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The extension of Rao, Wilton and Glisson basis functions on flat-faceted triangular elements to different element shapes, higher-order geometry and higher-order function support is outlined. A curvilinear coordinate formulation is used to obtain a family of finite elements for more accurate method of moments (MoM) computations. Results for the first two orders of geometry and function support in 1, 2 and 3 dimensions are presented. The basis functions are hierarchical, in that mixed orders of geometry and of function support can be used together in a single calculation to allow efficient local refinement. Practical issues of element assembly, treatment of singular and non-singular MoM integrals and of loop basis functions are addressed.