Computation of the orthogonal distance from a given point to an ellipse is the basis of orthogonal distance based ellipse fitting methods. The problem of this orthogonal distance and the corresponding orthogonal contacting point on the ellipse is investigated, and two algorithms, the exact one and the convergent iterative one, are proposed. The exact algorithm utilizes the closed form solution of quartic equations, but is numerically unstable. The iterative algorithm, however, uses Newton’s method to solve the equation, and starts from an initial solution that is proven to lead to a convergent iteration. The proposed algorithms are compared in experiments with an existing rival. Although the rival algorithm is slightly faster and more accurate in realistic scenarios, divergence is likely to occur. On the other hand, both our exact and iterative algorithms can reliably produce the solution needed. While the exact algorithm encounters numeric instability, the iterative algorithm is only slightly outperformed by the existing rival in speed and accuracy, but at the same time provides more reliable computation process, thus making it a preferable method for the task.