Well I don''t know what the hell danz''s function is supposed to do. I think he made some assumptions about your problem that are not correct, and so his answer doesn''t really make sense geometrically.

I''ll try to give you something that will work. Lets first clarify your definitions. You cannot merely say that "a" is the angle between the x and y axes. It must be measured relative to something, so lets say a is measured from the positive x axis, within the xy plane. Also, since your force vector is a full 3D vector, I''m going to say that a is the angle of the force vector as projected into the xy plane.

We''ll clarify b in the same way. Lets say b is measured from the positive x axis, and lies within the xz plane. b is the angle of the projection of force into the xz plane.

I think my clarification is probably close to what you really want. This problem setup is not really the best (because finding x, y, z is more complex than it could be), but it is something we can work with. The final answer is fairly simple, but involves an atan2() calculation.

I provide below a DERIVATION of the equations, then a SOLUTION to your problem. I''ll caveat this by saying I haven''t actually tested my derivation or calcs. But they should be correct unless I made a careless algebra error.

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DERIVATION
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If length is the magnitude of force, then we can say the following:

(Eq. 1) x = length*cos(c)*cos(a);
(Eq. 2) y = length*cos(c)*sin(a);
(Eq. 3) z = length*sin(c);

Here, I''ve introduced a new angle, c, which is the angle between the force vector and the xy plane, measured from the xy plane. If we knew c, then we could immediately calculate (x,y,z). But we don''t know c. Note that c is not the same as b! How do we find c? By using what we know about b and the xz plane:

(Eq. 4) x = length*cos(d)*cos(b);
(Eq. 5) z = length*cos(d)*sin(b);
(Eq. 6) y = length*sin(d);

Whoa, yet another angle, d! This is the angle between the force vector and the xz plane, measured from the xz plane. Now we have 5 unknowns: (x,y,z), c, and d. But we also have 5 equations, so we can solve them for c, d, and (x,y,z). Its quite easy to do. We take the two equations (1 and 4) for x and set them equal:

x = length*cos(c)*cos(a) = length*cos(d)*cos(b)

Divide out "length" to get:

(Eq. 7) cos(c)*cos(a) = cos(d)*cos(b)

We know a and, thus cos(a) and cos(b). We can convert this into an equation we can solve for either c or d. To do this, equate the two z equations (equation 3 and 6):

z = length*sin(c) = length*cos(d)*sin(b)

Or, simplified:

(Eq. 8) sin(c) = cos(d)*sin(b)

Solve equation 8 for cos(d):

cos(d) = sin(c)/sin(b)

and plug that into equation 7 to get:

(Eq. 9) cos(c)*cos(a) = sin(c)*cos(b)/sin(b)

Divide by cos(c) and multiply by sin(b)/cos(b) to get:

(Eq. 10) tan(c) = sin(b)*cos(a)/cos(b);

Since you know b and a, you can calculate tan(b) and cos(a), and thus you can calculate tan(c). You could just to c = atan(tan(b)*cos(a)), but that''s not advisable. You have to be careful calculating tan(c). Its best to use atan2() instead of atan(). I''ll just mention that cos(a) is a scaling factor that can go in the numerator or denominator. Lets pick numerator (to avoid a divide), so:

(Eq. 11) c = atan2(sin(b)*cos(a), cos(b));

Actually, the following equation gives the same result:

(Eq. 12) c = atan2(sin(b), cos(b)/cos(a));

So, how do you use all of this?

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SOLUTION
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STEP 1: Calculate the angle c itself using either equation 11 or 12.
STEP 2: Calculate (x,y,z) using equations 1, 2, and 3.
STEP 3: You''re done!

That should work for any values of a and b, without concern or problem when they are 90 degrees.

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.