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Cellular location-based services need to use a variety of positioning methods to meet requirements on availability, response time, and accuracy. Due to the geometrical properties of the different positioning methods, the number of standardized reporting formats is large. Because the user requirements on the reporting formats are as scattered as the geometrical properties of the positioning methods, there is a need for shape conversion algorithms that transform between different formats. This paper discusses a case that transforms between a polygon and an ellipse while preserving the confidence between the two shapes. The transformation is required for emergency positioning outside North America, and when fingerprinting, localization is used in the Global System for Mobile Communications (GSM). The algorithm has a number of technically interesting properties, e.g., with regard to the selection of the optimization criterion and the fact that the solution is computed by an analytical solution of a cubic equation. [ABSTRACT FROM PUBLISHER]