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The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number. For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more ...
The four-vertex theorem was first proved for convex curves (i.e. curves with strictly positive curvature) in 1909 by Syamadas Mukhopadhyaya. [8] His proof utilizes the fact that a point on the curve is an extremum of the curvature function if and only if the osculating circle at that point has fourth-order contact with the curve; in general the osculating circle has only third-order contact ...
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by different amounts in different directions at that point.
For most points on most “smooth” surfaces, different normal sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures, call these κ 1, κ 2. The Gaussian curvature is the product of the two principal curvatures Κ = κ 1 κ 2.
Inflection points in differential geometry are the points of the curve where the curvature changes its sign. [2] [3] For example, the graph of the differentiable function has an inflection point at (x, f(x)) if and only if its first derivative f' has an isolated extremum at x. (this is not the same as saying that f has an extremum).
The selection of lane width affects the safety, maximum capacity, and cost, of a highway. Safety is best at a width of 3.0 to 3.1 metres (9.8 to 10.2 ft) in urban settings, where both narrow (less than 2.8 metres (9 ft 2 in)) and wide (over 3.1 metres (10 ft)) lanes have higher crash risks.
In the geometry of plane curves, a vertex is a point of where the first derivative of curvature is zero. [1] This is typically a local maximum or minimum of curvature, [2] and some authors define a vertex to be more specifically a local extremum of curvature. [3]
Dubins proved his result in 1957. In 1974 Harold H. Johnson proved Dubins' result by applying Pontryagin's maximum principle. [4] In particular, Harold H. Johnson presented necessary and sufficient conditions for a plane curve, which has bounded piecewise continuous curvature and prescribed initial and terminal points and directions, to have minimal length.