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An important class of two-dimensional (2D) cloaks with polygonal geometries is presented. Explicit expressions of complex medium parameters are derived based on the direct transformation in the Cartesian coordinates. The generalisation of the transformation to design 2D cloaks with arbitrary geometries has also been presented. The medium parameters are then analysed in their eigen basis, where their physical properties can be described more clearly. The electromagnetic behaviours of the designed cloaks with irregular shapes are studied based on the finite-element method (FEM) combined with the Huygen's principle. The methods and results greatly improve the flexibility of the cloak design.