Let the function determined by y equation = f(x) and the corresponding schedule be set. It is required to find the radius of its curvature, that is to measure degree of a bend of the schedule of this function in some point of x0.
Instruction
1. The curvature of any line is defined by the speed of turn of its tangent in a point x at the movement of this point on a curve. As the tangent of angle of an inclination of a tangent is equal to value derivative of f(x) in this point, the speed of change of this corner has to depend on the second derivative.
2. A standard of curvature it is logical to accept a circle as it is evenly bent on all the extent. Radius of such circle is a measure of its curvature. By analogy, the radius of curvature of the set line in a point of x0 is called the radius of a circle most of which precisely measures degree of its bend in this point.
3. The required circle has to adjoin to the set curve in x0 point, that is be located from its concavity so that the tangent to a curve in this point was as well a tangent to a circle. It means that if F(x) is the circle equation, then equalities have to be carried out: F(x0) = f(x0), F ′ (x0) = f ′ (x0). Such circles, obviously, exists infinitely much. But for measurement of curvature it is necessary to choose that most of which precisely corresponds to the set curve in this point. As the curvature is measured by the second derivative, it is necessary to add to these two equalities also the third: F ′′ (x0) = f ′′ (x0).
4. Proceeding from these ratios, the radius of curvature is calculated on a formula: R = ((1 + f ′ (x0) ^2) ^ (3/2)) / (| f ′′ (x0) |). Size, the return to curvature radius, is called curvature of the line in this point.
5. If f ′′ (x0) = 0, then the radius of curvature is equal to infinity, that is the line in this point is not bent. It is always right for straight lines and also for any lines in inflection points. The curvature, respectively, is equal in such points to zero.
6. The center of the circle measuring curvature of the line in the set point is called the center of curvature. The line which is a locus for all centers of curvature of the set line is called its evolute.