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A Visual Intro to Curves and the Frenet Frame

Our submission for the Summer of Math Exposition 2 #some2. Topics: An introduction to the Mathematics of differential geometry of plane and space curves, leading up to the Frenet Frame, and Frenet-Serret Formulas and the Fundamental Theorem of Space Curves. Content: 0:00 Introduction, Motivation, and Applications 0:45 Overview 1:23 Circles and the Idea Behind Curvature 4:19 Definition of Curvature and Examples 6:54 Moving into the Third Dimension and the Frenet Frame 10:16 Derivation of the Frenet-Serret Equations and tau 13:51 Visualization and Conceptualization of the Frenet Frame 16:23 Frenet Frame in Popular Culture 16:56 The Remarkable Fundamental Theorem of Space Curves Prerequisites: Derivatives, Dot Product, Cross Product, Basis of a Vector Space Credits: Dan Walsh: Narration and Animations Franciscus Rebro: Concept and Script Bibliography / Works Consulted: Needham, Tristan - Visual Differential Geometry and Forms Pressley, Andrew - Elementary Differential Geometry, 2nd Ed. Tu, Loring W. - Differential Geometry Hidden Figures, 2016, Directed by Theodore Melfi