Meusnier, Monge and Dupin - Differential Geometry - Lecture 1

Explore the mathematical contributions of French differential geometer J. Meusnier in this 49-minute lecture on differential geometry. Delve into Meusnier's investigations of lines of curvature and his famous result on computing sectional curvature of surfaces cut by non-normal planes, presented using Rational Trigonometry. Examine concepts like elliptic, hyperbolic, and parabolic points, as well as umbilic points. Learn about Euler's work on normal curvatures and their maximum values in two perpendicular directions. Investigate asymptotic directions, centers of curvature, and principal curvatures. Study an ellipsoid with unequal axes as an example, and review the properties of parabolas. Gain insights into the mathematical foundations of differential geometry through this in-depth exploration of Meusnier's work and related concepts.

Introduction
J.B Meusnier 1754-1793
Euler - normal curvatures had a maximum value in two per .Directions
Asymptotic directions
Center of curvature
One of principal curvatures
Umbilic point
An Ellipsoid with unequal axes
Proof- Assumption
Recall: Parabola