The concept of curvature sensing is reviewed, and a comprehensive derivation of the curvature polynomials is given, whose inner products with the wavefront curvature data yield the Zernike aberration coefficients of an aberrated circular wavefront. The data consist of the Laplacian of the wavefront across its interior and its outward normal slope at its circular boundary. However, we show that the radial part of the curvature polynomials and their slopes at the boundary of the wavefront have a value of zero, except when the angular frequency of the corresponding Zernike polynomial is equal to its radial degree. As a result, the effect of noise on the corresponding Zernike coefficients is lower because the noisy data at the boundary of the wavefront is not used to determine their values. The use of the curvature polynomials to determine the Zernike coefficients is demonstrated with simulated noisy curvature data of an aberration function consisting of 10 Zernike coefficients, namely defocus, primary, secondary, and tertiary astigmatism, coma, and spherical aberrations.