We present a method for deriving a parametric description of a conic section (quadratic curve) in an image from the moments of the image with respect to several specially-constructed kernel functions. In contrast to Hough-transform-type methods, the moment approach requires no large accumulator array. Judicious implementation allows the parameters to be determined using five multiplication operations and six addition operations per pixel. The use of moments renders the calculation robust in the presence of high-frequency noise or texture and resistant to small-scale irregularities in the edge. Our method is generalizable to more complex classes of curves with more parameters as well as to surfaces in higher dimensions.