How Do I Know The Radius Of Curvature | The Science



The Science

How do I know the radius of curvature

Suppose that a function defined by the equation y = f (x) and a corresponding plot. Required to find the radius of curvature, that is, to measure the degree of curvature of the graph of the function at some the point x0.

How do I know the radius of curvature

Instruction how to find the radius of curvature

Step 1:

The curvature of each line is determined by the speed of rotation of its tangent at x when driving this point on the curve. Since the slope of the tangent equal to the value of the derivative f (x) at this point, the rate of change of this angle will depend on the second derivative.

Step 2:

The standard curvature is logical to take the circle, because it is uniformly curved along its entire length. The radius of this circle is a measure of its curvature. By analogy, the radius of curvature of a given line at a point x0 is called the radius of the circle which most accurately measures the degree of curvature at that point.

Step 3:

The desired circumference to be in contact with a given curve at x0, that is located with its concave side, so that the tangent to the curve at this point was also tangent to the circle. This means that if F (x) - equation of a circle, the equation must be met: F (x0) = f (x0), F '(x0) = f' (x0). Such circles, obviously, there are infinitely many. But to measure the curvature of the need to choose the one that most closely matches a given curve at this point. Since the curvature of the second derivative is measured, then these two equations need to add a third: F '' (x0) = f '' (x0).

Step 4:

Based on these ratios, the radius of curvature is calculated by the formula: R = ((1 + f '(x0) ^ 2) ^ (3/2)) / (| f' '(x0) |). The reciprocal of the radius of curvature, the curvature of the line is called at this point.

Step 5:

If f '' (x0) = 0, the radius of curvature is infinite, that is, the line at this point is not curved. This is always true straight lines, but also for any lines at inflection points. The curvature at these points, respectively, is zero.

Step 6:

Circle center, measuring the curvature of the line at a given point, called the center of curvature. The line is the locus of all centers of curvature of a given line, it is called the evolute.