Secrets of the Swallows
a. concave up
b. concave down
c. straight line
The answer is C.
First you must know that the swallows return to San Juan Capistrano yearly at spring. The shadow tip path cast by any prominence generates a family of conic curves during the course of the year and is concave up during the winter degenerating to a straight line on the vernal and autumnal equinox and becoming concave down during the summer (the reverse is true in the southern hemisphere). At spring or the fall, the plane containing the axis of the Earth is perpendicular to its orbital plane and its radius to the Sun. At this time there is no tilt toward the Sun and the shadow tip path curve has degenerated to a straight line. The rest of the year there is a varying tilt component toward the Sun that results in a unique conic curve (hyperbolas, parabolas and ellipses) as seen in Figure 1. Figure 1 illustrates the conic curve that is generated depending upon the relationship of the declination of the Sun and the latitude of the observer or the declination of the Sun and the inclination of the receiving surface to the horizontal plane.
Figure 2. Shadow tip path geometry guide
The shadow tip path cast by the Sun resulting in a hyperbola is shown in Figure 3.
The locus of the shadow tip path of the Sun was recognized as the basis of a sun dial and calendar early in history. Additionally the properties of the shadow tip path can be used as the basis for a solar compass as well which is depicted in Figure 4. The solar compass depicted allows the observer to determine true North and the local apparent time. The solar compass is held level and rotated until the tip of the shadow, cast by the raised index, is touching the day of the year curve shown on the face of the compass enabling the observer to read the direction of true North and the local apparent time.
Figure 4. Solar Compass for 35° to 45° north latitude
30° - 40°
30° - 40°
Dorrie, Heinrich. 100 Great Problems of Elementary Mathematics Their History and Solution. New York: Dover Publications, Inc.1965
Rohr, Rene. J. Sundials, History, Theory and Practice. Toronto: University of Toronto Press, 1965