Roger, The curve (of the gnomon of the Appingedam-sundial) is NOT a catenary curve.
For the catenary curve there is an equation independent from the mass per unit length of the cabel, the acceleration due to gravity and the tension in the cable. It is: y = a * cosh (x/a) or y = a/2 * (e^(x/a) + e^(-x/a)) where a = the distance between the x-axis and the lowest point of the catenary. Willy Leenders Hasselt, Flanders in Belgium Roger Bailey wrote: > Is this curve also described as a catenary curve, the hyperbolic cosine > function which defines the shape of a uniform cable hanging between two > points? > > My axes are different but a catenary is usually described by y = (H/?g) Cosh > (?gx/H)+C where ? is the mass per unit length of the cable, g the > acceleration due to gravity and H the tension in the cable and C a constant. > > Roger Bailey > > -----Original Message----- > From: [EMAIL PROTECTED] > [mailto:[EMAIL PROTECTED] Behalf Of Willy Leenders > Sent: September 12, 2002 4:24 AM > To: [email protected] > 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 > > > > > > > > > > > > > > > > > > > > - > > > > > > > > > > - > > > > > > > - > > > > - > > - > > - -
