differential geometry - Understanding the formula for curvature . . . How would we motivate that when speaking of curvature of the intuitive idea of curvature (how much you need to turn) as the above equatoion? And, even after all this one issue remains for me still, we define unit tangent vector using parameterizations, so the tangent vector in itself is reliant on a property outside the curve
如何简明地解释曲率（curvature）？ - 知乎 来自本材料得图片不做另外介绍 6 1 3 Definition of Curvature 曲率 (Curvature)是衡量曲线陡峭程度的量 (quantity that measures the sharpness of a curve)，与加速度密切相关 (closely related to the acceleration)。 想象一下，你正沿着弯曲的道路驾驶汽车。
Intrinsic and Extrinsic curvature - Mathematics Stack Exchange I want to understand the basic conceptual idea about intrinsic and extrinsic curvature If we consider a plane sheet of paper (whose intrinsic curvature is zero) rolled into a cylindrical shape, th
Purpose of sectional curvature - Mathematics Stack Exchange The Riemann curvature tensor doesn't contain any more information than all sectional curvatures The only intrinsic curvature we really define is Gaussian curvature of a surface at a point
differential geometry - Meaningfulness of Curvature for Smooth . . . Yes, you are right- curvature is meaningless without Riemannian metrics or adjacent structures, which exactly introduce the idea of "curvature" 3 You are wrong in the wrong direction- not only are tangent spaces and the definition of smooth manifolds insufficient to define curvature, Riemannian metrics are too, and curvature is an even
Is there any easy way to understand the definition of Gaussian Curvature? The Gaussian curvature is the ratio of the solid angle subtended by the normal projection of a small patch divided by the area of that patch The fact that this ratio is based totally on the definition of distance within the surface (independent of the embedding of the surface; that is, bending and twisting, etc ) is Gauss' Theorema Egregium
graphing functions - Difference between Slope and Curvature . . . The curvature, on the other hand, is the inverse of the radius of the circle that best approximates the curve at that point, a k a the osculating circle What makes for the “best” approximation is given a precise mathematical definition in calculus Usually, curvature, like slope, is a signed quantity
differential geometry - Radius of Curvature when dy dx is undefined . . . @JoonasD6, the radius of curvature is a geometric invariant that can be thought of as the radius of the osculating circle at the point Alternatively, it is the length of the second derivative with respect to an arc length parametrisation