Hi Frank:
I'm still pondering all of the information you gleaned from my experiment.
You have given your comments a lot of thought and I don't want to quickly
dismiss anything you said as being inaccurate without being sure first.
But I need to point out something important about the light projection (or
antishadow) cast by the hole-in-disk nodus:
As you can clearly see in the long shadow photo, the spot of light (the
antishadow) is not there at all. When tilting the board, the antishadow
became smaller and smaller as the shadow lengthened. In fact, when I was
tiltling the board, it disappeared when the shadow was 3 times longer than
the gnomon. This I think was due to the thickness of the cardboard that I
used to make the disk. The cardboard of the disk, in fact, was what shaded
the hole. If I had used a paper thin metal to make the disk, then I think
the antishadow would not have disappeared so quickly.
My conclusion about this is that the hole-in-disk works fine and is
extremely precise at high solar angles when you can see the anti-shadow, but
is completely useless at low solar angles when the antishadow disappears.
Do you agree?
John
p.s. I love your term "antishadow" to describe the projected sunlight
through an aperture nodus. Do you invent this term?
pps. I will try to bring the experiment to Vancouver if it will fit in my
suitcase.
----- Original Message -----
From: "Frank King" <[EMAIL PROTECTED]>
To: <[EMAIL PROTECTED]>
Cc: <sundial@uni-koeln.de>; <[EMAIL PROTECTED]>
Sent: Sunday, August 06, 2006 11:45 AM
Subject: Re: Expanded Nodi Shadow Experiment
Dear John,
That is a splendid experiment...
http://advanceassociates.com/WallDial/NodusShadowExperiment.pdf
It illustrates all kinds of interesting aspects of nodus design
The Purpose, Setup and Execution all earn top marks. The Conclusion,
though, is subject to a little caveat...
Let's concentrate on just three of your designs: the disc with the
0.25" hole at the top (or leftmost), the cone at the bottom (or
rightmost) and the 1" ball next to the cone. Now look at the two
sets of shadows:
1. When the shadows are short...
(a) the centre of the anti-shadow of the disc with a hole is
about 6.1" along your board.
(b) the centre of the shadow of the ball is just a little
shorter. It seems to be almost spot on the 6" mark. [This
is possibly because the supporting stick is not quite vertical.
This is not important.]
(c) the shadow of the tip of the cone is almost exactly in line
with the anti-shadow of the disc with a hole, about 6.1" along
your board.
2. When the shadows are long...
(a) the anti-shadow of the disc with a hole is no longer clear
(as you say) but because there is an equal amount of fuzz at
the extremities of the shadow of the disc as a whole you can
fairly easily estimate the centre. It seems to be about 23.3".
(b) the centre of the shadow of the 1" ball is just a little
less easy to estimate because the supporting stick disturbs
the fuzz at one of the extremities but one can see that the
centre is about the 23" mark. This, as expected, is shorter
than the shadow to the centre of the disc and is consistent
with 1(b). So far everything ties up.
(c) the shadow of the tip of the cone though has now fallen
behind the shadow of the centre of the disc. The shadow may
be easier to read but IT IS GIVING A FALSE RESULT.
The big big trouble with any asymmetric nodus is that you cannot
cancel out the fuzz. You have to decide just where in the fuzz
is the point of interest. This is difficult. Different people
will estimate different points.
As noted at 2(b), each of your ball nodi is slightly asymmetric
because of the supporting sticks. If you had mounted the balls
sideways (as you have the disc with the hole) it would be easier
to estimate the centre of the shadow.
To my mind, the disc with the hole gives the most accurate
result even if its shadow isn't the prettiest!
Incidentally, it is worth analysing the hole in your disc in
the long shadow case...
Diameter of hole 0.25"
Height of hole above the board 4"
Approximate length of shadow 23.3"
Distance of centre of anti-shadow from centre of hole 23.6"
Angle of incidence arctan(23.3/4) = 80.3 degrees
Now consider the hole viewed from the centre of the anti-shadow.
Given that the disc is parallel to the board, the hole will appear
as an ellipse whose angular dimensions in radians are:
Major axis 0.25/23.6 approx 1/94.4 radians
Minor axis 0.25 x cos(80.3) / 23.6 approx 1/558 radians
This last figure should be compared with the angular diameter of the
sun which (by a diallist's rule of thumb) is about 1/107.5 radians.
Now imagine an insect (wearing eye protection) at the point where
the centre of the anti-shadow should be. As seen by the insect,
the minor-axis of the hole appears to be less than one-fifth the
diameter of the sun. The anti-shadow is entirely penumbra and
impossible to detect.
In my opinion this is NOT a design error. My eccentric view is
that a disc with a hole IS the best form of nodus, especially for
big sundials (well ALMOST the best) because...
when the angle of incidence is small (short shadows) you
observe the centre of the anti-shadow and...
when the angle of incidence is high (long shadows) you
observe the centre of the shadow of the disc as a whole.
I say ALMOST the best because there is a special case of a disc
with a hole, and that is the great camera obscura sundials which
one comes across in Europe. There the angular diameters of the
holes are even smaller than yours. [According to Gianni Ferrari,
Cassini took the view that the hole should be 1/1000th of its
distance to the floor, half the size of your hole at 23.6".].
The disc though is effectively of infinite diameter because
the entire building surrounds the hole and you DO see a
splodge of light. This is not anti-shadow though; it is
actually an image of the sun and you can estimate its centre
VERY precisely.
Try making your disc of infinite size and you will see how
this works :-)
MORAL: Sticks and cones are seductive but should be resisted!
I hope this hasn't been too tiresome a message for this list!
Frank H. King
Cambridge, U.K.
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