Show how to distribute nine points around the surface of a sphere in such a way that each point is equidistant from its four nearest neighbors.
(In reply to
re(2): Solution -- Using spherical trig by Jer)
The two big formulae of Spherical Trigonometry are quite analogous to those of plane trig:
Law of cosines:
cos(c) = cos(a)*cos(b) + sin(a)*sin(b)*cos(C)
Law of sines:
sin(a)/sin(A) = sin(b)/sin(B) = sin(c)/sin(C)
There area variations of the cosine law based on interchanging sides with angles, since all are measured in angle/arc measure.
In the current problem I used for one triangle
a=arc segment from point on equator to equator's intersection with small circle point's meridian.
b=segment along that meridian from small circle's point to the equator
C= the right angle between the above two
c= the arc distance between the equatorial pt and its nearest on small circle
For the other triangle:
a and b are the two segments from the north pole to each of two successive points on the small circle.
C is the the 120° angle at the pole between those segments
c is the distance between those pts on the small circle.
The distance here had to match the first distance.
Edited on June 4, 2021, 2:10 pm
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Posted by Charlie
on 2021-06-04 13:45:17 |
re(3): Solution -- Using spherical trig
