On Tue, Dec 9, 2014 at 10:56 AM, Michael Torrie <[email protected]> wrote: > What I'm wondering is, given a > particular shape of double-napped cone (cone's angle is > 180-2*sun_relative_declination, where sun_relative_declination is > between 0 and 23.45 degrees), the distance between the points of the > cones and the plane (the height the gnomon shadow-caster), and the angle > of the plane through the cones (latitude), can I calculate the formula > of the resulting hyperbola?
I am not an math expert by any means, but the problem seemed interesting enough to look in to. So you may already know all this, but typing it up was educational for me and maybe you'll find it useful as well. I ran across a few diagrams that had hyperbolas that weren't sliced parallel to the cone's axis, but they remain symmetrical even when they're not sliced parallel, so it's not necessarily obvious from the appearance of the hyperbola whether it was sliced parallel or not. The angle at which you slice the cone is what determines "eccentricity", which is what changes the section from a circle (sliced exactly perpendicular to the cone's axis) to an ellipse (sliced somewhere between perpendicular to the cone's axis and the angle of the cone itself) to a parabola (parallel to the cone's angle) and finally to a hyperbola (anywhere beyond parallel to the cone's axis). The equations for eccentricity give a circle an eccentricity of 0, ellipses an eccentricity between 0 and 1, parabolas exactly 1, and hyperbolas greater than 1. See http://en.wikipedia.org/wiki/Eccentricity_%28mathematics%29 for more detail. Another bit that might be helpful if you want to understand the equations is this page: http://xahlee.info/SpecialPlaneCurves_dir/ConicSections_dir/conicSections.html and especially the section on "Sections of a cone and Dandelin Sphere" which shows a fairly intuitive way to understand how the cone along with the eccentricity (and linear eccentricity in the case of ellipses and hyperbolas) of the cutting plane relate to the typical conic parameters. If you haven't seen this site yet, it might be helpful: http://wvaughan.org/sundials.html It includes parametric equations for the conic sections of a gnomonic-projection sundial, and an interesting derivation of how they work based on an idealized spherical sundial. I found it very enlightening to view the model of a sundial as a spherical model with a point shadow-caster along with a projection of the spherical model onto a plane. --Levi /* PLUG: http://plug.org, #utah on irc.freenode.net Unsubscribe: http://plug.org/mailman/options/plug Don't fear the penguin. */
