Formula for sat dish offset
Java Jive
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
following pages on my site have been updated:
http://www.macfh.co.uk/JavaJive/AudioVisualTV/SatelliteTV/SatelliteGeneral.html
There is now an option to use the new formula in:
http://www.macfh.co.uk/JavaJive/AudioVisualTV/SatelliteTV/SatelliteCalculator.php
Gives a diagram and proof:
http://www.macfh.co.uk/JavaJive/AudioVisualTV/SatelliteTV/SatelliteAnalysisSettings.html
On Thu, 22 Sep 2011 00:00:08 +0100, Java Jive <ja...@evij.com.invalid>
wrote:
> 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?
>
> It's:
> Offset = asin[ (dT - dB) / d ]
>
> Where:
> 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.
--
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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.
seems to involve measurement to a vague point, rather than the definite
width and height ...
Java Jive
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
<usenet....@adslpipe.co.uk> wrote:
>
> 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
>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.
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...
>
>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.
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.
>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.
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.
--
John Legon
J G Miller
> 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 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
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
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
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.
Java Jive
<jo...@nospam.demon.co.uk> wrote:
> At 23:07:20 Wed, 5 Oct 2011, Java Jive <ja...@evij.com.invalid> wrote:
> >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.
> 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.
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.
> 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.
them, even if only a scan of handwritten notes.
> 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....
was an example somewhere in here:
http://www.qsl.net/n1bwt/chap1.pdf
> 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.
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
>On Thu, 6 Oct 2011 05:47:37 +0100, John Legon
><jo...@nospam.demon.co.uk> wrote:
>
>> At 23:07:20 Wed, 5 Oct 2011, Java Jive <ja...@evij.com.invalid> wrote:
>> >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.
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.
>
>> 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.
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.
>
>> 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.
based on my actual measurements, with a reconstruction of the focus:
http://www.john-legon.co.uk/temp/parab.jpg
>> 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
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.
>
>> 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)
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.
>
>So how does this compare with my formula which you tested, and your
>own settings?
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!
>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
Java Jive
<jo...@nospam.demon.co.uk> wrote:
>
> 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.
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?
> 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.
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.
> >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.
prove this formula, to save me working it out for myself.
> >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!
would change the effective offset of the dish anyway.
> >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.
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!
Demonic
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 :-)
John Legon
>On Fri, 7 Oct 2011 08:54:36 +0100, John Legon
><jo...@nospam.demon.co.uk> wrote:
>>
>> 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?
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.
>> 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.
>> >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.
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.
--
John Legon
Davey
--
Davey.
Java Jive
wrote:
> Did you ask for feedback?
wrote:
> 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
Java Jive <ja...@evij.com.invalid> wrote:
> On Fri, 7 Oct 2011 19:18:09 +0100, Davey <da...@example.invalid>
> wrote:
> >
> > Did you ask for feedback?
>
> On Thu, 22 Sep 2011 13:19:44 +0100, Java Jive <ja...@evij.com.invalid>
> wrote:
>
> > 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.
Java Jive
<jo...@nospam.demon.co.uk> wrote:
>
> 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.
> 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)
> 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)
> 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)
> 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
> 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).
> 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.
> 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
>On Fri, 7 Oct 2011 19:09:59 +0100, John Legon
><jo...@nospam.demon.co.uk> wrote:
>> 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.
any reason to doubt its validity.
>However, going with the flow for the mo ...
>
>For future reference let O be the origin (0,0)
>
>> 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)
>
>> 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)
>
>> 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
>
>> 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).
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.
--
John Legon
PeterC
> 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!
http://www.nonags.com/funimg/sat.jpg
--
Peter.
The gods will stay away
whilst religions hold sway
Java Jive
<jo...@nospam.demon.co.uk> wrote:
>
> At 23:20:02 Fri, 7 Oct 2011, Java Jive <ja...@evij.com.invalid> 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
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 <ja...@evij.com.invalid> wrote in
article <2211979n6o7iu78lj...@4ax.com>:
--
John Legon
Java Jive
<jo...@nospam.demon.co.uk> 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.
John Legon
>On Sat, 8 Oct 2011 20:07:50 +0100, John Legon
><jo...@nospam.demon.co.uk> 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.
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. :-)
>
>> ... 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.
trial and error...
>However, I sense that you're getting tired of the discussion,
--
John Legon
Java Jive
> At 16:52:19 Sun, 9 Oct 2011, Java Jive <ja...@evij.com.invalid> wrote:
> >On Sat, 8 Oct 2011 20:07:50 +0100, John Legon
> ><jo...@nospam.demon.co.uk> 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 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.
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. :-)
unknown dish!
Thanks again.
John Legon
>On Mon, 10 Oct 2011 07:56:38 +0100, John Legon
><jo...@nospam.demon.co.uk> 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
Java Jive
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
<jo...@nospam.demon.co.uk> 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.
--
John Legon
>On Mon, 10 Oct 2011 17:46:06 +0100, John Legon
><jo...@nospam.demon.co.uk> 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
Java Jive
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
<jo...@nospam.demon.co.uk> 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. :-)
--
John Legon
>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
Java Jive
<jo...@nospam.demon.co.uk> 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.
John Legon
>On Thu, 13 Oct 2011 15:27:45 +0100, John Legon
><jo...@nospam.demon.co.uk> 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?
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.
>
>> 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...
>
>>, 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).
>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...
>
>> 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.
>
>> 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.
>
>> 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.
--
John Legon
Java Jive
<jo...@nospam.demon.co.uk> wrote:
>
> 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.
> 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.
> 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.
> 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.
> 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
>On Fri, 14 Oct 2011 07:58:57 +0100, John Legon
><jo...@nospam.demon.co.uk> wrote:
>>
>> 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!
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.
[...]
>> 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.
fixed parameter of the dish itself, and the effective or working offset
which your formula provides.
>
>> 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.
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.
> 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.
alignment, for whatever reason.
>
>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?
the axis, just the top half of the feed horn intrudes, and in practice
this makes no difference at all...
>
>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.
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!
>
>> 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.
>
>> 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.
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.
>
>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.
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.
--
John Legon