Anselmo, Thanks for your equations. I like to add anoter special case for the inclinatin z. The result is an "ordinary" horizontal sundial.
The case is: z = phi. The gnomon than is a polestyle. b = a / zin z = a / sin phi d = 0 tan hourlineangle = y / x = a * sin H / ( b * cos H ) = sin phi * tan H This is the usual formula for a horizontal dial. And the footpoint of the gnomon always is at the same place with d = 0. Best, Fer. Fer J. de Vries mailto:[EMAIL PROTECTED] http://www.iae.nl/users/ferdv/ Eindhoven, Netherlands lat. 51:30 N long. 5:30 E ----- Original Message ----- From: "Anselmo Pérez Serrada" <[EMAIL PROTECTED]> To: "Sundial, Mailinglist" <[email protected]>; "Sonderegger Helmut" <[EMAIL PROTECTED]> Sent: Wednesday, April 17, 2002 5:15 PM Subject: Inclined gnomon alemmatics > 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 > > > > - -
