## Introduction

In the differential geometry, the countdown of the curvature is the radius of curvature, ie r = 1 / k. The curvature of the planar curve is to define the rotation rate of the arc length for a certain point in the curve, defined by differential, indicating that the curve deviates from the straight line. For a curve, it is equal to the radius of the arc closer to the curve at this point. For surface, the radius of curvature is the radius of a circle that is suitable for normal cross sections or thereof.

The radius of curvature is mainly used to describe the degree of curve change in curve on the curve. Special as: the degree of bending degree on the circle is the same, the radius of curvature is the radius of the circle; straight line is not Bending, and the radius of the circle in this point can be arbitrarily, so the curvature is 0, so the line does not have a radius of curvature, or the radius of the curvature is

.

The larger the radius of the circle, the smaller the degree of bending, the more like a straight line. Therefore, the larger the radius of curvature, the smaller the curvature, and vice versa.

If a point equal to the curvature is found for a certain point, the radius of the curvature on the curve is the radius of the circle (note, the radius of curvature of this point) Other points have other radius of curvature). It can also be understood that it is as possible to differentiates the curve until the last approximation is a circular arc, which is the radius of the arc is the radius of curvature on the curve.

## Formula Detecting

In the case of a space curve, the radius of curvature is the length of the curvature vector. In the case of a planar curve, then R is to take an absolute value.

where S is the arc length of the fixed point on the curve, α is tangential angle, K is the curvature.

If the curve is expressed as

, the radius of curvature is (assuming curve differential)

If the curve is given by the function

and
parameter, the curvature is

If

is the parameter curve in
, the radius of curvature at each point is
is given by the following formula:

As special case, if f (t) is a function from R to R, the radius γ (T) of the figure thereof is = (t, f (t))

### semicircle

For the half round of the upper half-plane radius A:

For the half circle of the upper half-plane radius A:

radius A circle of curvature is equal to A.

### Ellipse

In the ellipse having long axis 2a and the short axis 2b, the vertices on the long axis have any point of the minimum curvature radius,

; / p>

and the vertices on the short axis have any point of the maximum radius of curvature

.

## Application

(1) For applications, see the CESàro equation;

(2) for the radius of curvature of the Earth (approximated by elliptical ellipse Please refer to the radius of curvature of the Earth;

(3) The radius of curvature is also used in the bending three parts equation of the beam;