Formula for sat dish offset
 

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. :-)


At 18:25:20 Sat, 8 Oct 2011, Java Jive wrote in
article :
On Sat, 8 Oct 2011 08:12:05 +0100, John Legon
wrote:

At 23:20:02 Fri, 7 Oct 2011, Java Jive wrote:

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:
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.

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.

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

Formula for sat dish offset
 

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

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.

... 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.
--
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Formula for sat dish offset
 

At 16:52:19 Sun, 9 Oct 2011, Java Jive wrote:
On Sat, 8 Oct 2011 20:07:50 +0100, John Legon
wrote:

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 we

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. :-)


... 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...

However, I sense that you're getting tired of the discussion,

Well twigged !

--
John Legon

Formula for sat dish offset
 

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

At 16:52:19 Sun, 9 Oct 2011, Java Jive wrote:
On Sat, 8 Oct 2011 20:07:50 +0100, John Legon
wrote:

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!

These we

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.

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.

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. :-)

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

Thanks again.
--
================================================== =======
Please always reply to ng as the email in this post's
header does not exist. Or use a contact address at:
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Formula for sat dish offset
 

At 11:57:00 Mon, 10 Oct 2011, Java Jive wrote:
On Mon, 10 Oct 2011 07:56:38 +0100, John Legon
wrote:

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.

--
John Legon

Formula for sat dish offset
 

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
wrote:

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.
--
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Formula for sat dish offset
 

At 18:03:16 Mon, 10 Oct 2011, Java Jive wrote:
On Mon, 10 Oct 2011 17:46:06 +0100, John Legon
wrote:

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. :-)

--
John Legon

Formula for sat dish offset
 

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
wrote:

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. :-)
--
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Formula for sat dish offset
 

At 19:04:49 Wed, 12 Oct 2011, Java Jive wrote:
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. :)


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.

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 :)

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.


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...

--
John Legon

Formula for sat dish offset
 

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

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?

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.

, 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.

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.

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.

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.
--
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