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Continuous thin-plate deformable mirrors with discrete actuators are widely used as adaptive mirrors. The performance of a thin-plate deformable mirror can be characterized by the in uence functions and the layout of the actuators. This paper rst derives equations that model the in uence functions of thin-plate deformable mirrors based on an analytic calculation and a nite- element analysis; then, it presents a performance analysis for the cases of triangular, rectangular, and hexagonal patterned-actuators. The results from this study may be used for rst-order design and analysis of thin-plate deformable mirrors.