In this paper it is provided a new, the first one as far as the authors knowledge, parametrization of the I-V curve associated to the photovoltaic (PV) single-diode model (SDM), which is the most common model in the literature to analyze the behavior of a PV panel. The SDM relates, through a transcendental equation with five parameters to be determined, the voltage with the current. There are many methodologies to extract the SDM parameters and, some of them, are based on obtaining the best fit of a voltage-current data through the ordinary least squares method, however, the fact that errors affect not only the current but also the voltage indicates that the maximum likelihood estimation (MLE) of the parameters is obtained by the total least squares method, also called orthogonal distance regression (ODR). The main difficulty in performing ODR lies in obtaining the Euclidean distance from a point to the I-V curve which is, in general, a hard mathematical problem but, in our particular case, it is noticeably more difficult due to the implicit nature of the SDM equation and the fact that solution candidates might not be unique.  The new parametrization will allow to reduce the calculus of the Euclidean distance from any point to the I-V curve to solve a single-variable equation. An in-depth mathematical analysis will determine the number of possible candidates where the Euclidean distance can be attained. Moreover, a full casuistry together with a geometrical study based on the curvature of the I-V curve and the Maximum Curvature Point, will allow to locate and classify all these candidates, enabling to develop the first complete algorithm able to compute the Euclidean distance from a point to an I-V curve in any condition and, as a consequence, to perform a reliable ODR to obtain the MLE of the SDM parameters. Using the obtained theoretical background, it will be demonstrated that two existing methodologies to compute the Euclidean distance fail in some conditions. It will be shown that the proposed algorithm is even faster than the previous methodologies despite it needs to verify with casuistry the right selection of the solution.