Tortuosity of retinal blood vessels is an important symptom of diabetic retinopathy or retinopathy of prematurity. In this paper, we propose an automatic image-based method for measuring single vessel and vessel network tortuosity of these vessels. Simplicity of the algorithm, low-computational burden, and an excellent matching to the clinically perceived tortuosity are the important features of the proposed algorithm. To measure tortuosity, we use curvature which is an indicator of local inflection of a curve. For curvature calculation, template disk method is a common choice and has been utilized in most of the state of the art. However, we show that this method does not possess linearity against curvature and by proposing two modifications, we improve the method. We use the basic and the modified methods to measure tortuosity on a publicly available data bank and two data banks of our own. While interpreting the results, we pursue three goals. First, to show that our algorithm is more efficient to implement than the state of the art. Second, to show that our method possesses an excellent correlation with subjective results (0.94 correlation for vessel tortuosity, 0.95 correlation for vessel network tortuosity in diabetic retinopathy, and 0.7 correlation for vessel network tortuosity in retinopathy of prematurity). Third, to show that the tortuosity perceived by an expert and curvature possess a nonlinear relation.