I believe I may have the answer here. Cant formally prove it yet, but it makes sense.
Returning to the most basic sundial, the equatorial, the hour angles are equal, and hours differing by 6 have hour angle lines at 90°. Inscribing any right triangle in a circle (the equatorial plane), the hypotenuse of the triangle will always be a diameter. All diameters intersect in a single point, the center of the circle. When the circular equatorial dial is projected on a non-equatorial plane, the equatorial circle becomes an ellipse. The hour angle pair chords must still intersect, even though the angles at the origin of the dial are no longer necessarily right angles. The common intersection point on the (horizontal, e.g.) dial is the projection of the center of the equatorial dial. Dave _____ From: sundial [mailto:[email protected]] On Behalf Of Willy Leenders Sent: Sunday, February 03, 2013 2:11 PM To: Sundial sundiallist Subject: I found out (graphic) a theorem In his book "Die Sonnenuhr und ihre Theorie" (The sundial and his theory) Jörg Meyer writes on page 200 (my English translation): I found the following remarkable theorem in the book of Heinz Schilt. 'Ebene Sonnenuhren' (Plane sundials) Through the point P where all the hour lines of any sundial come together, a circle is drawn. The center M and the radius of the circle are irrelevant. The hour lines whose hour angle differ from each 6 hours or 90 ° were grouped into pairs. The points where the hour lines of a pair intersect the circle, are connected with a chord (a straight line joining the ends of an arc ) Then is applicable: all these chords pass through a common point Q. In the study of this case and assuming that the projection of a circle is an ellipse and vice versa, I found out (graphic) the theorem: Of all right triangles, inscribed in an ellipse, of which the right angle point (point P) is common, the hypotenuses intersect in the same point (point S). See drawing. I found that theorem never formulated, certainly no proof. Who can prove this theorem? Willy Leenders Hasselt in Flanders (Belgium) Visit my website about the sundials in the province of Limburg (Flanders) with a section 'worth knowing about sundials' (mostly in Dutch): http://www.wijzerweb.be
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