Intuitively, a cylinder is clearly in some sense more curved than a flat piece of paper: they have different extrinsic curvature. Note that such surfaces have the same intrinsic (Gaussian) curvature, but have very different extrinsic curvature. This is possible because these surfaces all have zero Gaussian curvature they are called " developable surfaces". It can be bent, rolled, cut, and folded (but not stretched) to form surfaces such as a cone, M öbius band, or cylinder. Curvature is split into two types: "intrinsic" and "extrinsic". To visualise the difference between intrinsic and extrinsic curvature, take a piece of paper. For a fun introduction to curvature in the real world, see this excellent Wired article. For a nice overview of the history of the study of curvature, see Michael Garman and Jessica Bonnie's paper. Intuitively, curvature describes how much an object deviates from being "flat" (or "straight" if the object is a line). Struik's account.Ĭurvature is an important notion in mathematics, studied extensively in differential geometry. So how do you find the shortest distance from one point to another? Another example: how do we characterise the difference between a cylinder, which you can make by bending a plane without stretching it, and a sphere, which you cannot? For a very readable introduction to the history of differential geometry, see D. For example, if you live on a sphere, you cannot go from one point to another by a straight line while remaining on the sphere. Differential geometry is the tool we use to understand how to adapt concepts such as the distance between two points, the angle between two crossing curves, or curvature of a plane curve, to a surface.
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