This paper presents an algorithm for segmenting and measuring retinal vessels, by growing a "Ribbon of Twins" active contour model, which uses two pairs of contours to capture each vessel edge, while maintaining width consistency. The algorithm is initialized using a generalized morphological order filter to identify approximate vessels centerlines. Once the vessel segments are identified the network topology is determined using an implicit neural cost function to resolve junction configurations. The algorithm is robust, and can accurately locate vessel edges under difficult conditions, including noisy blurred edges, closely parallel vessels, light reflex phenomena, and very fine vessels. It yields precise vessel width measurements, with subpixel average width errors. We compare the algorithm with several benchmarks from the literature, demonstrating higher segmentation sensitivity and more accurate width measurement.