, occasionally rotate the tool in the same direction as you are walking around the barrel and the mirror in the opposite direction. With time the mirror will become concave and the tool will become convex.
2] Sub-diameter Tool on Top Method. Same grit as in method 1, only this time your tool is smaller than your mirror. We do that sometimes. The tool is on top, and as you pass over the center of the mirror with the near-edge of the tool, your weight is on the edge of the tool. This will wear away at the edge of the tool and the center of the mirror, making the mirror eventually concave and the tool convex.
3] Mirror on Top Method. This should be done with a full-sized tool. The mirror is passed over the tool so that the center of the mirror passes over the edge of the tool. Weight is over the middle of the mirror, which will eventually wear away at the center of the mirror and edge of the tool, making the mirror concave and the tool convex.
4] Horizontal Pendulum Arm Method. This is the greatest way we have used so far for removing large amounts of glass from large, fast telescope mirrors. Mike Lavieri supplied us with a 1 HP motor, reducing pulleys, pillow blocks and shaft, as well as the diamond grinding wheel. At the end of a 2X6 arm the motor turns the diamond grinding wheel at 3500 RPM. With a water feed, the system removes glass quickly and cooly, and “quite safely”. The 2X6 is attached to another 2X6 with alignment holes which enable us to adjust the system to any radius of curvature we desire. Dave Wilson supplied us with the stainless steel shaft mounted with pillow block to the wall as a pivot. The mirror is held solidly with its face pointing horizontally facing the grinding wheel. The grinding wheel pivots across and back revolving about the pivot in a “perfect” circle grinding a depth of no more than 1/20 inch into the glass. Then the mirror is rotated slightly and the grinding wheel is passed again. Once the mirror has had a complete rotation accomplished, a small ground circle exists on it and it is moved forward 1/20 inch and it is repeated. Each time we finish a series, the mirrors spherical cut is larger, and this is continued until the mirror has been fully ground spherical edge to edge. It is smooth, clean, accurate, and takes zillions of years out of your hogging out time. Jeff’s 24” f/3.6 was hogged out in a day and was extremely spherical when finished. Here is a shot of that assembly.
Update: I now have a mill and I connected the shaft to the mill stand. Below is a more current photo.
Here are three videos of us working the horizontal pendulum.
7] Metal Ring Method. A iron ring, such as a pipe end cap, can be used to curve generate mirrors. The ring ought to be roughly 1/3 to ½ the diameter of the mirror, and grinding center over center strokes with it using large grits, such as 60 grit, will curve generate a mirror. You will have to cast a tool for it when finished. Below is a photo of using a barbell weight on our mirror making machine to curve a 32” f/3.67 mirror to an f/3.1. The weight is 1/3 the diameter of the mirror and the center over center strokes are performed 5.5 times per mirror rotation. We also move the center over center condition over to 16% off-center and 25% off-center now and then to keep the middle from deepening faster than the rest of the mirror. Click here to see a movie.
When you use your tool to curve generate, then you already will have a matching tool to continue to use. If you use one of the methods that doesn’t use a tool, then you will have to make a matching tool to continue working your mirror. See the section on making a tool.