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We introduce a class of path optimization problems, which we call "sweeping path problems," found in a wide range of engineering applications. The question is how to find a family of curve segments on a free-form surface that optimizes a certain objective or a cost while respecting specified constraints. For example, when machining a free-form surface, we must ensure that the surface can be machined or swept as quickly as possible while respecting a given geometric tolerance, and while satisfying the speed and the acceleration limits of the motors. The basic requirement of engineering tasks of this type is to "visit" or "cover" an entire area, whereas conventional optimal control theory is largely about point-to-point control. Standard ordinary differential equation-based Lagrangian description formulations are not suitable for expressing or managing optimization problems of this type. We introduce a framework using an Eulerian description method, which leads to partial differential equations. We show that the basic requirement is expressed naturally in this formulation. After defining the problem, we show the connection between the two perspectives. Using this reasoning, we develop the necessary conditions for the optimality of the problem. Finally, we discuss computational approaches for solving the problem.