The curvature of a curve is specific to a point on the curvetangentdirectionhornyesarc lengthOfturnRate, passingdifferentialTo define and indicate the curvedeviateThe degree of straightness.A numerical value that mathematically indicates the degree of curvature of a curve at a point.
The greater the curvature, the greater the curvature of the curve.The reciprocal of curvature isRadius of curvature。
It can be obtained by using the derivative method of parameter equation
Curvature circle and radius
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The reciprocal of the curvature at point M on the curve is called the radius of curvature of the curve at this point, and is recorded as
, then
Take a point D on one side of the normal of the curve at point M, so that
, with D as the center
Make a circle for the radius.This circle is called the curvature circle of the curve at point M, and the center D is called the curvature center of the curve at point M.
Curvature circles have the following properties:
(1) The curvature circle and curve have the same tangent and curvature at point M;
(2) The concave direction adjacent to point M is the same as that of the curve;
Therefore, in practical engineering design problems, an arc near the point M of curvature circle is often used to approximate the curved arc, so as to simplify the problem.[2]
significance
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curvatureIt is a measure of the unevenness of geometry.Flattening has different meanings for different geometries.
In this paper, we consider the basic case, the curvature of curves and surfaces in Euclidean space.For general curvature, please refer toCurvature tensor。
According to the interpretation of general relativity, in the gravitational field,space-timeThe property of is determined by the distribution of the "mass" of the object, which makes the space-time property uneven and causes the curvature of space-time.Because an object with mass will bend space-time, and you can think that with speed, the object with mass will become heavier, and the curvature of space-time will be larger.
