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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.

Thank you.

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1 Answer

A couple of vague pieces of advice ...

(1) Try to work with objects that have the desired constraints built into them from the start. So, in your spline example, work with splines that are guaranteed to be C2. Don't work with more general splines and try to constrain them to be C2. Or, saying this another way -- don't introduce extraneous variables that you will then have to lock down by using constraints.

(2) Try to work with linear systems, because then it's much easier to decide whether the constraints are sufficient. Take your spline studies as an example, again. Many of the constraints needed in editing spline are linear -- constraint equations involving derivatives often are, for example.

Broad sweeping generalizations, unfortunately. But, you asked a broad sweeping question.

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