You are right because all the following analyticals equations are the same
x = (1 - z) * ((1 - z^2)^0.5) / z
x^2 = (1 - 2z + 2z^3 - z^4) / z^2.
x^2=[(1-z)^3 * (1+z)]/z^2
z^4 - 2z^3 +x^2*z^2+2z-1=0
Kind regards, Jose Luis
----- Original Message -----
From: Willy Leenders <[EMAIL PROTECTED]>
To: <[email protected]>
Sent: Thursday, September 12, 2002 12:23 PM
Subject: Re: On bifilar polar sundial
> Dear Frans, Jose, Anselmo and all,
>
> There is an other analytical equation for the curved gnomon of the
> Appingedam-sundial::
>
> x = (1 - z) * ((1 - z^2)^0.5) / z
>
> Willy Leenders
> Hasselt, Flanders in Belgium
>
>
>
> "Frans W. Maes" wrote:
>
> > You are right, Jose. Thanks for checking!
> > Frans
> >
> > ----- Original Message -----
> > From: "Jose Luis Diaz" <[EMAIL PROTECTED]>
> > To: <[email protected]>
> > Sent: Tuesday, September 10, 2002 2:10 PM
> > Subject: RE: On bifilar polar sundial
> >
> > 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: <[email protected]>
> > 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 Pérez Serrada" <[EMAIL PROTECTED]>
> > > To: <[email protected]>
> > > 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
> > > >
> > > >
> > > >
> > > >
> > > > -
> > > >
> > >
> > > -
> > >
> >
> > -
> >
> > -
>
> -
>
-
