Re: Spherical Triangle

On 17 Apr., 15:31, Scott Hemphill <hemph...@xxxxxxxxxxxxx> wrote:
hans.mil...@xxxxxxxxxx writes:
On 16 Apr, 20:46, Scott Hemphill <hemph...@xxxxxxxxxxxxx> wrote:

OK, I have solved this problem manually. I confirm that there is exactly
one solution:

One solution, I agree. But the value for C?

In order to form a triangle, the length of C must be in the
range between 67 Deg 20' and 200 Deg.
|A-B| < C < A+B
If outside this range, A and C can not meet each other.

I don't agree with this. You can have spherical triangles with sides
greater than 180 degrees. For example, choose a 270 degree arc along
the equator. Now connect the endpoints with arcs to the north pole.
The arcs to the north pole are each 90 degrees long and their sum is
180 degrees, which is less than 270 degrees. This is a perfectly valid
spherical triangle. The angle at the north pole is 270 degrees, also.

Scott
--
Scott Hemphill hemph...@xxxxxxxxxxxxxxxxxx
"This isn't flying. This is falling, with style." -- Buzz Lightyear


The reason it is not necessary to have sides greater than 180, angles
greater
than 180, etc. is that ALL the properties of the larger triangle can
be
found by using the smaller triangle with the side and angle of 360 --
270 = 90.
Hence, none of the books on spherical trigonometry I have seen
consider the larger angles and sides as solutions to the spherical
triangle problems. So even though your example is a spherical
triangle,
such a triangle appears not to be needed
to solve physical problems that arise. So your second solution to
the
original problem would not be accepted in the textbooks. Sort of an
Occam's Razor situation: Do not use more than needed to solve
problems. This would just complicate things unnecessarily.
And the textbooks would have to be rewritten!

In PART 1, the failure of the left side to equal the right side of
the Gauss Analogy using the values from Hans and from Shibli's
book, shows those values are incorrect.

Charles A. Huffer

.

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