bend

Curvature of a curve (Q ū l ǜ) is the rotation rate of the tangent direction angle of a point on the curve to the arc length, which is defined by differentiation and represents the degree of deviation of the curve from a straight line. A numerical value that mathematically represents the degree of curvature of a curve at a certain point.

The greater the curvature, the greater the curvature of the curve. The reciprocal of curvature is the radius of curvature.

Take a plane curve as an example. Make a circle through a point A and two adjacent points B 1, B2 on the plane curve. When B 1 and B2 infinitely approach a, the limit position of this circle is called the curvature circle at point A of the curve. The center and radius of the curvature circle are called the curvature center and radius of the curve at point A respectively.

When an arc is a part of a circle, the radius of curvature of the arc is the radius of the circle. The greater the radius of curvature, the smoother the arc, and the smaller the radius of curvature, the steeper the arc. The reciprocal of the radius of curvature is curvature. Curvature k = (rotation angle/corresponding arc length). When the angle and arc length approach zero at the same time, it is the standard definition of curvature of smooth curve with arbitrary shape. For a circle, the curvature does not change with the position.

In dynamics, generally speaking, when an object moves at variable speed relative to another object, it will also produce curvature. This is caused by the distortion of time and space. Combined with the equivalence principle of general relativity, an object with variable speed motion can be regarded as being in a gravitational field, resulting in curvature.

According to the explanation of general relativity, in the gravitational field, the nature of space-time is determined by the "mass" distribution of objects, which makes the nature of space-time uneven and causes the curvature of space-time. Because objects with mass will bend space-time, and you can think that with speed, objects with mass will become heavier and the curvature of space-time will be greater.