If you were to for example call the arm between joints 2 and three 2_3 and you wanted joint 3 at the xyz coordinate of (2,4,-8) in relation to joint 2, you would need to compute the inverse tangent to find the angle. To do so you need a right triangle with one angle at joint 2 and the other joint 3. Because this triangle is not aligned with the axis's you will need to break it down into 2 other triangles, one for y and the other for xz with their hypotenuses forming the legs of your final triangle. so if you solve for the Pythagorean theorem for x and z you get 8.246. Then because the xz leg is the opposite of the angle your solving for, you put it above the y leg which is the adjacent. So you would solve for the inverse tangent of 8.246/4 which equals 64.1 degrees. Now that is only part of the solution as you still need to find the rotation degree which is defined by x and z. Solving the inverse tangent for z over x yields a result of -75.96 degrees which is in the clockwise direction. So your final movement needs to be a clockwise rotation of 75.96 degrees and the arm needs to be bent to an angle of 64.1 degrees. The full formula for the bending angle is tan(-1)( sqrt((x)^2 + (z)^2) / y ) and the formula for rotation is tan(-1)( z / x ) All of this calculated with my trusty Prizm

Well, say the base frame is (0,0,0) and the tool frame is (2,4,-8), each of the articulations is, say, 20cm, and that's all I know (I don't care of XYZ positions of the angles in between), there's some way to calculate the 4 angles?

Hmm that's far beyond my own knowledge of physics. It seems like either way you will have to calculate the xyz coordinate of each joint and have them all add up to your destination. You know that the points joint 3 for example could be forms a hollow sphere and each point upon that sphere is the center of another hollow sphere with each of those points being a possible position of joint 4 and the total shape is now a solid sphere with a diameter of 40 cm. How to calculate the individual angles though is something I can't figure out.

Bump. I guess I'll just download a C++ library for that.

I guess that you have to take 2 fixed points so you can simply solve the last one.
but obviously, you have a looooot of solutions for a given (x,y) point.

Update: Forget inverse kinematics, I don't need it anymore. My final project will involve controlling the arm with a Wiimote, which doesn't need any XYZ coordinate. It also have the side effect to be rather awesome.

Yeah I don't need help either, just telling you guys it'll be awesome ^_^ I'll try to record videos.