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Euclid's Parallel Postulate and the Origins of Hyperbolic Geometry in Differential Geometry Implicit and Parametric Definitions of a Curve in Differential Geometry Curves, Tangent Vectors, and Curvature: Fundamental Concepts in Differential Geometry Coordinate Systems in Differential Geometry Describing Curves with Implicit Equations and Level Sets in Differential Geometry Calculating the Speed of a Curve in Differential Geometry Velocity Vectors and Tangent Lines of the Astroid Curve in Differential Geometry Parameterizing Curves with Circles and Parabolas in Differential Geometry