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