I think the result is x^2 = (1 - 2z + 2z^3 - z^4) / z^2.
Kind regards,
----- Original Message -----
From: Frans W. Maes <[EMAIL PROTECTED]>
To: <sundial@rrz.uni-koeln.de>
Sent: Tuesday, September 10, 2002 9:13 AM
Subject: Re: On bifilar polar sundial
> Dear Anselmo and all,
>
> With respect to the "bifilar polar dial" in Appingedam (NL), my site does
> not specify the shape of the curved gnomon. It is certainly no ellipse,
nor
> a hyperbola. Do you like some math? Have a look at the article by Fer de
> Vries in the NASS Compendium 8 (4), in particular fig. 4.
>
> Point E (the center of the dial face, for those who don't have the
> Compendium at hand) has been taken as the origin of the coordinate system,
> EB (the east-west line) as the x-axis and EF (perpendicular to the dial
> face, intersecting the pole-style) as the z-axis. The coordinates of a
point
> Q on the curved gnomon were derived as:
> x = EC = g.tan(t) - g.sin(t), and z = CQ = g.cos(t),
> in which t is the hour angle of the sun and g the height of the pole-style
> above the dial face. Scaling x and z in units of g, the shape of the
curved
> gnomon is given by the parametric equations:
> x(t) = tan(t) - sin(t) and z(t) = cos(t).
>
> Your question actually is to convert this pair of equations into an
analytic
> expression z(x). This can be done by making the usual substitutions:
> tan(t) = sin(t) / cos(t) and cos(t) = sqrt[1-sin(t)^2],
> but it is not going to look very nice. In case you would like to probe
this
> route, it is perhaps easier to swap the axes and calculate x(z). My result
> is (please check):
> x^2 = (1 - 2z + z^2 + 2z^3 - z^4) / z^2.
> Definitely not the equation of a conic section!
>
> Some properties of the curve can be obtained from looking at the
parametric
> equations. For t->90 degrees (6 hr local time), x->infinity and z->0. The
> curve thus approaches the dial face asymptotically when moving out.
>
> In the center, the curved gnomon touches the pole-style. The slope dz/dx
of
> the curve at this point (x=0) is infinite. There are several ways to
arrive
> at this result. You love calculus, do you?
>
> 1) For t->0 degrees (local noon), x->0 and z->1, as expected for a polar
> dial. The slope of the curve at x=0 is:
> dz/dx = (dz/dt) / (dx/dt) = -sin(t) / [1/cos(t)^2 - cos(t)].
> For t=0, this unfortunately gives 0/0, an indeterminate value. According
to
> the rule of Bernoulli (or De l'Hopital) one may take the derivatives of
the
> numerator and the denominator, which at t=0 leads to 1/0, or infinite.
>
> 2) Make a Taylor series expansion of the quotient:
> x(t) = tan(t) - sin(t) = t^3/2 + (t^5)/8 + ..., and:
> z(t) = cos(t) = 1 - t^2/2 + (t^4)/24 + ...
> Hence dz/dx (t->0) = 2/(t^3), which approaches infinity for t->0.
>
> 3) Differentiate the analytic expression given above (your homework for
> today ;-).
>
> 4) The intuitive approach: if the slope were finite, the pole-style and
the
> initial part of the curved gnomon would span an inclined plane. As long as
> the sun would be below this plane, an intersection point of the two shadow
> edges would be formed, which would fall on the straight, perpendicular
date
> line. When the sun would rise above this plane, the intersection point
would
> disappear and the initial part of the curved gnomon would cast an oblique
> shadow, which is incompatible with the existence of a perpendicular date
> line. Hence, the slope should be infinite.
>
> Kind regards,
>
> Frans Maes
> 53.1 N, 6.5 E
> www.biol.rug.nl/maes/sundials/
>
> ----- Original Message -----
> From: "Anselmo PÈrez Serrada" <[EMAIL PROTECTED]>
> To: <sundial@rrz.uni-koeln.de>
> Sent: Sunday, September 08, 2002 10:42 PM
> Subject: On bifilar polar sundial
>
>
> > Hoi, Frans!
> >
> > I have been playing a bit with the equations for a bifilar dial trying
> to
> > reproduce the bifilar polar dial that I saw in your web. There you say
> > that the transversal gnomon is a piece of hyperbola, but I have found
> > that it is really a piece of ellipse (there are more solutions, but none
> > is an hyperbolic arc). I suppose my calculations are wrong but I can't
> > find the mistake in them. Can you please provide me more
> > information about this topic? Do you know the exact equation for this
> > curve?
> >
> > Hartelijke bedankt,
> >
> > Anselmo P. Serrada
> >
> >
> >
> >
> > -
> >
>
> -
>
-
