Hi Bill, This may not be the solution that you are looking for but it is one that simplifies matters. You did not specify which angles are the right angles. It looks like the slider creates the right angles but a simpler solution has a right angle at the gnomon at the top end of a. This creates a plane at right angles to the gnomon. Angle C is now the time angle t, measured from the vertical noon plane = 0 and increasing 15 degrees per hour of time. Angle A from the gnomon to the plane is the latitude if the plane is horizontal. Angle B is the hour angle for a dial on that plane. Besides solving for the lengths you wanted, we can easily prove the horizontal dial design equation Tan B = Tan C x Sin A.
With a right angle at the gnomon, b/c = Tan C where C is the time angle. Define length e on the middle rod from the intersection to the point where length a intersects. a/e = Sin A and a/c = Tan A. This gives the required length c in terms of length a where c = a / Tan A. Similarly b/e = Tan B and b/d = Sin B. Length d = b / Sin B. Since b=a Tan C, this simplifies for length d in terms of a as d = a Tan C / Sin B >From the above, Tan B = b/e = a Tan C /(a/Sin A), so Tan B = Tan C x Sin A. QED. Roger Bailey N 511 W 115 At 02:21 PM 12/16/99 -0500, Debra Lopez & William Gottesman wrote: >12/16 >I have been trying, on and off, for the past year to create a really nifty >algorithm to easily achieve perfect alignment of a sundial with the earth's >axis, just by comparing 3 time readings of a slighly misaligned dial with >your watch. I have never seen this done elsewhere. I won't bother you all >with details, except that I am fairly well stymied by one component of the >construction. Originally it was a problem in shperical trig, but I have >simplified it to planar trig, with the hope that I can then translate the >solution back to spherical trig. The attached .gif will present the >problem, for any takers out there! If any one wants the problem in it's >original spherical format, let me know. > >Bill Gottesman >Burlington, VT > >Attachment Converted: "c:\eudora\attach\trig prob.gif" >
