john hoy's question on the type of curve defining the limit of useful area on a Capuchin dial is excellent. I agree with the other opinions that it is a hyperbola but I do not see an obvious proof. But the question did introduce the general topic of lines that define useful areas on dials.
The usual lines of this type are declination lines that show the path of the gnomon shadow tip. Although it is not obvious from the trigonometric equations for altitude and azimuth that usually define these lines, they are hyperbolae. They curve towards the sun in the summer and away in the winter. To be more accurate, it depends on depends on whether the signs of the declination and latitude are the same or contrary. The equinox is the exception. When the solar declination is zero, the declination line is a straight line from due east to west. This is true for all latitudes, any plane, everywhere in the world. Except perhaps at the poles, where is the problem of marking the end of an infinitely long shadow. Prove it yourself today or tomorrow. The vernal equinox is on 21 March at 1:46 AM GMT. Find a convenient gnomon. Anything casting a shadow will do. Mark the tip of the shadow at various times through the day. On a horizontal surface these points will define a straight east west line. Roger Bailey Walking Shadow Designs N 51 W 115 In the heart of the Rocky Mountains where the first day of spring is truly an exceptional day.
