Ruby Bojorquez wrote:
> To all:
>
> Could someone please help a floundering beginner in the fascinating
> world of gnomonics? I have two questions.
>
> 1. First, what is the purpose of the horizon ring in an armillary
> sphere? and
>
> 2. How does one calculate the zodiac lines on a horizontal dial?
>
Hello,
The rings on the sphere are meant to represent the Equator,
Tropics ect...
Regarding the calculation of declination or date lines,
essentially this is an exercise of transforming a set of Polar Coordinates
into Rectangular Coordinates. First Polar Coordinates, comprise a radius
of given length and an angle, the radius in this case is the length of a
cast shadow and is calculated by the equation:
stylus_height * cot(theta) where theta is equal to the Apparent
Solar Altitude.
The first illustration of the attached image shows how this is derived
using similar triangles given.
length A = tan(theta)
length a = the stylus height
length B = 1 i.e., the unit circle axis
length b = shadow length
Whence
B/A = b/a -> 1 / tan(theta) = shadow length / stylus height
-> shadow length = stylus height *
cot(theta)
As an example lets assume that the Solar Altitude is 45deg at an
Azimuth of 135deg and the stylus height is 2.5 inches, the shadow length
would then be 2.5 * 1/tan(45) -> 2.5 *1/1 = 2.5 inches. The Polar
Coordinates are then (2.5,135).
Now that we have the radius magnitude (shadow length) and its angle
(the Azimuth) we can now transform them into their rectangular equivalent.
The x coordinate is calculated using sin(Aziumth) * shadow_length ->
sin(135) * 2.5 = 1.77 and the y coordinate is cos(135) * 2.5 = -1.77,
therefore the rectangular coordinates (in the Solar Azimuth coordinate
frame) are (1.77,-1.77) and this matches the position of the shadow in the
attached illustration. Now the final step is to multiply the Y coordinate
by -1 and the coordinates are now (1.77,1.77) and in the more standard
cartesian frame.
I suppose there are alot of ways of looking at it but this is how I
see it. A final reduction to get what you're after might then be the
following.
x = stylus_height * sin(Azi) / tan(Alt)
y = stylus_height * -cos(Azi) /tan(Alt)
To get a very accurate set of Apparent Altitude and Azimuth values
(horizontal coordinates) for a given day, try my Solar Calculator at the
URL below. The online doc is way out of date, I hope to update it soon.
http://www.groknet.com/suncalc.html
Regards,
Luke Coletti
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