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Explore the concept of canonical parameterizations for two-dimensional shapes in this 24-minute conference talk presented at GSI. Delve into the mathematical foundations and practical applications of these parameterizations, gaining insights into how they can be used to represent and analyze various 2D geometric forms. Learn about the advantages of canonical representations in shape analysis, computer graphics, and related fields. Discover techniques for deriving and implementing these parameterizations, and understand their role in simplifying complex geometric problems.