{'fraction_non_alphanumeric': 0.02268041237113402, 'fraction_numerical': 0.010309278350515464, 'mean_word_length': 5.657534246575342, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The Projective Linear Transition Map for Constructing Smooth Surfaces

We exhibit the essentially unique projective linear (rational linear) reparameterization for constructing C^s surfaces of genus g>0. Conversely, for quadrilaterals and isolated vertices of valence 8, we show constructively for s=1,2 that this map yields a projective linear spline space for surfaces of genus greater or equal to 1. This establishes the reparametrization to be the simplest possible transition map.