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Euler's Gem by David S. Richeson

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CHAPTER 21

THE TOPOLOGY OF CURVY SURFACES

If others would but reflect on mathematical truths as deeply and as continuously as I have, they would make my discoveries.—Carl Friedrich Gauss1

One of the most fundamental topics in the geometry of planar curves is curvature. The curvature at a point x is a number, k, that measures the “sharpness” of the turn at x—it measures how quickly the tangent vectors change direction. Roughly speaking, given a normal vector images to a curve at x, if the curve bends in the direction of images, then k > 0, if it bends away, then ...

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