In this research, Zernike polynomials for a unit annular elliptical aperture (ellipse inscribed by a unit circle of unit radius obscured by elliptical obscuration) have been studied in Cartesian coordinates and in polar coordinates. These polynomials have been shown to form a complete basis orthogonal on a unit annular ellipse aperture, and they represent balanced classical aberrations just as Zernike circular polynomials in a unit circle.