Are Cartesian coordinates considered to be curvilinear coordinates?

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In the wikipedia page on curvilinear coordinates it is said:

"Well-known examples of curvilinear coordinate systems in three-dimensional Euclidean space (R^3) are Cartesian, cylindrical and spherical polar coordinates."

Is this generally accepted as correct?

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Yes. "Curvilinear" means that the coordinate lines are allowed to be more general curves, but they can also just be straight lines.