In differential geometry,curvatureOfreciprocalIs the radius of curvature, that is, R=1/K.The curvature of a plane curve is specific to a point on the curvetangentBearing pairarc lengthRate of rotation, viadifferentialTo indicate how far the curve deviates from the straight line.For a curve, it is equal to the radius of the arc closest to the curve at that point.For surfaces, the radius of curvature is the radius of the circle that best fits the normal section or its combination.
In differential geometry,curvatureOfreciprocalIs the radius of curvature, that is, R=1/K.The curvature of a plane curve is specific to a point on the curvetangentBearing pairarc lengthRate of rotation, viadifferentialTo indicate how far the curve deviates from the straight line.For a curve, it is equal to the radius of the arc closest to the curve at that point.For surfaces, the radius of curvature is the radius of the circle that best fits the normal section or its combination.[1]
The radius of curvature is mainly used to describe the degree of bending change of a curve somewhere on the curve. For example, the degree of bending is the same everywhere on the circle, so the radius of curvature is the radius of the circle;The straight line is not bent, and the radius of the circle tangent to the straight line at this point can be arbitrarily large, so the curvature is 0, so the straight line has no radius of curvature, or the radius of curvature is recorded as
。
The larger the radius of the circle, the smaller the degree of bending, and the closer it is to a straight line.Therefore, the larger the radius of curvature, the smaller the curvature, and vice versa.
If a circle with the same curvature can be found for a point on a curve, the radius of curvature of this point on the curve is the radius of the circle (note that it is the radius of curvature of this point, and other points have other radii of curvature).It can also be understood as follows: that is, the curve is differentiated as much as possible until it is finally approximated to an arc, and the radius corresponding to this arc is the radius of curvature of the point on the curve.
Formula derivation
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In the case of space curves, the radius of curvature is the length of the curvature vector.In the case of plane curve, R should be taken as absolute value.
Where s is the arc length of the fixed point on the curve, α is the tangential angle, and K is the curvature.
If the curve is expressed in Cartesian coordinates as
, then the radius of curvature is (assuming that the curve is differentiable)
If the curve consists of a function
and
If the parameter is given, the curvature is
If
yes
The radius of curvature at each point of the curve
It is given by the following formula:
As a special case, if f (t) is a function from R to R, then the radius of curvature of the graph γ (t)=(t, f (t)) is
give an example
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Semicircle
For the semicircle of radius a of the upper half plane:
For the semicircle of radius a of the upper half plane:
The radius of curvature of a circle with radius a is equal to a.
ellipse
In an ellipse with major axis 2a and minor axis 2b, the vertices on the major axis have the minimum radius of curvature of any point,
；
And the vertex on the minor axis has the maximum radius of curvature of any point
。
application
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The concept of radius of curvature and its application are very extensive, which play an important role in lens imaging, machining parts, highway and railway curve design and construction.[3]
(1) For the application of difference geometry, please refer to the Ces à ro equation;
(2) For the radius of curvature of the earth (approximated by an elliptical ellipse), see Radius of curvature of the earth;
(3) The radius of curvature is also used in the three part equation of beam bending;
(4) Radius of curvature (optical).
(5) Stress in semiconductor structure:
The stresses in semiconductor structures that involve evaporating thin films usually come from thermal expansion (thermal stress) in the manufacturing process.Thermal stress occurs because film deposition is usually above room temperature.When cooling from deposition temperature to room temperature, the difference of thermal expansion coefficient between substrate and film causes thermal stress.
When atoms are deposited on the substrate, the inherent stress is caused by the microstructure formed in the film.Due to the attractive interaction of atoms passing through the pores, tensile stress is generated in the micropores of the film.
The stress in the thin film semiconductor structure leads to the warping of the wafer.The curvature radius of the stress structure is related to the stress tensor in the structure, which can be described by the modified Stoney formula.An optical scanner can be used to measure the shape of the stress structure including the radius of curvature.Modern scanner tools have the ability to measure the overall view of the substrate and the two main curvature radii, and provide 0.1% accuracy for curvature radii 90 meters and above.[2]
