Dear dialists,
I do not have time enough (that is, skills enough! :-) to modify Helmut
& Roger's spreadsheet to include
the equations for an inclined gnomon. Some of you have given some useful
equivalences, but I thought it'd
be good to give the full equations for an arbitrarily inclined gnomon on an
horizontal surface. It's very easy
to scribble them on a minimal spreadsheet and they're very nice to play
with:
Let's define the following input data:
a is the East-West semiaxis of our ellipse (it's not necesarily the
major axis)
z the zenithal angle of the gnomon (ie., it's inclination: 90º if it
is vertical, 0º if it lies on the ground)
phi the latitude of the place where the dial is drawn
dec the Sun's declination for a given day
H the hour angle of the Sun for a certain moment (negative in the
mornings, positive in the evenings)
L is the length of the gnomon
We have to derive the equations for the following variables
b the North-South semiaxis of the ellipse
x the Northern distance from a point in the ellipse to the WE axis
(passing through its center)
y the Eastern distance from a point in the ellipse to the NS axis
(passing through its center)
d distance from the foot of the gnomon to the center of the ellipse
We can calculate them applying the following equations
b = a * cos(phi - z) / sin(z)
d = a * sin(phi - z) * tan(dec) / sin(z)
x = b * cos(H)
y = a * sin(H)
[*] Here are some particular cases for the inclination (z)
z = 90º ===> analemmatic dial
z = 90º + phi ===> Parent's dial (the ellipse is just a segment)
z = 45º + phi/2 ===> Foster-Lambert's dial (the ellipse is a
circumference). It's the same for 135º + phi/2
[**] Addendum: If the EW axis is 'a' then the minimal longitude of the
gnomon should be at least
L >= a * ( cos(phi) + tan(eps) * sin(phi) ) / sin(z)
where eps = 23.46º.
They're easy to undersand but not so easy to include in Helmut&Roger's
spreadsheet because the spreadsheet
has got many whistles & doodles to take care of.
Best regards,
Anselmo Perez Serrada
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