We prove that when a line is approximated using digital line, the error in the slope of the digital line has a definite upper bound and is strongly dependent on the two pixels chosen for defining the digital line. Thus, an analytical expression of the maximum deviation of the pixels from the digital line can be derived. Using this, the conventional line fitting methods that use maximum tolerable deviation as the optimization goal can be made control-parameter independent. This error bound can be used to make the most recent and sophisticated line fitting methods parameter independent and more robust to digitization noises. In our knowledge, this is the first line fitting method completely devoid of any control parameter. Such control-parameter independent line fitting algorithm retains the characteristics of the digital curve with sufficient reliability and precision and provides good dimensionality reduction in representing the digital curves. Extensive results have been generated for 9 datasets comprising of about a hundred thousand images. The proposed method shows robust and repeatable performance across all the datasets with low standard deviation in the performance.