differential geometry - Understanding the formula for curvature . . . 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
calculus - Why is the radius of curvature = 1 (curvature . . . 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)
How to know when a curve has maximum curvature and why? 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 defined locally, i e it varies from place to place
Deriving curvature formula - Mathematics Stack Exchange $\begingroup$ Sure, but ${\bf T}' = \kappa {\bf N}$ there means ${\bf T}'(s) = \kappa(s) {\bf N}(s)$ and so curvature is defined in terms of arc length It sounds to me like he is asking for a derivation divorced from arc length, but maybe I am hearing him wrong $\endgroup$
Intrinsic and Extrinsic curvature - Mathematics Stack Exchange The best way I have had it put to me is that, extrinsic curvature corresponds to everyone's layman understanding of curvature before we were ever introduced to differential geometry If the difference in dimension (or co-dimension) is greater than one, we can define multiple normal vectors to the manifold $\Sigma$, and there is a third notion
differential geometry - What the curvature $2$-form really represents . . . 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
Purpose of sectional curvature - Mathematics Stack Exchange The Riemann curvature indeed contains all information The other way around as well, you can reconstruct the Riemann curvature from the sectional curvature One problem with the Riemann curvature is that it is very abstract It contains a lot of information about the manifold and its Riemannian structure, but not always in a very
Relation between the Hessian matrix and curvature 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
geometry - What does the definition of curvature mean? - Mathematics . . . There are a number of characterizations of curvature The "most intuitive" and geometric one, in my opinion, is the inverse of the radius of curvature That is, intuitively, the radius of the circle that lies tangent to the curve in a neighborhood of the point (more formally, the osculating circle