Formula for sat dish offset

[Andy Burns]

Although I'd really have liked a lot more feedback than the single
example I did get (thanks for the good work, John), the single
underlying assumption is so reasonable and the proof is so simple that
I felt confident enough to release this live.

I couldn't see that it was any easier than your old formula, in fact it
seems to involve measurement to a vague point, rather than the definite
width and height ...

[Java Jive]

Yes, the actual previous measurements and calculation were as simple
as the new calculation, but that calculation has the drawback of an
underlying assumption, which may not always be true - the assumption
is the so-called 'boresight' assumption, that, seen from the satellite
and/or the LNB, the dish appears to be circular.

Certainly, with a conventional taller than wide offset dish, this
assumption makes sense, because that is the way to use the dish to
maximum efficiency, but nevertheless the manufacturer cannot be
absolutely relied upon to have made it so, while with minidishes the
resulting formula cannot be used at all.

The ONLY assumption underlying the new system of calculation is that
the dish is parabolic - surely a cast-iron assumption - AND, best
of all, the formula will work for minidishes too.

I agree that the precise position of the focal point a little behind
the front face of the LNB is debatable, but I still think that even
after errors in measurement to this point, the result is likely to be
more accurate than the boresight calculation.

I'm also hoping to do further research to determine exactly where the
focal point of a 'standard' LNB is. Ultimately, whatever their
manufacturing differences, they all, when mounted at a known pont of
reference in the LNB holder, must all focus the beam, or they simply
wouldn't work, so, acknowledging a possible difference between
conventional LNBs and minidish LNBs, those in each group must each
share a common focal point to an acceptable level of accuracy.

Therefore, if we can work out how to calculate where the focal point
is for each of the two types, it should be possible to adapt the
method of calculation so the user can just measure to a definitely
known point such as the front face of the LNB or, perhaps more likely,
the middle of the holder, and the calculator page will correct the
measurements automatically.

On Wed, 05 Oct 2011 21:02:03 +0100, Andy Burns

Post by Andy Burns
I couldn't see that it was any easier than your old formula, in fact it
seems to involve measurement to a vague point, rather than the definite
width and height ...

[John Legon]

Post by Java Jive
The ONLY assumption underlying the new system of calculation is that
the dish is parabolic - surely a cast-iron assumption - AND, best
of all, the formula will work for minidishes too.

Actually, you are also assuming that the design and manufacture of the
LNB arm and bracket are such that the LNB is located in the correct
position. With some cheap dishes, I think this is quite unlikely, and
in fact I bent the arm on one of my dishes to put the LNB where my
analysis showed that it ought to be...

Post by Java Jive
I agree that the precise position of the focal point a little behind
the front face of the LNB is debatable, but I still think that even
after errors in measurement to this point, the result is likely to be
more accurate than the boresight calculation.
I'm also hoping to do further research to determine exactly where the
focal point of a 'standard' LNB is. Ultimately, whatever their
manufacturing differences, they all, when mounted at a known pont of
reference in the LNB holder, must all focus the beam, or they simply
wouldn't work, so, acknowledging a possible difference between
conventional LNBs and minidish LNBs, those in each group must each
share a common focal point to an acceptable level of accuracy.

But in fact, the LNB can be moved some distance away from the true focal
point and it will still work, if the dish is aligned accordingly. After
all, multiple LNBs can be mounted on a dish for different satellites.
One of my dishes has three LNBs, but only one could possibly give the
correct result for the offset angle using your formula.

Post by Java Jive
Therefore, if we can work out how to calculate where the focal point
is for each of the two types, it should be possible to adapt the
method of calculation so the user can just measure to a definitely
known point such as the front face of the LNB or, perhaps more likely,
the middle of the holder, and the calculator page will correct the
measurements automatically.

A couple of year ago, I analysed curvature of a dish by taking offsets
to the surface from a straight edge laid across from top to bottom, and
worked out that the best fit to a parabolic curve was obtained when the
axis of rotation of the paraboloid was located on the lower rim.

Consequently, it turns out that the offset angle theta can be calculated
using the formula

sin(theta) = 4p/d(cos(theta))

where d is the length of the chord across the dish from top to bottom,
and p is the maximum perpendicular distance from the chord to the dish.

This can be solved by assuming an initial value for theta in the RHS,
say zero, to obtain a more accurate value in the LHS, and then repeating
the calculation with the new value....

The exact solution can be obtained by developing a quadratic equation in
sin^2(theta) and finding the roots in the usual manner, but as the maths
is a bit messy I'll leave that as an exercise for the reader.

Once the offset angle has been found, the "correct" position for the LNB
can be calculated.

[J G Miller]

Post by John Legon
Actually, you are also assuming that the design and manufacture of the
LNB arm and bracket are such that the LNB is located in the correct
position. With some cheap dishes, I think this is quite unlikely, and
in fact I bent the arm on one of my dishes to put the LNB where my
analysis showed that it ought to be...

This should be a primary criterion in buying a satellite dish.

If the LNB arm is cheap and flimsy, do not buy the dish and look for
brands and models which have a solidly built and attached LNB support arm.

[John Legon]

Post by J G Miller

Post by John Legon
Actually, you are also assuming that the design and manufacture of the
LNB arm and bracket are such that the LNB is located in the correct
position. With some cheap dishes, I think this is quite unlikely, and
in fact I bent the arm on one of my dishes to put the LNB where my
analysis showed that it ought to be...

This should be a primary criterion in buying a satellite dish.
If the LNB arm is cheap and flimsy, do not buy the dish and look for
brands and models which have a solidly built and attached LNB support arm.

The dish in question may have been cheap, but it certainly isn't flimsy!
In fact it took me as much strength as I could muster to bend the LNB
arm - by an inch or so - and I'm not a weakling!

The issue with this dish was one of manufacturing accuracy, although it
worked well enough as supplied for the major satellites. My interest
was partly theoretical, but I also wanted to get the best performance
possible with a view to bringing in some of the weaker signals.

I would also suggest that there is no guarantee that an expensive dish
will necessarily have accurate geometry - it's something worth checking
from both theoretical and practical viewpoints.

[Demonic]

Post by John Legon

Post by J G Miller

Post by John Legon
Actually, you are also assuming that the design and manufacture of the
LNB arm and bracket are such that the LNB is located in the correct
position. With some cheap dishes, I think this is quite unlikely, and
in fact I bent the arm on one of my dishes to put the LNB where my
analysis showed that it ought to be...

This should be a primary criterion in buying a satellite dish.
If the LNB arm is cheap and flimsy, do not buy the dish and look for
brands and models which have a solidly built and attached LNB support arm.

The dish in question may have been cheap, but it certainly isn't flimsy!
In fact it took me as much strength as I could muster to bend the LNB
arm - by an inch or so - and I'm not a weakling!
The issue with this dish was one of manufacturing accuracy, although it
worked well enough as supplied for the major satellites. My interest was
partly theoretical, but I also wanted to get the best performance
possible with a view to bringing in some of the weaker signals.
I would also suggest that there is no guarantee that an expensive dish
will necessarily have accurate geometry - it's something worth checking
from both theoretical and practical viewpoints.

The deviation from a true parabolic shape is quite small if the LNB is a
bit high or low or even off to the left or right - that is why it is
possible to use a standard dish with 3 or 4 LNBs to get, for example 13,
19.2E and 28.2E as many of us do. Yes, there is a slight loss of gain
for the LNBs that are farthest from the intended focus but not enough to
negate the technique.

[John Legon]

Post by Demonic

Post by John Legon
The issue with this dish was one of manufacturing accuracy, although it
worked well enough as supplied for the major satellites. My interest was
partly theoretical, but I also wanted to get the best performance
possible with a view to bringing in some of the weaker signals.
I would also suggest that there is no guarantee that an expensive dish
will necessarily have accurate geometry - it's something worth checking
from both theoretical and practical viewpoints.

The deviation from a true parabolic shape is quite small if the LNB is a
bit high or low or even off to the left or right - that is why it is
possible to use a standard dish with 3 or 4 LNBs to get, for example 13,
19.2E and 28.2E as many of us do. Yes, there is a slight loss of gain
for the LNBs that are farthest from the intended focus but not enough to
negate the technique.

As I mentioned earlier, I have one dish with three LNBs for just those
three satellites, so I appreciate that the technique works. But I also
have a dish on a motor, and found that tweaking the position of the LNB
on that dish gave a distinct improvement. I don't think it's just a
question of signal strength - having the LNB at the true focal point
will almost certainly improve the focusing power and resolution of the
dish, and hence give a increase in signal quality for satellites that
are close to others in the arc.

[Demonic]

Post by John Legon

Post by Demonic

Post by John Legon
The issue with this dish was one of manufacturing accuracy, although it
worked well enough as supplied for the major satellites. My interest was
partly theoretical, but I also wanted to get the best performance
possible with a view to bringing in some of the weaker signals.
I would also suggest that there is no guarantee that an expensive dish
will necessarily have accurate geometry - it's something worth checking
from both theoretical and practical viewpoints.

The deviation from a true parabolic shape is quite small if the LNB is
a bit high or low or even off to the left or right - that is why it is
possible to use a standard dish with 3 or 4 LNBs to get, for example
13, 19.2E and 28.2E as many of us do. Yes, there is a slight loss of
gain for the LNBs that are farthest from the intended focus but not
enough to negate the technique.

As I mentioned earlier, I have one dish with three LNBs for just those
three satellites, so I appreciate that the technique works. But I also
have a dish on a motor, and found that tweaking the position of the LNB
on that dish gave a distinct improvement. I don't think it's just a
question of signal strength - having the LNB at the true focal point
will almost certainly improve the focusing power and resolution of the
dish, and hence give a increase in signal quality for satellites that
are close to others in the arc.

Agreed. The acceptance angle of a dish improves as the dish size gets
larger. That is one of the problems with the mini-dish and why it is
wider than its height. AFAIR a 60cm dish has an acceptance angle of a
bit under 3 degrees which is about the spacing of N. Europe Sats around
the Clarke Belt.

Naturally getting the focus spot-on must improve things. The parabola
has two benefits - it accepts a signal from an infinite distance
(parallel beam) and it has the same path length from everywhere on the
source to the focus, thus eliminating destructive phasing.

I have an 80cm with 3 LNBs and a 120 cm for feed hunting. Great hobby
but almost too many channels to waste time watching them :-)

[PeterC]

Post by Demonic
I have an 80cm with 3 LNBs and a 120 cm for feed hunting. Great hobby
but almost too many channels to waste time watching them :-)

So many channels, so little time!
Loading Image...

[Java Jive]

On Thu, 6 Oct 2011 05:47:37 +0100, John Legon

Post by John Legon

Post by Java Jive
The ONLY assumption underlying the new system of calculation is that
the dish is parabolic - surely a cast-iron assumption - AND, best
of all, the formula will work for minidishes too.

Actually, you are also assuming that the design and manufacture of the
LNB arm and bracket are such that the LNB is located in the correct
position. With some cheap dishes, I think this is quite unlikely, and
in fact I bent the arm on one of my dishes to put the LNB where my
analysis showed that it ought to be...

In that case, depending on amount of the manufacturer's error:

:-) If only slight, it would be even more essential that we use
the most accurate formula possible, in order to keep the total error
to a minimum.

:-( If not, we're f***ed whatever the formula we use - we'd be
reduced to trial and error.

Post by John Legon
But in fact, the LNB can be moved some distance away from the true focal
point and it will still work, if the dish is aligned accordingly. After
all, multiple LNBs can be mounted on a dish for different satellites.
One of my dishes has three LNBs, but only one could possibly give the
correct result for the offset angle using your formula.

Yes, but that's simply because talk of focal point is a convenient but
inexact simplification, it would be more correct to talk in terms of
focal surface. With a multi-LNB set up, the LNBs must all lie in the
focal surface, and because the Clarke Belt effectively forms a line in
front of the dish, the LNBs will further be constrained to lie in one
horizontal 'line' of the focal surface.

Post by John Legon
A couple of year ago, I analysed curvature of a dish by taking offsets
to the surface from a straight edge laid across from top to bottom, and
worked out that the best fit to a parabolic curve was obtained when the
axis of rotation of the paraboloid was located on the lower rim.

That's good, I'd be interested to see your workings, if you still have
them, even if only a scan of handwritten notes.

Post by John Legon
Consequently, it turns out that the offset angle theta can be calculated
using the formula
sin(theta) = 4p/d(cos(theta))
where d is the length of the chord across the dish from top to bottom,
and p is the maximum perpendicular distance from the chord to the dish.
This can be solved by assuming an initial value for theta in the RHS,
say zero, to obtain a more accurate value in the LHS, and then repeating
the calculation with the new value....

Yes, I've seen a similar style of calculation somewhere, ISTR there
was an example somewhere in here:
http://www.qsl.net/n1bwt/chap1.pdf

Post by John Legon
The exact solution can be obtained by developing a quadratic equation in
sin^2(theta) and finding the roots in the usual manner, but as the maths
is a bit messy I'll leave that as an exercise for the reader.

Errm? Multiplying through by cos(theta) suggests use of the double
angle formula:

sind(theta)*cos(theta) = (1/2)*sin(2*theta) = 4p/d

Therefore sin(2*theta) = 8p/d
Therefore 2*theta = asin(8p/d)
Therefore theta = (1/2)*asin(8p/d)

So how does this compare with my formula which you tested, and your
own settings? Oh, and just for the record, is yours a conventional
taller than wide offset dish, in which case it would be also
interesting to know what the 'boresight' method gives, or a wider than
tall minidish?

[John Legon]

Post by Java Jive
On Thu, 6 Oct 2011 05:47:37 +0100, John Legon

Post by John Legon

Post by Java Jive
The ONLY assumption underlying the new system of calculation is that
the dish is parabolic - surely a cast-iron assumption - AND, best
of all, the formula will work for minidishes too.

Actually, you are also assuming that the design and manufacture of the
LNB arm and bracket are such that the LNB is located in the correct
position. With some cheap dishes, I think this is quite unlikely, and
in fact I bent the arm on one of my dishes to put the LNB where my
analysis showed that it ought to be...

:-) If only slight, it would be even more essential that we use
the most accurate formula possible, in order to keep the total error
to a minimum.
:-( If not, we're f***ed whatever the formula we use - we'd be
reduced to trial and error.

The fundamental starting point is the dish itself. If the curvature
isn't accurate then nothing can be done, but at least the position of
the LNB can be excluded from the calculation of the offset angle.

Post by Java Jive

Post by John Legon
But in fact, the LNB can be moved some distance away from the true focal
point and it will still work, if the dish is aligned accordingly. After
all, multiple LNBs can be mounted on a dish for different satellites.
One of my dishes has three LNBs, but only one could possibly give the
correct result for the offset angle using your formula.

Yes, but that's simply because talk of focal point is a convenient but
inexact simplification, it would be more correct to talk in terms of
focal surface. With a multi-LNB set up, the LNBs must all lie in the
focal surface, and because the Clarke Belt effectively forms a line in
front of the dish, the LNBs will further be constrained to lie in one
horizontal 'line' of the focal surface.

I don't follow. The belt is inclined upwards from (say) Astra 28E in
the east to (say) Hotbird at 13 E, and the reflection of the arc off the
dish is inclined downwards accordingly. The LNBs are not placed in a
horizontal line but rather above and below that line with (in my set up)
only the LNB for Astra 19 E on the (presumed) focal surface.

Post by Java Jive

Post by John Legon
A couple of year ago, I analysed curvature of a dish by taking offsets
to the surface from a straight edge laid across from top to bottom, and
worked out that the best fit to a parabolic curve was obtained when the
axis of rotation of the paraboloid was located on the lower rim.

That's good, I'd be interested to see your workings, if you still have
them, even if only a scan of handwritten notes.

It would be good if I could find them! I have, however, found this plot
based on my actual measurements, with a reconstruction of the focus:
Loading Image...

Post by Java Jive

Post by John Legon
Consequently, it turns out that the offset angle theta can be calculated
using the formula
sin(theta) = 4p/d(cos(theta))
where d is the length of the chord across the dish from top to bottom,
and p is the maximum perpendicular distance from the chord to the dish.
This can be solved by assuming an initial value for theta in the RHS,
say zero, to obtain a more accurate value in the LHS, and then repeating
the calculation with the new value....

Yes, I've seen a similar style of calculation somewhere, ISTR there
http://www.qsl.net/n1bwt/chap1.pdf

It's actually in the appendix at the end of:

http://www.qsl.net/n1bwt/chap5.pdf

Significantly, the method proposed there assumes that the offset angle
of the dish is known, and then proceeds to find the focal length and
location of the origin.

Post by Java Jive

Post by John Legon
The exact solution can be obtained by developing a quadratic equation in
sin^2(theta) and finding the roots in the usual manner, but as the maths
is a bit messy I'll leave that as an exercise for the reader.

Errm? Multiplying through by cos(theta) suggests use of the double
sind(theta)*cos(theta) = (1/2)*sin(2*theta) = 4p/d
Therefore sin(2*theta) = 8p/d
Therefore 2*theta = asin(8p/d)
Therefore theta = (1/2)*asin(8p/d)

Excellent! It really is that simple :-)

Now using this formula with the data supplied in the above pdf file,
namely for a dish with d = 500 and p = 43, we get an offset angle of
21.7 degrees and hence a complement of 68.3 degrees.

The pdf assumes 66.9 degrees initially, but then - realizing that the
axis of the parabola intersects the lower rim of the dish - obtains the
better result of 68.3 degrees by trial and error, in perfect agreement
with my calculation.

Post by Java Jive
So how does this compare with my formula which you tested, and your
own settings?

Measuring inside the raised edge on the rim of the dish I get d = 644 mm
with a maximum depth of 54 mm. Hence the offset angle will be

(1/2)*asin(8*54/644) = 21.1 degrees

As mentioned earlier in this thread, using your method gives an offset
angle of 20.7 degrees, so that's quite close. However, I only get this
agreement because I had already bent the LNB boom arm to place the LNB
where I thought it ought to be!

Post by Java Jive
Oh, and just for the record, is yours a conventional
taller than wide offset dish, in which case it would be also
interesting to know what the 'boresight' method gives, or a wider than
tall minidish?

The outer dimensions are 605 x 655 mm, giving 22.5 degrees.

[Java Jive]

On Fri, 7 Oct 2011 08:54:36 +0100, John Legon

Post by John Legon
The fundamental starting point is the dish itself. If the curvature
isn't accurate then nothing can be done, but at least the position of
the LNB can be excluded from the calculation of the offset angle.

Except, surely, the position of the LNB *affects* the offset angle, in
the sense that where the position of the LNB is slightly out of true
but is not corrected, which it won't be by most installers, the
elevation of the dish will have to be slightly different than it would
be if the LNB were in the correct position?

Post by John Legon
I don't follow. The belt is inclined upwards from (say) Astra 28E in
the east to (say) Hotbird at 13 E, and the reflection of the arc off the
dish is inclined downwards accordingly. The LNBs are not placed in a
horizontal line but rather above and below that line with (in my set up)
only the LNB for Astra 19 E on the (presumed) focal surface.

Ok, I worded that loosely, partly because for some reason or other I
was thinking you have a rotor like myself, thus making the Clarke belt
'horizontal' wrt to the face of the dish no matter at which bit of the
belt the dish is pointing. But what I meant is still true, the LNB's
will lie on a line lying in the focal surface of the dish.

Post by John Legon

Post by Java Jive
Therefore theta = (1/2)*asin(8p/d)

Excellent! It really is that simple :-)
Now using this formula with the data supplied in the above pdf file,
namely for a dish with d = 500 and p = 43, we get an offset angle of
21.7 degrees and hence a complement of 68.3 degrees.
The pdf assumes 66.9 degrees initially, but then - realizing that the
axis of the parabola intersects the lower rim of the dish - obtains the
better result of 68.3 degrees by trial and error, in perfect agreement
with my calculation.

Good. If possible, I'd like to see at least an outline of how to
prove this formula, to save me working it out for myself.

Post by John Legon

Post by Java Jive
So how does this compare with my formula which you tested, and your
own settings?

Measuring inside the raised edge on the rim of the dish I get d = 644 mm
with a maximum depth of 54 mm. Hence the offset angle will be
(1/2)*asin(8*54/644) = 21.1 degrees
As mentioned earlier in this thread, using your method gives an offset
angle of 20.7 degrees, so that's quite close. However, I only get this
agreement because I had already bent the LNB boom arm to place the LNB
where I thought it ought to be!

But, as mentioned above, if the LNB arm is positioned wrongly, that
would change the effective offset of the dish anyway.

Post by John Legon

Post by Java Jive
Oh, and just for the record, is yours a conventional
taller than wide offset dish, in which case it would be also
interesting to know what the 'boresight' method gives, or a wider than
tall minidish?

The outer dimensions are 605 x 655 mm, giving 22.5 degrees.

Yes, it's definitely looking to me as though the 'boresight'
calculation is the least reliable of the three so far discussed. The
other two show good agreement though, which is encouraging for both. I
will have to give some thought, and maybe some maths, as to how an LNB
holder being slightly out of true will effect the accuracy of each
method.

But again, thanks for the detailed trouble you're going to, to help.
It really is much appreciated!

[John Legon]

Post by Java Jive
On Fri, 7 Oct 2011 08:54:36 +0100, John Legon

Post by John Legon
The fundamental starting point is the dish itself. If the curvature
isn't accurate then nothing can be done, but at least the position of
the LNB can be excluded from the calculation of the offset angle.

Except, surely, the position of the LNB *affects* the offset angle, in
the sense that where the position of the LNB is slightly out of true
but is not corrected, which it won't be by most installers, the
elevation of the dish will have to be slightly different than it would
be if the LNB were in the correct position?

Of course, the position of the LNB will determine whether the tilt of
the dish when mounted corresponds to the theoretical ideal, and most
people won't care if it isn't, as long as the system works. It's still
the case, however, that the "correct" offset angle is defined by the
geometry of the dish, and not by the position of the LNB.

Post by Java Jive

Post by John Legon
I don't follow. The belt is inclined upwards from (say) Astra 28E in
the east to (say) Hotbird at 13 E, and the reflection of the arc off the
dish is inclined downwards accordingly. The LNBs are not placed in a
horizontal line but rather above and below that line with (in my set up)
only the LNB for Astra 19 E on the (presumed) focal surface.

Ok, I worded that loosely, partly because for some reason or other I
was thinking you have a rotor like myself, thus making the Clarke belt
'horizontal' wrt to the face of the dish no matter at which bit of the
belt the dish is pointing. But what I meant is still true, the LNB's
will lie on a line lying in the focal surface of the dish.

I do have a dish with a rotor, and also a dish with three LNBs...

Post by Java Jive

Post by John Legon

Post by Java Jive
Therefore theta = (1/2)*asin(8p/d)

Excellent! It really is that simple :-)
Now using this formula with the data supplied in the above pdf file,
namely for a dish with d = 500 and p = 43, we get an offset angle of
21.7 degrees and hence a complement of 68.3 degrees.
The pdf assumes 66.9 degrees initially, but then - realizing that the
axis of the parabola intersects the lower rim of the dish - obtains the
better result of 68.3 degrees by trial and error, in perfect agreement
with my calculation.

Good. If possible, I'd like to see at least an outline of how to
prove this formula, to save me working it out for myself.

Don't know about proof - this is just something I sketched out the other
day on a scrap of paper. But since you ask...

Take the vertical section through an offset dish with origin (0,0) of
the parabolic curve x = y^2/4a located on the lower rim. Take a chord,
length d, from the origin to a point (xt,yt) located on the upper rim.
The offset angle of the dish will be atan(xt/yt).

From the mid-point of the chord (xt/2, yt/2), drop a perpendicular
(parallel to the x-axis) on to the y-axis. It will intersect the
parabola at the point (xt/4, yt/2). Hence the distance along this line
from the mid-point of the chord to the parabola is also xt/4.

This distance, which I call k, could be measured from the dish itself if
the offset angle theta was known, but the angle isn't known and we can
only measure the distance p perpendicular to the chord.

Now to a close approximation, p = k*cos(theta). This is because the
gradient of the parabolic curve at the chosen half-way point is parallel
to the chord. For the same reason, the distance p is effectively the
maximum distance perpendicularly from the chord to the curve.

Hence the distance of 2k from the mid-point of the chord to the y-axis,
measured parallel to x-axis, is 2p/cos(theta). The mid-point is also at
a distance of d/2 along the chord from the origin, and the offset angle
can be calculated as

sin(theta) = 2p/(cos(theta)) / (d/2)

and theta = (1/2)*asin(8p/d)

As regards the approximation to k in the above procedure, I've worked
out that it amounts to about 0.2 mm in the actual measurement, or about
0.1 degree in the final result.

[Java Jive]

On Fri, 7 Oct 2011 19:09:59 +0100, John Legon

Post by John Legon
Of course, the position of the LNB will determine whether the tilt of
the dish when mounted corresponds to the theoretical ideal, and most
people won't care if it isn't, as long as the system works. It's still
the case, however, that the "correct" offset angle is defined by the
geometry of the dish, and not by the position of the LNB.

Well, I guess that depends on how you actually define the offset. As
we are interested in knowing it because of how it effects what
elevation we set on the dish, I would suggest that the useful
definition is the difference in elevation between the type of dish
under discussion and an axi-symmetric equivalent. Under that
definition, the position of the LNB does indeed affect the offset.

Post by John Legon
Don't know about proof - this is just something I sketched out the other
day on a scrap of paper. But since you ask...
Take the vertical section through an offset dish with origin (0,0) of
the parabolic curve x = y^2/4a located on the lower rim.

I don't like this assumption. Although, like the boresight
assumption, it makes sense, I don't think we can absolutely rely on
all manufacturers to produce dishes that we think makes sense.

However, going with the flow for the mo ...

For future reference let O be the origin (0,0)

Post by John Legon
Take a chord,
length d, from the origin to a point (xt,yt) located on the upper rim.
The offset angle of the dish will be atan(xt/yt).
From the mid-point of the chord (xt/2, yt/2)

Let M be midpoint of the chord (xt/2, yt/2)

Post by John Legon
drop a perpendicular
(parallel to the x-axis) on to the y-axis. It will intersect the
parabola at the point (xt/4, yt/2).

Let P this point of intersection with the parabola (xt/4, yt/2)

Post by John Legon
Hence the distance along this line
from the mid-point of the chord to the parabola is also xt/4.
This distance, which I call k,

So MP = k

Post by John Legon
could be measured from the dish itself if
the offset angle theta was known, but the angle isn't known and we can
only measure the distance p perpendicular to the chord.
Now to a close approximation, p = k*cos(theta).

Surely it's actually EXACTLY that?

The gradient of the parabola at any point is given by ...
d sqrt(4ax) / dx
... which evaluates to ...
sqrt(a/x)

Therefore, to find x where the tangent is parallel to the chord, we
equate the gradients:

sqrt(a/x) = (yT/xT)

... which after squaring and substituting yT^2 = 4a.xT gives ...

x = xT/4

So the tangent is parallel exactly at P. If we drop a perpendicular
from P onto the chord and call the point where it meets it Q, then PQ
is p. The angle QMP = 0MP = 90 - theta, so MPQ = theta, and ...
p = k.cos(theta).

Post by John Legon
Hence the distance of 2k from the mid-point of the chord to the y-axis,
measured parallel to x-axis, is 2p/cos(theta). The mid-point is also at
a distance of d/2 along the chord from the origin, and the offset angle
can be calculated as
sin(theta) = 2p/(cos(theta)) / (d/2)
and theta = (1/2)*asin(8p/d)

Yes, agreed.

Post by John Legon
As regards the approximation to k in the above procedure, I've worked
out that it amounts to about 0.2 mm in the actual measurement, or about
0.1 degree in the final result.

I don't think any approximation is actually involved. I'd be
interested to hear your thoughts after reading mine.

I think the next step with this is to see if it can be generalised it
to remove the assumption that the bottom of the dish B is at (0,0). If
we could do that, and produce equations for the offset and perhaps the
focal point that only rely on the two dimensions measured, that would
have great potential use. However, if we can't remove that
assumption, I would favour my new formula over this method.

[John Legon]

Post by Java Jive
On Fri, 7 Oct 2011 19:09:59 +0100, John Legon

Post by John Legon
Take the vertical section through an offset dish with origin (0,0) of
the parabolic curve x = y^2/4a located on the lower rim.

I don't like this assumption. Although, like the boresight
assumption, it makes sense, I don't think we can absolutely rely on
all manufacturers to produce dishes that we think makes sense.

I don't much care for the assumption myself, but haven't as yet found
any reason to doubt its validity.

Post by Java Jive
However, going with the flow for the mo ...
For future reference let O be the origin (0,0)

Post by John Legon
Take a chord,
length d, from the origin to a point (xt,yt) located on the upper rim.
The offset angle of the dish will be atan(xt/yt).
From the mid-point of the chord (xt/2, yt/2)

Let M be midpoint of the chord (xt/2, yt/2)

Post by John Legon
drop a perpendicular
(parallel to the x-axis) on to the y-axis. It will intersect the
parabola at the point (xt/4, yt/2).

Let P this point of intersection with the parabola (xt/4, yt/2)

Post by John Legon
Hence the distance along this line
from the mid-point of the chord to the parabola is also xt/4.
This distance, which I call k,

So MP = k

Post by John Legon
could be measured from the dish itself if
the offset angle theta was known, but the angle isn't known and we can
only measure the distance p perpendicular to the chord.
Now to a close approximation, p = k*cos(theta).

Surely it's actually EXACTLY that?

Not quite. See below.

Post by Java Jive
The gradient of the parabola at any point is given by ...
d sqrt(4ax) / dx
... which evaluates to ...
sqrt(a/x)
Therefore, to find x where the tangent is parallel to the chord, we
sqrt(a/x) = (yT/xT)
... which after squaring and substituting yT^2 = 4a.xT gives ...
x = xT/4
So the tangent is parallel exactly at P. If we drop a perpendicular
from P onto the chord and call the point where it meets it Q, then PQ
is p. The angle QMP = 0MP = 90 - theta, so MPQ = theta, and ...
p = k.cos(theta).

The tangent is exactly parallel at P, but the position of P depends on
the offset angle which we're trying to find. For this reason, my
distance p isn't measured from P to the chord at point Q, but is instead
the distance along the perpendicular bisector to the chord through the
midpoint M. This line intersects the curve at a point where the tangent
isn't exactly parallel to the chord, and there is a slight approximation
in taking k to be p/cos(theta).

However, the error can be eliminated by taking p to be the maximum
distance from the chord to the curve. This being the case, it is isn't
necessary to specify exactly where the measurement of p is made, but it
should in theory correspond to PQ.

[Java Jive]

On Sat, 8 Oct 2011 08:12:05 +0100, John Legon

Post by John Legon

Post by Java Jive
I don't like this assumption. Although, like the boresight
assumption, it makes sense, I don't think we can absolutely rely on
all manufacturers to produce dishes that we think makes sense.

I don't much care for the assumption myself, but haven't as yet found
any reason to doubt its validity.

Let me give you a possible one ...

I've managed to find two pictures of my last dish, which I no longer
have. I've temporarily put them up on my site so that anyone
interested can check my working:
Loading Image...
Loading Image...

From these, by knowing the rest of the dimensions which I measured
previously to getting rid of it, and counting pixels in the pictures
(in PSP, this is easily done by choosing selection areas exactly
enclosing the item of interest, reading off the dimensions of the
selection as it is being made, and doing a Pythagoras calculation), I
have been able to estimate the perpendicular depth p.

In the first the dish is mounted in use, the picture being taken from
the ground vertically underneath the mounting. The scale of the
picture is determined by the known width of the dish as below. The
min figure is from the number of pixels between the parallel lines,
the max from the full length of the perpendicular line.

In the second, the dish is taken from exactly edge on, but not from
exactly in the middle of the side. The scale was therefore assumed to
be that the line across the rim was the average of the height and the
width, 825.

Dimensions (mm):
Width: 800
Height: 850
B2LNB: 535
T2LNB: 860
First pic ...
Min estimated depth: 65
Max estimated depth: 90
Ave estimated depth: 75 (approx)
Second pic ...
Estimated depth: 65
(all rounded to nearest 5mm)

Offset Calculations (deg):
Yours min: 18.86
Yours max: 28.95
Yours ave: 22.45
Yours #2: 18.86
Boresight: 19.75
Universal: 22.48

You can see that for your method, while the average from the first pic
agrees closely and encouragingly with mine, the min and max are each
further out even than the boresight method, itself a long way out.
However, I think the second pic is more reliable, and this is very
close to the min estimate from the first pic. All this suggests that:
:-( In this case, the underlying assumption, that the bottom of
the dish B is coincident with the origin O, is likely to be wrong;
:-( Your method is sensitive to errors in measuring the depth;
:-( Therefore it is probably also sensitive to the correctness of
the assumption that B is coincident with O.

I think the sensitivity problem lies in the 8 times factor in the asin
argument.

Post by John Legon
However, the error can be eliminated by taking p to be the maximum
distance from the chord to the curve. This being the case, it is isn't
necessary to specify exactly where the measurement of p is made, but it
should in theory correspond to PQ.

I must have misunderstood what you originally wrote, I thought the
above was what you were actually doing.

[John Legon]

Frankly, I doubt that it is realistically possible to draw meaningful
conclusions from the photos you provide. The depth measurement with my
method has to be made with reasonable accuracy off the surface of the
dish itself. An error of 1 mm in the measurement for a 65 cm dish will
result in an error of about 0.5 degrees in the final result. However, I
think it is entirely possible to achieve an accuracy of that order with
careful measurement.

In my view, based upon actual measurements, your LNB method is subject
to similar uncertainties, but the matter is largely theoretical and in
practice most people don't care or need to know what the offset angle of
their dish might be. :-)

Post by Java Jive
On Sat, 8 Oct 2011 08:12:05 +0100, John Legon

Post by John Legon

Post by Java Jive
I don't like this assumption. Although, like the boresight
assumption, it makes sense, I don't think we can absolutely rely on
all manufacturers to produce dishes that we think makes sense.

I don't much care for the assumption myself, but haven't as yet found
any reason to doubt its validity.

Let me give you a possible one ...
I've managed to find two pictures of my last dish, which I no longer
have. I've temporarily put them up on my site so that anyone
http://www.macfh.co.uk/Temp/OldDish1.png
http://www.macfh.co.uk/Temp/OldDish2.png
From these, by knowing the rest of the dimensions which I measured
previously to getting rid of it, and counting pixels in the pictures
(in PSP, this is easily done by choosing selection areas exactly
enclosing the item of interest, reading off the dimensions of the
selection as it is being made, and doing a Pythagoras calculation), I
have been able to estimate the perpendicular depth p.
In the first the dish is mounted in use, the picture being taken from
the ground vertically underneath the mounting. The scale of the
picture is determined by the known width of the dish as below. The
min figure is from the number of pixels between the parallel lines,
the max from the full length of the perpendicular line.
In the second, the dish is taken from exactly edge on, but not from
exactly in the middle of the side. The scale was therefore assumed to
be that the line across the rim was the average of the height and the
width, 825.
Width: 800
Height: 850
B2LNB: 535
T2LNB: 860
First pic ...
Min estimated depth: 65
Max estimated depth: 90
Ave estimated depth: 75 (approx)
Second pic ...
Estimated depth: 65
(all rounded to nearest 5mm)
Yours min: 18.86
Yours max: 28.95
Yours ave: 22.45
Yours #2: 18.86
Boresight: 19.75
Universal: 22.48
You can see that for your method, while the average from the first pic
agrees closely and encouragingly with mine, the min and max are each
further out even than the boresight method, itself a long way out.
However, I think the second pic is more reliable, and this is very
:-( In this case, the underlying assumption, that the bottom of
the dish B is coincident with the origin O, is likely to be wrong;
:-( Your method is sensitive to errors in measuring the depth;
:-( Therefore it is probably also sensitive to the correctness of
the assumption that B is coincident with O.
I think the sensitivity problem lies in the 8 times factor in the asin
argument.

Post by John Legon
However, the error can be eliminated by taking p to be the maximum
distance from the chord to the curve. This being the case, it is isn't
necessary to specify exactly where the measurement of p is made, but it
should in theory correspond to PQ.

I must have misunderstood what you originally wrote, I thought the
above was what you were actually doing.

[Java Jive]

On Sat, 8 Oct 2011 20:07:50 +0100, John Legon

Post by John Legon
Frankly, I doubt that it is realistically possible to draw meaningful
conclusions from the photos you provide.

Perhaps, but I've just remembered how my formula of a year or two back
worked ...

For a dish making the same assumption as your formula, that the bottom
of the dish is at the origin O, TO is the line of the dish chord, TOY
is the offset, so TOX = TOF = 90-Offset. The triangle TOF can thus be
solved using the cosine rule to find the offset ...

dT^2 = d^2 + dB^2 - 2.d.dB.cos(90-Offset)
dT^2 = d^2 + dB^2 - 2.d.dB.sin(Offset)

Therefore

2.d.dB.sin(Offset) = d^2 + dB^2 - dT^2
sin(Offset) = (d^2 + dB^2 - dT^2) / (2.d.dB)
Offset = asin[ (d^2 + dB^2 - dT^2) / (2.d.dB) ]

This gives an offset of 17.21, even further out than the boresight
calculation. So I really think the assumption that B is at O is
unsound.

Post by John Legon
... but the matter is largely theoretical and in
practice most people don't care or need to know what the offset angle of
their dish might be. :-)

Well, it's only likely to be of importance when someone is installing
a dish with no scale or an uncorrected scale.

However, I sense that you're getting tired of the discussion, which is
fair enough. You've been an enormous help, John. Thank you.

[John Legon]

Post by Java Jive
On Sat, 8 Oct 2011 20:07:50 +0100, John Legon

Post by John Legon
Frankly, I doubt that it is realistically possible to draw meaningful
conclusions from the photos you provide.

Perhaps, but I've just remembered how my formula of a year or two back
worked ...
For a dish making the same assumption as your formula, that the bottom
of the dish is at the origin O, TO is the line of the dish chord, TOY
is the offset, so TOX = TOF = 90-Offset. The triangle TOF can thus be
solved using the cosine rule to find the offset ...
dT^2 = d^2 + dB^2 - 2.d.dB.cos(90-Offset)
dT^2 = d^2 + dB^2 - 2.d.dB.sin(Offset)
Therefore
2.d.dB.sin(Offset) = d^2 + dB^2 - dT^2
sin(Offset) = (d^2 + dB^2 - dT^2) / (2.d.dB)
Offset = asin[ (d^2 + dB^2 - dT^2) /
(2.d.dB) ]
This gives an offset of 17.21, even further out than the boresight
calculation. So I really think the assumption that B is at O is
unsound.

Well, I don't know what measurements you used in that calculation, but
they are not the ones I gave at the start of this thread. These were:

d = 65, dB = 40, dT = 63 cm

Plugging those figures into the above equation gives an offset angle of
20.91 degrees.

Now using my method, I gave a chord length d of 644 mm as measured from
top to bottom inside the lip on the rim of the dish, and a maximum depth
p of 54 mm from that chord to the surface of the dish. The offset angle
is calculated using the formula:

Offset = (1/2).asin(8p/d)

giving an angle of 21.06 degrees. Hence the difference between the two
methods is only 0.15 degree...

Contrary to your conclusion, therefore, the above analysis demonstrates
the accuracy of my method and the validity of the assumption that point
O is located on the lower rim at B.

Having thus found the offset from just two measurements - concerning
which, unlike the position of the focal point, there can be no doubt -
we can proceed to find the focal length, which gives the distance dB
from the bottom of the dish to the LNB:

dB = (d/4) * (1/sin(offset) - sin(offset))

while the distance from the top of the dish to the LNB will be:

dT = d.sin(offset) + dB

Hence taking a 65 cm dish with the 21 degree offset, we get

dB = 39.5 and dT = 62.7 cm

which only goes to show that the LNB on my motorised dish is more or
less where it should be. :-)

Post by Java Jive

Post by John Legon
... but the matter is largely theoretical and in
practice most people don't care or need to know what the offset angle of
their dish might be. :-)

Well, it's only likely to be of importance when someone is installing
a dish with no scale or an uncorrected scale.

It's by no means essential, since a result can always be obtained by
trial and error...

Post by Java Jive
However, I sense that you're getting tired of the discussion,

Well twigged !

[Java Jive]

On Mon, 10 Oct 2011 07:56:38 +0100, John Legon

Post by John Legon

Post by Java Jive
On Sat, 8 Oct 2011 20:07:50 +0100, John Legon

Post by John Legon
Frankly, I doubt that it is realistically possible to draw meaningful
conclusions from the photos you provide.

Perhaps, but I've just remembered how my formula of a year or two back
worked ...
For a dish making the same assumption as your formula, that the bottom
of the dish is at the origin O, TO is the line of the dish chord, TOY
is the offset, so TOX = TOF = 90-Offset. The triangle TOF can thus be
solved using the cosine rule to find the offset ...
dT^2 = d^2 + dB^2 - 2.d.dB.cos(90-Offset)
dT^2 = d^2 + dB^2 - 2.d.dB.sin(Offset)
Therefore
2.d.dB.sin(Offset) = d^2 + dB^2 - dT^2
sin(Offset) = (d^2 + dB^2 - dT^2) / (2.d.dB)
Offset = asin[ (d^2 + dB^2 - dT^2) /
(2.d.dB) ]
This gives an offset of 17.21, even further out than the boresight
calculation. So I really think the assumption that B is at O is
unsound.

Well, I don't know what measurements you used in that calculation, but
they are not the ones I gave at the start of this thread.

No, no! I'm referring to my old dish!

Post by John Legon
d = 65, dB = 40, dT = 63 cm
Plugging those figures into the above equation gives an offset angle of
20.91 degrees.
Now using my method, I gave a chord length d of 644 mm as measured from
top to bottom inside the lip on the rim of the dish, and a maximum depth
p of 54 mm from that chord to the surface of the dish. The offset angle
Offset = (1/2).asin(8p/d)
giving an angle of 21.06 degrees. Hence the difference between the two
methods is only 0.15 degree...
Contrary to your conclusion, therefore, the above analysis demonstrates
the accuracy of my method and the validity of the assumption that point
O is located on the lower rim at B.

The bottom at origin assumption obviously works with your dish, but my
point was and is that it doesn't with my old one. Thus it cannot be
generally relied upon.

Post by John Legon
Having thus found the offset from just two measurements - concerning
which, unlike the position of the focal point, there can be no doubt -
we can proceed to find the focal length, which gives the distance dB
dB = (d/4) * (1/sin(offset) - sin(offset))
dT = d.sin(offset) + dB
Hence taking a 65 cm dish with the 21 degree offset, we get
dB = 39.5 and dT = 62.7 cm
which only goes to show that the LNB on my motorised dish is more or
less where it should be. :-)

Yes, I can't imagine that you'd ever have trouble installing an
unknown dish!

Thanks again.

[John Legon]

Post by Java Jive
On Mon, 10 Oct 2011 07:56:38 +0100, John Legon

Post by John Legon
Well, I don't know what measurements you used in that calculation, but
they are not the ones I gave at the start of this thread.

No, no! I'm referring to my old dish!

Oh, I see! Did you use the measurements that you took off the photos
you posted? If so, I think that perspective distortion etc will skew
the results quite considerably.

[Java Jive]

No, the only measurements I made via the photos were to estimate the
depth of the dish to use in your formula. All the other measurements
were taken while I still had the dish.

On Mon, 10 Oct 2011 17:46:06 +0100, John Legon

Post by John Legon
Oh, I see! Did you use the measurements that you took off the photos
you posted? If so, I think that perspective distortion etc will skew
the results quite considerably.

[John Legon]

Post by Java Jive
On Mon, 10 Oct 2011 17:46:06 +0100, John Legon

Post by John Legon
Oh, I see! Did you use the measurements that you took off the photos
you posted? If so, I think that perspective distortion etc will skew
the results quite considerably.

No, the only measurements I made via the photos were to estimate the
depth of the dish to use in your formula. All the other measurements
were taken while I still had the dish.

In that case, I'm inclined to think that your formulas for the offset
don't give the same results because the LNB was not in the correct
position. If it had been just 3 cm higher and 2.5 cm inwards, then I
think you would have got the same offset angle with both calculations,
consistent with the axis being on the lower rim. :-)

[Java Jive]

For someone who's admitted to being tired of the discussion, you're
certainly putting in a great deal of work, John !-)

That's interesting, certainly. I suppose the 2.5cm further in could
be accounted for if I'd measured the distance to the centre of the LNB
holder, and the true focal point is near the face of the LNB. However,
although I can't now definitely remember what I did, I have a feeling
that I put the LNB back in the holder to make the measurements. Also,
looking at a photo of the LNB, which I also sold, it does seem to me
to be entirely possible that the focal point is approximately in the
centre of the holder ring! Either way, I don't we can tell much more
without access to the dish.

But the real point at issue is that, whatever the reason the formulae
don't agree, which of the four we now have is likely to be the most
accurate?

I think that, in practice, what I have chosen to call my 'universal'
formula is likely to be the most accurate, because it uses the actual
position of an LNB as mounted on a given dish, rather than its
theoretically optimum position, and it does not rely on any other
assumptions which may, but may not, be true.

However, I think that if you wanted to actually check and if necessary
correct the position of the LNB arm to ensure that it was optimal,
then the Legon formula would be a good starting point. I think it
would be even better if it could be generalised not rely on the bottom
at origin assumption, but I couldn't see a way of doing that when I
had a quick look at it.

I admit that the above claims are merely hunches, which really one
ought to do some work, probably using some calculus for small changes
and errors, to prove, but until I have time to do so, I'm willing to
assume that, for the reasons given which I find persuasive, they are
correct.

On Tue, 11 Oct 2011 21:25:34 +0100, John Legon

Post by John Legon
In that case, I'm inclined to think that your formulas for the offset
don't give the same results because the LNB was not in the correct
position. If it had been just 3 cm higher and 2.5 cm inwards, then I
think you would have got the same offset angle with both calculations,
consistent with the axis being on the lower rim. :-)

[John Legon]

Post by Java Jive
For someone who's admitted to being tired of the discussion, you're
certainly putting in a great deal of work, John !-)

I think we've both displayed an unhealthy obsession with satellite dish
calculations. :)

Post by Java Jive
That's interesting, certainly. I suppose the 2.5cm further in could
be accounted for if I'd measured the distance to the centre of the LNB
holder, and the true focal point is near the face of the LNB. However,
although I can't now definitely remember what I did, I have a feeling
that I put the LNB back in the holder to make the measurements. Also,
looking at a photo of the LNB, which I also sold, it does seem to me
to be entirely possible that the focal point is approximately in the
centre of the holder ring! Either way, I don't we can tell much more
without access to the dish.

It's a pity you don't still have that dish, but looking at your photos,
I think the curvature is consistent with my view that the axis of the
parabola is located on the lower rim. The LNB arm ought therefore to
have been bent slightly upwards to give optimum results.

Post by Java Jive
But the real point at issue is that, whatever the reason the formulae
don't agree, which of the four we now have is likely to be the most
accurate?

I think the four formulae are complementary rather than competing.
The boresight method probably shows what the manufacturer intended the
offset to be, my formula shows what the offset actually is, and your
two formulae show what offset could be assuming that the LNB arm was
accurately constructed :)

Post by Java Jive
I think that, in practice, what I have chosen to call my 'universal'
formula is likely to be the most accurate, because it uses the actual
position of an LNB as mounted on a given dish, rather than its
theoretically optimum position, and it does not rely on any other
assumptions which may, but may not, be true.

Certainly, your 'universal' formula can give a useful result, but unless
the LNB is at the focus of the dish, there can be no single solution for
the offset angle. It will depend on the part of the dish that the beam
is reflected off from.

Post by Java Jive
However, I think that if you wanted to actually check and if necessary
correct the position of the LNB arm to ensure that it was optimal,
then the Legon formula would be a good starting point. I think it
would be even better if it could be generalised not rely on the bottom
at origin assumption, but I couldn't see a way of doing that when I
had a quick look at it.

A more generalised method might be to measure the depth of the curvature
at several points, and use an interpolation formula to construct the
equation of the curve, which may or may not be strictly parabolic...

[Java Jive]

On Thu, 13 Oct 2011 15:27:45 +0100, John Legon

Post by John Legon
It's a pity you don't still have that dish, but looking at your photos,
I think the curvature is consistent with my view that the axis of the
parabola is located on the lower rim. The LNB arm ought therefore to
have been bent slightly upwards to give optimum results.

So the pictures aren't good enough to test your formula when I do it,
but are when you do it?

Post by John Legon
I think the four formulae are complementary rather than competing.
The boresight method probably shows what the manufacturer intended the
offset to be, my formula shows what the offset actually is

No, it doesn't show what it actually is, because that's determined by
the actual position of the LNB. Your formula shows what the offset
would be if the LNB were the dish accurately constructed with the LNB
where it should be.

Post by John Legon
, and your
two formulae show what offset could be assuming that the LNB arm was
accurately constructed :)

No, the 'universal' one shows what it actually is, as determined by
the actual position of the LNB. The bottom-at-origin-assumption one
will only agree with the 'universal' one if the bottom of the dish is
actually at the origin.

Post by John Legon
Certainly, your 'universal' formula can give a useful result, but unless
the LNB is at the focus of the dish,

No, as above, because my formula uses the actual rather than the
theoretically optimum position of the LNB, it measures the offset as
it actually is.

Post by John Legon
there can be no single solution for the offset angle.
It will depend on the part of the dish that the beam
is reflected off from.

I suspect that in practice effectively parallel rays from the sat will
as near as dammit focus to a point even when arriving slightly above
or below the dish axis.

Post by John Legon
A more generalised method might be to measure the depth of the curvature
at several points, and use an interpolation formula to construct the
equation of the curve, which may or may not be strictly parabolic...

Yes, that would be the most accurate method, but it probably get us
into the messy iterative procedures that I was trying to avoid.

[John Legon]

Post by Java Jive
On Thu, 13 Oct 2011 15:27:45 +0100, John Legon

Post by John Legon
It's a pity you don't still have that dish, but looking at your photos,
I think the curvature is consistent with my view that the axis of the
parabola is located on the lower rim. The LNB arm ought therefore to
have been bent slightly upwards to give optimum results.

So the pictures aren't good enough to test your formula when I do it,
but are when you do it?

The photos aren't good enough to test my formula by calculation, since
the depth of the dish can't be estimated with sufficient accuracy.
I have, however, plotted the curve using your measurements, with the
focus placed where I think it should be, and the result matches up
nicely with the curvature of the dish as seen in the photos.

Post by Java Jive

Post by John Legon
I think the four formulae are complementary rather than competing.
The boresight method probably shows what the manufacturer intended the
offset to be, my formula shows what the offset actually is

No, it doesn't show what it actually is, because that's determined by
the actual position of the LNB. Your formula shows what the offset
would be if the LNB were the dish accurately constructed with the LNB
where it should be.

Any given dish has only one specific offset angle, which is given by my
formula regardless of the LNB. That in my view is what the offset angle
of the dish actually is! What the tilt of the dish will be when aligned
to a satellite is another matter...

Post by Java Jive

Post by John Legon
, and your
two formulae show what offset could be assuming that the LNB arm was
accurately constructed :)

No, the 'universal' one shows what it actually is, as determined by
the actual position of the LNB.

I don't think so (see below).

Post by Java Jive
The bottom-at-origin-assumption one
will only agree with the 'universal' one if the bottom of the dish is
actually at the origin.

Agreed. And since the origin is located at the bottom in the general
case, a discrepancy between the two methods will indicate that the LNB
is in the wrong place...

Post by Java Jive

Post by John Legon
Certainly, your 'universal' formula can give a useful result, but unless
the LNB is at the focus of the dish,

No, as above, because my formula uses the actual rather than the
theoretically optimum position of the LNB, it measures the offset as
it actually is.

I don't think so.

Post by Java Jive

Post by John Legon
there can be no single solution for the offset angle.
It will depend on the part of the dish that the beam
is reflected off from.

I suspect that in practice effectively parallel rays from the sat will
as near as dammit focus to a point even when arriving slightly above
or below the dish axis.

But unless the parallel rays from the satellite meet the dish at the
geometrically correct offset angle, as given by my formula, then there
can be no point of focus, and neither your formulae nor mine will show
what the effective working tilt of the dish might be.

The path lengths for rays reflected off different parts of the dish will
be different, the signals from top and bottom will become out of phase,
and only trial and error will give a result that can at best be only
sub-optimal.

Taking your old dish and measurements as an example, with the origin at
the bottom and the LNB as the source of the beam, I estimate that rays
reflected off the top and bottom of the dish will not be parallel but
will diverge by about three degrees. What, then, is the offset angle?

Your formulae for the offset are only valid when the LNB is located at
the focus of the parabola.

Post by Java Jive

Post by John Legon
A more generalised method might be to measure the depth of the curvature
at several points, and use an interpolation formula to construct the
equation of the curve, which may or may not be strictly parabolic...

Yes, that would be the most accurate method, but it probably get us
into the messy iterative procedures that I was trying to avoid.

Agreed.

[Java Jive]

On Fri, 14 Oct 2011 07:58:57 +0100, John Legon

Post by John Legon
The photos aren't good enough to test my formula by calculation, since
the depth of the dish can't be estimated with sufficient accuracy.
I have, however, plotted the curve using your measurements, with the
focus placed where I think it should be, and the result matches up
nicely with the curvature of the dish as seen in the photos.

You can't possibly justify that!

There is no view of the actual profile taken from the side from
half-way up its height. The only side view is distorted by being
taken from a vantage point well above the middle of it. In fact,
although the dish is tilted away from the camera, it looks to me as
though the vantage point was as high as or higher than even the top of
the dish.

From that photo, I'd have far more confidence in my depth measurement
than any attempt to obtain a profile of the dish.

Post by John Legon
Any given dish has only one specific offset angle, which is given by my
formula regardless of the LNB. That in my view is what the offset angle
of the dish actually is! What the tilt of the dish will be when aligned
to a satellite is another matter...

Well, again, it depends on how you define the offset. As we're
interested in knowing it in order to align the dish initially well
enough to get a signal to use for fine adjustment, the only definition
that makes sense to me is the difference in elevation between the dish
we're trying to align and an axi-symmetric equivalent which would
point directly at the sat.

Post by John Legon
Agreed. And since the origin is located at the bottom in the general
case, a discrepancy between the two methods will indicate that the LNB
is in the wrong place...

You can not justify such a sweeping claim.

Between the dishes we've measured and the ones in the literature we've
examined, we seem to have as many examples of the origin apparently
not being coincident with the bottom of the dish as being so, more if
we include your own before you 'corrected' it. You may argue that
it's actually the LNB that's in the wrong position, but that is your
assumption, which others are free to accept or not. I am prepared to
accept that it may sometimes be true, and possibly was in the case of
your own dish. It may also have been true of my old dish - I bought
it second-hand, so it may be that some idiot youth thought it would be
cool to have a swing on the LNB arm when it was under earlier
ownership. However, it may also be that it was built deliberately
that way, and that the LNB is actually at the correct focus. Without
access to the dish, either theory could be correct, and neither
accepts the other's interpretations of the photographs of it.

But even if it's true for both those dishes, that is too small a
sample to make such a sweeping claim as you make above.

Consider, why offset a dish? There are a number of advantages to
doing so, but probably the principal one is so that the LNB does not
shade, and therefore effectively waste, part of the reflecting
surface. But if you put both the bottom of the dish and the LNB on
the axis of the parabola, then the top of the LNB will be shading the
bottom of the dish. Why do half a job, why not make the bottom of the
dish a little higher so that NONE of the LNB shades the dish?

So I can see very good reasons why the bottom of the dish might not
always be at the origin of the parabola, and consequently that your
sweeping claim above is fundamentally unsound.

Post by John Legon
But unless the parallel rays from the satellite meet the dish at the
geometrically correct offset angle, as given by my formula, then there
can be no point of focus, and neither your formulae nor mine will show
what the effective working tilt of the dish might be.
The path lengths for rays reflected off different parts of the dish will
be different, the signals from top and bottom will become out of phase,
and only trial and error will give a result that can at best be only
sub-optimal.
Taking your old dish and measurements as an example, with the origin at
the bottom and the LNB as the source of the beam, I estimate that rays
reflected off the top and bottom of the dish will not be parallel but
will diverge by about three degrees. What, then, is the offset angle?

At that point between the two extremes that gives the best signal,
which - at a guess, I haven't checked - will probably be about
half-way between.

Post by John Legon
Your formulae for the offset are only valid when the LNB is located at
the focus of the parabola.

No, given the position of the LNB, whether it's where it ideally
should be or not, my formula should give something sufficiently close
to the optimum alignment of the dish to obtain an initial signal.

By contrast, yours, by taking no account of the actual position of the
LNB, and by making an unsound assumption which might not always be
true, is quite liable to be out, perhaps even badly enough out to be
unable to tune the sat.

[John Legon]

Post by Java Jive
On Fri, 14 Oct 2011 07:58:57 +0100, John Legon

Post by John Legon
The photos aren't good enough to test my formula by calculation, since
the depth of the dish can't be estimated with sufficient accuracy.
I have, however, plotted the curve using your measurements, with the
focus placed where I think it should be, and the result matches up
nicely with the curvature of the dish as seen in the photos.

You can't possibly justify that!

It wasn't my intention to attempt to prove anything by it! Ironically,
the reconstruction of the focal point fell on the centre of the LNB
holder in the photo, thus showing that perspective distortion has indeed
skewed the result.

[...]

Post by Java Jive

Post by John Legon
Any given dish has only one specific offset angle, which is given by my
formula regardless of the LNB. That in my view is what the offset angle
of the dish actually is! What the tilt of the dish will be when aligned
to a satellite is another matter...

Well, again, it depends on how you define the offset. As we're
interested in knowing it in order to align the dish initially well
enough to get a signal to use for fine adjustment, the only definition
that makes sense to me is the difference in elevation between the dish
we're trying to align and an axi-symmetric equivalent which would
point directly at the sat.

It's useful, though, to make a distinction between the offset which is a
fixed parameter of the dish itself, and the effective or working offset
which your formula provides.

Post by Java Jive

Post by John Legon
Agreed. And since the origin is located at the bottom in the general
case, a discrepancy between the two methods will indicate that the LNB
is in the wrong place...

You can not justify such a sweeping claim.
Between the dishes we've measured and the ones in the literature we've
examined, we seem to have as many examples of the origin apparently
not being coincident with the bottom of the dish as being so, more if
we include your own before you 'corrected' it.

I don't know what these examples of dishes with the origin apparently
not on the bottom of the dish might be. In the pdf cited earlier in
this thread, the designer of the RCA offset dishes was quoted as saying
that the origin was on the lower rim. In the General Dynamics offset
dish geometry webpage, the vertex or origin of the parabola is shown to
be on the lower rim, and this applies to a wide range of dishes of
different sizes.

As regards my own dishes, my conclusion that the origin is located on
the lower rim is based upon measurements of the curvature at several
points across the dish, and an iterative curve-fitting procedure to find
the best fit to a parabolic curve.

Dishes of this standard circular type are made in vast numbers, and may
well be said to represent the "general case". I'm not saying that all
dishes are like this, simply that if your two formulae for the offset
don't give the same angle, then there's a good chance that it's because
the LNB is not accurately mounted.

In the case of your old dish, for which the curvature isn't known, the
'boresight' method will give a good indication of the intended offset
angle. The formula works, not simply because the LNB sees the dish as
being circular, but because the plane section through the paraboloid of
rotation describes an ellipse, and the projection of that ellipse onto a
plane at right angles to the axis gives a circle. Since the outer rim
of the dish represents a plane surface (unless the dish is warped or is
a Sky dish), the rim must be elliptical, and the boresight calculation
will give the intended offset.

However, your formulae based on the LNB position give offset values that
conflict with the ellipse calculation. It follows that the LNB on your
old dish could not have been where it should have been - regardless of
whether the origin is at the bottom of the dish or somewhere else.

Post by Java Jive
You may argue that
it's actually the LNB that's in the wrong position, but that is your
assumption, which others are free to accept or not. I am prepared to
accept that it may sometimes be true, and possibly was in the case of
your own dish. It may also have been true of my old dish - I bought
it second-hand, so it may be that some idiot youth thought it would be
cool to have a swing on the LNB arm when it was under earlier
ownership. However, it may also be that it was built deliberately
that way, and that the LNB is actually at the correct focus. Without
access to the dish, either theory could be correct, and neither
accepts the other's interpretations of the photographs of it.

As I say, the boresight calculation shows that the LNB really was out of
alignment, for whatever reason.

Post by Java Jive
But even if it's true for both those dishes, that is too small a
sample to make such a sweeping claim as you make above.
Consider, why offset a dish? There are a number of advantages to
doing so, but probably the principal one is so that the LNB does not
shade, and therefore effectively waste, part of the reflecting
surface. But if you put both the bottom of the dish and the LNB on
the axis of the parabola, then the top of the LNB will be shading the
bottom of the dish. Why do half a job, why not make the bottom of the
dish a little higher so that NONE of the LNB shades the dish?

Because the body of the LNB and the holder and supporting arm are below
the axis, just the top half of the feed horn intrudes, and in practice
this makes no difference at all...

Post by Java Jive
So I can see very good reasons why the bottom of the dish might not
always be at the origin of the parabola, and consequently that your
sweeping claim above is fundamentally unsound.

I still haven't seen any real evidence to contradict my contention that
the great majority of circular offset dishes in general use for domestic
satellite TV are constructed as half a paraboloid. It doesn't matter to
me if this should turn out not to be the case - I just want to see the
evidence!

Post by Java Jive

Post by John Legon
But unless the parallel rays from the satellite meet the dish at the
geometrically correct offset angle, as given by my formula, then there
can be no point of focus, and neither your formulae nor mine will show
what the effective working tilt of the dish might be.
The path lengths for rays reflected off different parts of the dish will
be different, the signals from top and bottom will become out of phase,
and only trial and error will give a result that can at best be only
sub-optimal.
Taking your old dish and measurements as an example, with the origin at
the bottom and the LNB as the source of the beam, I estimate that rays
reflected off the top and bottom of the dish will not be parallel but
will diverge by about three degrees. What, then, is the offset angle?

At that point between the two extremes that gives the best signal,
which - at a guess, I haven't checked - will probably be about
half-way between.

Post by John Legon
Your formulae for the offset are only valid when the LNB is located at
the focus of the parabola.

No, given the position of the LNB, whether it's where it ideally
should be or not, my formula should give something sufficiently close
to the optimum alignment of the dish to obtain an initial signal.

Well, that's probably true. Your "universal" formula should give a
result that works well enough in practice when the LNB isn't exactly
where it should be. But from the point of view of optimising the
performance of a dish, I would want to know whether the effective
working offset is the same as the offset of the dish itself, because
only then will the dish function as intended.

Post by Java Jive
By contrast, yours, by taking no account of the actual position of the
LNB, and by making an unsound assumption which might not always be
true, is quite liable to be out, perhaps even badly enough out to be
unable to tune the sat.

But I never suggested that my formula should be used to calculate the
working offset of a dish when the LNB is in the wrong position! On the
contrary, the objective was to determine what the offset angle of the
dish itself really is as an entity, and from that result to work out
what the correct position for the LNB should be.

[Davey]

On Tue, 04 Oct 2011 21:35:06 +0100

Although I'd really have liked a lot more feedback than the single
example I did get (thanks for the good work, John), the single
underlying assumption is so reasonable and the proof is so simple that
I felt confident enough to release this live. Accordingly the
http://www.macfh.co.uk/JavaJive/AudioVisualTV/SatelliteTV/SatelliteGeneral.html
http://www.macfh.co.uk/JavaJive/AudioVisualTV/SatelliteTV/SatelliteCalculator.php
http://www.macfh.co.uk/JavaJive/AudioVisualTV/SatelliteTV/SatelliteAnalysisSettings.html

One of the few advantages of having to sit around while vinyls are
recording in real time is that you can do some mathematical
doodling, as a result of which I now have a formula for the offset
of any sat dish, even a minidish, as long as it is parabolic is
section, which surely they must all be?
Offset = asin[ (dT - dB) / d ]
d = chordal distance across dish from top to bottom
dB = distance of bottom of dish to focal point
dT = distance of top of dish to focal point
It really should be as simple as that. What I like about this
formula is that, unlike the 'boresight' one that's currently on my
site and another I derived a year or two ago, the ONLY assumption
it relies upon is that the dish is parabolic in section.

Did you ask for feedback?

[Java Jive]

Post by Davey
Did you ask for feedback?
I'd be interested to hear from anyone else willing to try and verify
this new formula by measurement, particularly against a dish where the
offset is known from the manufacturer's specifications.
I'd also be interested in people's views on where the focus of the
dish is in relation to the LNB, where it's present, or just the holder
where it is not. Although LNBs must surely vary at least a little,
they must sit at the correct focal point, and as the only given is the
LNB holder, it must be possible to say that the focal point must be x
mm directly in front of the holder, but what is x?

[Davey]

On Fri, 07 Oct 2011 19:28:59 +0100

Post by Davey
Did you ask for feedback?
I'd be interested to hear from anyone else willing to try and verify
this new formula by measurement, particularly against a dish where
the offset is known from the manufacturer's specifications.
I'd also be interested in people's views on where the focus of the
dish is in relation to the LNB, where it's present, or just the
holder where it is not. Although LNBs must surely vary at least a
little, they must sit at the correct focal point, and as the only
given is the LNB holder, it must be possible to say that the focal
point must be x mm directly in front of the holder, but what is x?

Ok. I was looking in the Original Post, not further down the thread.