The study of geometry is among the first we face in our younger years. Rightly so. Any study of the physical world necessitates reflection about geometry... shape, distance, curvature. As we grow older, our formulation of geometry becomes more "analytical" and abstract. A circle is no longer a "shape", but rather a locus of ordered pairs \( (x, y) \) such that \( x^2 + y^2 = 1 \). This feels foreign at first, but it allows us to reason more generally: it is trivial to imagine the "shape" of a circle, but what is the "shape" of all possible rotations (the "3D rotation group")? and why does the answer to this first question explain the plate trick?
Here I have collected various resources about geometry, especially those applicable to the other ideas summarized on Linotype, including groups, manifolds, tangent spaces. Our modern understanding of geometry is particularly suited to geometric mechanics. Spacecraft attitude determination, perception, control, robotic manipulation, general relativity, computational geometric processing, even deep learning for logical reasoning... all of these ideas are made possible by these analytic theories of geometry.
The theory of Lie groups is fundamental to contemporary differential geometry, control, and robotics.