differential geometry - Understanding the formula for curvature . . . In the post they write, "However, we don't want differences in the rate at which we move along the curve to influence the value of curvature since it is a statement about the geometry of the curve itself and not the time-dependent trajectory of whatever particle happens to be traversing it For this reason, curvature requires differentiating T (t) with respect to arc length, S (t), instead of
calculus - Why is the radius of curvature = 1 (curvature . . . @RockyRock considering curvature was defined like that (definition in my textbook), a problem arises because radius of curvature is the radius of an imaginary circle of which the arc of the curve is a part of, and it seems that radius of curvature is a more basic property
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
Deriving curvature formula - Mathematics Stack Exchange What are you taking as your definition of curvature? Typically it is defined as the magnitude of the derivative of the unit tangent vector with respect to arc length, right?
geometry - What does the definition of curvature mean? - Mathematics . . . 3 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
differential geometry - What the curvature $2$-form really represents . . . We call the curvature 2 2 -form then the differential form Ω = Dω Ω = D ω where ω ω is the connection 1 1 -form Although the definition is perfectly clear I can't understand what this object 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
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
Relation between Curvature and Radius of Curvature [duplicate] The radius of curvature is the radius of the osculating circle, the radius of a circle having the same curvature as a given curve and a point So the inverse relationship of a circle's curvature to its radius transfers to arbitrary curves