Thanks to the work of people like David Lewis, and Richard Schwartz - analysis of mirror support schemes has improved dramatically for the amateur telescope maker. It's great to get output from David's software (PLOP) that tells you that your large, thin mirror can be supported adequately with only 12 points (and you were worried that 27 points would not be enough). However...how do you go about making a mirror cell that uses 12 points? Sure, you need to end up with three collimation bolts, but what about all the mechanism in between...how do you support 12 points on three points? Below are some examples. Hopefully the pattern of analysis and construction shown here can be applied to just about any number of support points.
(For a great example of other possible support schemes, with 18 or more points, see:
http://www.cs.berkeley.edu/~jonah/18plus/
and for analysis of PLOP and good/bad ways to use the software, see:
http://www.cs.berkeley.edu/~jonah/18plus/p18.html)
Before we jump into a 12 point support, let's look at a four
point support as a warm-up and review. Note the colors used for
various components and functions. The largest black circle is the
mirror. The four smaller black circles are the points at which
the cell will support the mirror. (In many amateur applications
the mirror will either rest on teflon pads or be glued to the
mirror cell with RTV silicone. If you do not glue the mirror to
the cell, you will need a sling to support the mirror from the
bottom.) The green rectangle is a beam (typically of steel or
aluminum). The red X's are the three points that will be
supported and adjusted by collimation bolts. (In some cases you
can use two collimation bolts, as long as the mirror cell is
supported at all three points. You can still adjust mirror
tilt/collimation this way, but you can't move the mirror nearer
to or farther from other optical elements.)
NOTE: In all diagrams the location of support points is notional...they have not been optimized with PLOP for the best possible location to minimize mirror distortion. Also, these designs have all points bearing an equal load of the mirror.
Here is an initial starting point for a 12 point support
scheme - 3 inner supports in one ring, and 9 supports in an outer
ring. All we see are the 12 locations at which the cell will
support the mirror. Now we must make a decision on how to 'divide
and conquer' the 12 support points onto the final three
collimation bolts. One approach would be to use three four-point
supports.... 
...and shown here are three (green) beams that will help
define the system of three four-point supports. 
Adding red X's (some are in the middle of the green beams)
show the three support points that will be needed...and three
support points imply.... 
...(black) triangles to support the three points. We have
three triangles to support, so...
...the blue symbols are the (approximate in this case)
locations for the three collimation bolts. NOTE. The location of
the collimation bolts must be at the centroid (the balance point)
of the forces that the triangles must support. Note that in this
case the triangles do *not* have equal force applied to all three
corners - one corner supports a beam, and the beam supports *two*
mirror points, so the ratio of three forces acting on the
triangle is *not* 1 : 1 : 1. Rather, the ratio of forces on the
triangle is 2 : 1 : 1. If you do not properly calculate the
centroid (balance point), your mirror cell will not provide equal
support to all 12 points, and you will distort your mirror more
than you had designed through PLOP software. (As a comparison,
the green beams do support equal forces on both ends...so the
centroid is simply the middle location between the two supported
points.) 
Are there other ways to skin this 12-point cat? Yes. Here is a second example. Instead of using three four-point supports, let's try four three-point supports.
Once again we start with the initial 12 point support scheme -
3 inner supports in one ring, and 9 supports in an outer ring.
Now we will define four three-point supports with....
...triangles. Each triangle will be supported at its
respective centroid. In this case, all three corners of all
triangles support equal loads, so the centroid is easier to
calculate...
...the centroids are shown with blue symbols. However, we have
four to support, so we need to... 
...use a green beam to support two of these four support
points. Again, the beam is supporting equal loads at each end, so
the centroid of the beam's loads is the middle point between the
two points it supports....
...which leaves you with three red X's for the location of the
collimation bolts. 
For easy comparison, here are the two final results next to each other. One uses three triangles and three beams. The other uses four triangles and one beam:
However, Nils
Olof Carlin asked me 'why use triangles?' Here
is the approach he points out, and it uses only beams. We start
with 12 points to support on the mirror...
...and then put six beams across six pairs of points... 
...which mean we have six centroids (balance points, shown as
red X's)... 
...that require three more beams... 
...which require three collimation bolts (blue symbols) at the
centroids of these three beams. One advantage of this approach is
that the centroids are easy to calcuate for beams...especially
for beams that bear equal loads at both ends. 
Nils also pointed out this this approach which I showed above
has very narrow triangles as the outermost supports. The problem
is that if you are imprecise in locating the pivot when you
fabricate these skinny triangles...you may not be pivoting about
the true centroid...and you will have unequal support forces
acting on the mirror...distorting it beyond what PLOP would
predict. Note. Nils has pointed out an important aspect of cell
design in this example - sensitivity analysis. These skinny
triangles require more precise fabrication for proper balancing
behavior. It is better to use triangles that are closer to
equilateral in shape, avoiding this problem with skinny
triangles.
Richard
Schwartz recommends I use 12 points in a slightly different
arrangement - an inner ring of four, and an outer ring of 8
points. Steve Vegos also points out that PLOP indicates that this
support arrangement may be somewhat better than an inner ring of
3 and an outer ring of 9 points. Here is the starting arrangement
of 12 points in rings of 4 and 8. 
We can 'divide and conquer' with four triangles. 
...which have four support points at their centroids (red
X's)...
...but four supports need a beam across one of them...
...which now set us up for three collimation bolts (blue
symbols)... 
In anticipation of comments from Nils...yes, you could support
this mirror using only beams. Here is a finished example....
NOTE: Some of these examples do not exhibit symmetry with respect to location of the three collimation bolts. This is not inherently bad, but you must be careful when you design and build your frame that carries the collimation bolts. If you do not pay attention to this, you may be surprised to see that your mirror sits in the cell/frame without being properly centered! ;-)
This does not come close to exhausting the design and support possibilities, but I hope this is enough to get some ideas across.
(Again, for a great example of other possible support schemes, with 18 or more points, see:
http://www.cs.berkeley.edu/~jonah/18plus/
and for analysis of PLOP and good/bad ways to use the software, see:
http://www.cs.berkeley.edu/~jonah/18plus/p18.html)
All feedback is encouraged!
email: t-k-r-a-j-c-i-@-s-a-n-.-o-s-d-.-m-i-l (remove the dashes)
Last update: 14 Dec 2002
