There is a class of problems that I need to investigate the constraints of the variables in the problem and build a linear system to solve it.
I find that I have no "feeling" about whether the constraints are chosen correctly and whether they can lead to a solution. The whole process of problem solving is frustrating before I solve the linear system and finally get the answer. In other words, the constraint based approach can not give me an "integral sense" of the problem.
For example, I am working on a spline curve editor and I want to keep $C^2$ continuity among the control points. I do not know whether the equations based on constraint are adequate to get a unique solution.
I understand that I can check if the number of unknowns equals the number of equations. But what if I modeled the system using more variables than it should required at the beginning?
My question is:
How can I know that I choose a correct set of variables to model a problem? And what is the condition that the set of constraints can lead to a (unique)solution? (I do not mean what is the condition that a linear system has solution. I mean how can I know a set of constraints is adequate to solve the problem, if there are redundant ones, how can I avoid them)
Finally, may I ask some problems that based on similar solving technique? I would like to practice more on these to build a better understanding.