This paper derives closed-form orthonormal polynomials over noncircular apertures using the Gram-Schmidt orthogonalization process. Isometric plots, interferograms, and point-spread functions are illustrated. Their use in wavefront analysis is discussed.In a recent paper 1 , we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. In this paper, we extend our work to elliptical, rectangular, and square pupils. Using the Zernike circle polynomials as the basis functions for the Gram-Schmidt orthogonalization process, we derive closed-form polynomials that are orthonormal over such pupils. The polynomials are given in both the polar and Cartesian forms. The polynomials are illustrated by their isometric plots, interferograms, and point-spread functions. Relationship between the orthonormal coefficients and the corresponding Zernike coefficients for a given pupil are also obtained. The error when Zernike circle polynomials are used in noncircular pupils is analyzed in a separate paper. 2 References 1. V. N. Mahajan and G.-m. Dai, "Orthonormal polynomials for hexagonal pupils," accepted by Opt. Lett. (2006). 2. G.-m. Dai and V. N. Mahajan, "Orthonormal polynomials in wavefront analysis: error analysis,"