Oct 11, 2014 · 如何简明地解释曲率（curvature）？ 曲率是啥，挠率（torsion）是啥，咋来的，有啥用？ 指的是对于函数 [公式] 显示全部

Apr 23, 2015 · I mean, when we define curvature for curves on space, the curvature is meant to represent how much the curve deviates from a straight line. On the other hand, when reading books about General Relativity some time ago, I read that the curvature of the Levi-Civita connection is intended to encode the information of the difference between a ...

Sep 16, 2018 · My textbook Thomas' Calculus (14th edition) initially defines curvature as the magnitude of change of direction of tangent with respect to the arc length of the curve (|dT/ds|, where T is the tangent vector and s is the arc length) and later by intuition conclude that κ = 1/ρ (where, κ=curvature,ρ = radius).

这个事实告诉我们，可以用密切圆的曲率来定义曲线的曲率（因为格式所限，详细推导请查看此处，还是挺有意思的，综合 ...

Oct 3, 2017 · The curvature is what makes the difference between a straight line and a curve, i.e. a measure of "non-straightness". And it is intuitive that a curve of constant curvature is a circle. For curves that are not circles, the curvature must be …

Feb 19, 2021 · Then mean curvature vector is defined to be the trace of the second fundamental form. I am confused as to what does it mean for a vector to be the trace of the second fundamental form. Usually contractions/trace will yield a real-valued function on the manifold, but here we want a vector.

The idea is that this is the inverse of the "circle of best fit" to the graph. It describes how quickly the graph curves in $\mathbb{R}^2$. The Hessian describes "intrinsic" curvature. This type of curvature (to my understanding) is inherently two dimensional. In particular any one-manifold is always flat with respect to this notion of curvature.

Jan 18, 2018 · I tried finding the curvature formed by 3 points using this equation: where a, b, c stands for distance between all three line segments. This worked fine in some cases with soft curve, but failed to detect some points with very less acute angle as attached in …

I see that we are lacking a definition of radius of curvature : I want to use the most obvious definition(to me) : Distance of point from centre of curvature at that point where the centre is defined as intersection of two infinitesimally close normals. Or we could use the physics like $\frac{v^2}{a_\perp}$

Aug 11, 2020 · A way to define curvature then would be to find the "tangent circle" (if it exists) at each point, then the curvature would be the reciprocal of the radius of this "tangent circle". It turns out that the equations needed to derive the tangent circle are simplified if the tangent vector at each point of the curve has length $1$ , which is the ...