An ellipse-based geometric construction for horizontal dials


From an article in Tech Directions magazine, by Mark Schwendau, "It's High Time to Make Sundials!"

The described technique is to start by drawing two circles at a common center, the first with radius 1.0, the second with radius equal to 1/sin(latitude).
From the center, draw up to 24 equatorial hour lines, with 15 degree spacing, and length sufficient to intersect both circles. Label the vertical line's intersection with the top of the outer circle 12", and the rest in clockwise fashion. In the accompanying drawings, I used Anselmo Pérez Serrada 's suggested labelling of T' for th einner circle intersections, T" for the outer intersections, and T for the final points.

Draw horizontal lines from the intersection of the initial hour lines with the outer circle. It is convenient to draw lines connecting the 11" point with 13", 10" with 14", etc.
Draw vertical lines from the intersection of the hour lines with the inner circle, extending them to intersect the corresponding horizontal lines. Label these points of intersection 11, 13, etc. as these will be the final hour line directions.
Finally, draw new hour lines from the common center through the intersections of the horizontals and verticals. These are the hour lines of the finished dial.

The following drawings were done in DeltaCad, based upon the original article's dimensions of 1.00 and 1.3426. This was said to be 1/ sin(42 deg), for the Latitude of Chicago, Illinois. Actually, that value is correct for a latitude of 48.1 degrees.

Initial layout with equatorial hour lines
Initial layout with equatorial hour lines
 Adding horizontal construction lines
Adding horizontal construction lines
Adding vertical construction lines
Adding vertical construction lines
 Closeup of intersections and new labels



Closeup of intersections and new labels
Drawing final hour lines
Drawing final hour lines
 After removing construction lines




After removing construction lines
Anselmo Pérez Serrada 's suggested alternate construction
Anselmo Pérez Serrada 's suggested alternate construction, with inverted radii