Curvature is defined by average curvature: k (average) =△α/△s, and the curvature of a point on the curve is the limit of average curvature with arc length approaching zero-k = | dα/ds |.
Question 2: What is the meaning of curvature? What is the meaning of curvature? Curvature of knowledge curve is the rotation rate of tangent direction angle of a point on the curve to arc length, which is defined by differential and represents the degree of curve deviation from 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.
Question 3: What is curvature? Curvature represents the degree of curve bending.
The curvature of a plane curve is defined by the differential of the rotation rate of the tangent direction angle to the arc length at a certain point on the curve, which indicates the degree to which the curve deviates from the straight line. The greater the curvature, the greater the curvature of the curve.
K = lim | Δ α/Δ s |, and when Δ s tends to 0, k is defined as curvature.
The reciprocal of curvature is the radius of curvature.
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.
Question 4: Curvature refers to the degree of bending.
Curvature of a curve 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. The greater the curvature, the greater the curvature of the curve.
We sometimes say that the radius of curvature (the reciprocal of curvature is the radius of curvature. ) how much, to illustrate the size of the bend.
Extension:
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 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.
In physics, curvature is usually obtained from normal acceleration, see normal acceleration for details.
Question 5: What is relative bending? What is the difference between curvature and curvature? There is a straight line and a curve (two reference systems). When you look at that curve from one end of a straight line to the other, the other end is really a curve in your eyes. But when you look at a straight line from one end of the curve to the other, it is not a straight line. It looks like a curve. You must regard that straight line as a curve, because you have no other frame of reference except the line you walk, which will lead you to subconsciously think that you are.
Relative bending requires two or more relatively different reference systems, and curvature describes the bending of a single reference system.
Question 6: How to find the curvature of a curve at a certain point? Suppose the curve is y=f(x), the center of curvature is (a, b) and the radius is r;
The essence of curvature circle is to require that the tangent and concavity of curve and circle are the same at this point.
Firstly, the equation of curvature circle is: (x-a) 2+(y-b) 2 = r 2;
Suppose the curve is concave at this point, b >；; y，y = b-(R2-(x-a)2)( 1/2)；
y ' =(- 1/2)[(r^2-(x-a)^2)^(- 1/2)]*(2)(x-a)=(x-a)(r^2-(x-a)^2)^(- 1/2)； -type a
y ' ' =(r^2-(x-a)^2)^(- 1/2)+(x-a)*(- 1/2)(r^2-(x-a)^2)^(-3/2)*(-2)(x-a)
= (R2-(x-a) 2) (-1/2)+(x-a) 2 (R2-(x-a) 2) (-3/2)-b formula
According to reasons A and B, (x-a) can be eliminated to obtain the expression of radius r, which is expressed by y' and y'';
However, it is troublesome to directly substitute the elimination method, which can be replaced as follows:
Know from A that (r 2-(x-a) 2) (-1/2) = y' (x-a) is substituted into B:
y ' ' = y '(x-a)+(x-a)^2(y'/(x-a))^3 = y '(x-a)+y'^3/(x-a)=(y '+y'^3)/(x-a)
= & gt(x-a) = (y'+y' 3)/y'' Substitute this formula into the formula:
y ' =((y '+y'^3)/y'')(r^2-((y '+y'^3)/y'')^2)^(- 1/2)
= & gtr^2 =(( 1+y'^2)/y'')^2+((y '+y'^3)
/ y'')^2
= (( 1 + y'^2)^3) / (y''^2)
= & gtr = ( 1 + y'^2)^(3/2)
/ y ' '
The curvature is1/r;
With the radius r and the normal slope (-1/y'), it is easy to find out the center of the circle of curvature, and then find out the equation of the circle of curvature.
I wonder if it will help you.
Question 7: How to calculate the curvature of the corresponding point on curve S? If the curve is represented by y=f(x),
Then the curvature formula is:
This is the second derivative of y.
The denominator is the square of the first derivative of y.