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Geometry plays an important role in phased array performance. There is thus an opportunity to improve array performance by including geometry as a design variable. This is the core idea in the emerging field of shape-changing phased arrays. Therefore, formal analytic tools are needed for assessing the performance of different geometries. This work establishes a solid theoretical foundation for a commonly used analysis technique, aperture projection analysis. Using this tool, the fundamental trade-offs between gain and steering range for common array geometries are explored. Analytic formulas of the maximum gain are established and numeric calculations directly compare the relationship between maximum gain and steering range. It is demonstrated that planar arrays have the highest gain, that spherical and cylindrical arrays have the highest steering range, and conic arrays offer the best compromise.