Lecture 10—Smooth Curves

After spending a great deal of time understanding some basic algebraic and analytic tools (exterior algebra and exterior calculus), we’ll finally start talking about geometry in earnest, starting with smooth plane and space curves. Even low-dimensional geometry like curves reveal a lot of the phenomena that arise when studying curved manifolds in general. Our main result for this lecture is the fundamental theorem of space curves, which reveals that (loosely speaking) a curve is entirely determined by its curvatures. Descriptions of geometry in terms of “auxiliary” quantities such as curvature play an important role in computation, since different algorithms may be easier or harder to formulate depending on the quantities or variables used to represent the geometry. Next lecture, for instance, we’ll see some examples of algorithms for curvature flow, which naturally play well with representations based on curvature!

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