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From: Roger Stafford on 25 May 2010 01:38
Walter Roberson <roberson(a)hushmail.com> wrote in message <VRHKn.21398$mi.9382(a)newsfe01.iad>...
> ........
> There are also solutions for arbitrary phi. They are, however, more than
> 1000 lines long.
- - - - - - - - -
How do you mean, Walter? Which of the five angles are to be arbitrarily fixed and which variable with this 1000 line solution?

Roger Stafford
From: Walter Roberson on 25 May 2010 02:15
Roger Stafford wrote:
> Walter Roberson <roberson(a)hushmail.com> wrote in message
> <VRHKn.21398$mi.9382(a)newsfe01.iad>...
>> ........
>> There are also solutions for arbitrary phi. They are, however, more
>> than 1000 lines long.
> - - - - - - - - -
> How do you mean, Walter? Which of the five angles are to be
> arbitrarily fixed and which variable with this 1000 line solution?

In the original posting, the poster indicated that solving for phi and
beta was required, and that's what I fed into Maple. The poster followed
up indicating it was psi and beta to be solved for, but since you had
posted a much better solution, I didn't bother to retry the solution.

The solution involved converting to exp and simplify()'ing that, which
spits out a bunch of lines of the form

- 1/8 cos(-delta - phi + beta + psi)

- 1/16 cos(-delta + phi + beta - psi + omega)

with sign changes for the various variables.

When I do the same thing for psi and beta instead, the solution is a
list of arctans for psi and arctans for beta, with some fairly long
expressions for some of them. One of the solutions has beta as a free
parameter rather than an arctan, and the expression for psi is by far
the longest for that possibility. The major component of it is the root
of a cubic... pages and pages of it.
From: Roger Stafford on 25 May 2010 03:44
Walter Roberson <roberson(a)hushmail.com> wrote in message <vWJKn.12303$7d5.1931(a)newsfe17.iad>...
> In the original posting, the poster indicated that solving for phi and
> beta was required, and that's what I fed into Maple. The poster followed
> up indicating it was psi and beta to be solved for, but since you had
> posted a much better solution, I didn't bother to retry the solution.
>
> The solution involved converting to exp and simplify()'ing that, which
> spits out a bunch of lines of the form
>
> - 1/8 cos(-delta - phi + beta + psi)
>
> - 1/16 cos(-delta + phi + beta - psi + omega)
>
> with sign changes for the various variables.
>
> When I do the same thing for psi and beta instead, the solution is a
> list of arctans for psi and arctans for beta, with some fairly long
> expressions for some of them. One of the solutions has beta as a free
> parameter rather than an arctan, and the expression for psi is by far
> the longest for that possibility. The major component of it is the root
> of a cubic... pages and pages of it.
- - - - - - - - -
You're right, Walter. Adjusting phi and beta is proving to be much more difficult than for psi and beta. I'll see if my brain works better tomorrow morning.

Roger Stafford
From: Roger Stafford on 25 May 2010 14:14
Walter Roberson <roberson(a)hushmail.com> wrote in message <vWJKn.12303$7d5.1931(a)newsfe17.iad>...
> Roger Stafford wrote:
> > Walter Roberson <roberson(a)hushmail.com> wrote in message
> > <VRHKn.21398$mi.9382(a)newsfe01.iad>...
> >> ........
> >> There are also solutions for arbitrary phi. They are, however, more
> >> than 1000 lines long.
> > - - - - - - - - -
> > How do you mean, Walter? Which of the five angles are to be
> > arbitrarily fixed and which variable with this 1000 line solution?
>
> In the original posting, the poster indicated that solving for phi and
> beta was required, and that's what I fed into Maple. The poster followed
> up indicating it was psi and beta to be solved for, but since you had
> posted a much better solution, I didn't bother to retry the solution.
>
> The solution involved converting to exp and simplify()'ing that, which
> spits out a bunch of lines of the form
>
> - 1/8 cos(-delta - phi + beta + psi)
>
> - 1/16 cos(-delta + phi + beta - psi + omega)
>
> with sign changes for the various variables.
>
> When I do the same thing for psi and beta instead, the solution is a
> list of arctans for psi and arctans for beta, with some fairly long
> expressions for some of them. One of the solutions has beta as a free
> parameter rather than an arctan, and the expression for psi is by far
> the longest for that possibility. The major component of it is the root
> of a cubic... pages and pages of it.
- - - - - - -
Walter, this morning I continued to have trouble showing that a phi and beta solution could always be found, given any delta, psi, and omega, but finally I stumbled onto a counterexample! If you set delta = 20 degrees, omega = 240 degrees, and psi = 175 degrees, then no matter what phi and beta are, the argument, x, of acosd above never climbs above about .65, so that no solution is possible.

To show this, let

A = sind(delta);
B = -sind(delta)*cosd(psi);
C = cosd(delta)*cosd(omega);
D = cosd(delta)*cosd(psi)*cosd(omega);
E = cosd(delta)*sind(psi)*sind(omega);

For each value of phi define

S = A*sind(phi)+C*cosd(phi);
T = B*cosd(phi)+D*sind(phi)+E;

Then the argument x as a function of phi and beta can be expressed as

x = S*cosd(beta)+T*sind(beta);

For each possible phi, if beta is set to 180/pi*atan2(T,S), the value of x for that fixed phi will be maximized at sqrt(S^2+T^2). This lets us plot this maximized x as a function of phi, and it is clear looking at the plot that it never gets anywhere near 1.

I think Maple must surely have slipped a cog somewhere in its 1000 lines. I don't see any hole in the above reasoning.

Roger Stafford
From: Walter Roberson on 25 May 2010 15:14
Roger Stafford wrote:
> this morning I continued to have trouble showing
> that a phi and beta solution could always be found, given any delta,
> psi, and omega, but finally I stumbled onto a counterexample! If you
> set delta = 20 degrees, omega = 240 degrees, and psi = 175 degrees, then
> no matter what phi and beta are, the argument, x, of acosd above never
> climbs above about .65, so that no solution is possible.

> I think Maple must surely have slipped a cog somewhere in its 1000
> lines. I don't see any hole in the above reasoning.

Maple doesn't assume that solutions are to be confined to real numbers :)


I will re-run the solution a bit later and see what answer maple comes up with.
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