Jacobian and IK
Not sure where this came from, but I received three e-mails yesterday asking ‘what is a Jacobian,’ one of which wanted to know how it was used in IK. When something is asked multiple times, I try to respond in the blog and my first response is always check Wikipedia for a basic definition. Way oversimplified, a Jacobian is a matrix of first partial derivatives. For IK, its construction allows a convenient representation of the change in end-effector position given a change in joint angles. Practical application is considerably more complex. You can get a good introduction here.
If your goal is to implement IK in Flash (perhaps Papervision), I would not use this approach. For most humanoid (bipedal) characters, a good limb solver will go a long way and you are better off thinking about multi-chain solvers. RFK or CCD are good (and much simpler) alternatives and there is a reasonable starting point in Peters’ Making Things Move book.
Singularity also has a set of rigging classes (in-plane solvers), including a limb solver with joint constraints. Even if your goal is IK in PV3D, working with in-plane solvers is a good starting point. My FITC 2007 presentation, Dynamic Skeletal Animation can be downloaded here. The presentation discusses FK and IK, and has several demos, including mixed IK-FK.
Enjoy and good luck with your IK efforts!