A Little More About Curvature of Curves

In the chapter entitled Curvature of Curves, we have learned how we can find out which way a curve is curved, that is, whether it curves upwards or downwards towards the right. This gave us no indication whatever as to how much the curve is curved, or, in other words, what is its curvature.

By curvature of a line, we mean the amount of bending or deflection taking place along a certain length of the line, say along a portion of the line the length of which is one unit of length (the same unit which is used to measure the radius, whether it be one inch, one foot, or any other unit). For instance, consider two circular paths of centre and and of equal lengths (see the following figure). When passing from to along the arc of the first one, one changes one’s direction from to , since at one faces in the direction and at one faces in the direction . In other words, in walking from to one unconsciously turns round through the angle , which is equal to the angle . Similarly, in passing from to , along the arc , of equal length to , on the second path, one turns round through the angle , which is equal to the angle , obviously greater than the corresponding angle . The second path bends therefore more than the first for an equal length.

This fact is expressed by saying that the curvature of the second path is greater than that of the first one. The larger the circle, the lesser the bending, that is the lesser the curvature. If the radius of the first circle is etc. times greater than the radius of the second, then the angle of bending or deflection along an arc of unit length will be 2, 3, 4 , ... etc. times less on the first circle than on the second, that is, it will be etc. of the bending or deflection along the arc of same length on the second circle. In other words, the curvature of the first circle will be etc. of that of the second circle. We see that, as the radius becomes 2, 3, 4, .. etc. times greater, the curvature becomes 2, 3, 4, ... etc. times smaller, and this is expressed by saying that the curvature of a circle is inversely proportional to the radius of the circle, or

where is a constant. It is agreed to take , so that always.

If the radius becomes indefinitely great, the curvature becomes zero, since when the denominator of a fraction is indefinitely large, the value of the fraction is indefinitely small. For this reason mathematicians sometimes consider a straight line as an arc of circle of infinite radius, or zero curvature.

In the case of a circle, which is perfectly symmetrical and uniform, so that the curvature is the same at every point of its circumference, the above method of expressing the curvature is perfectly definite. In the case of any other curve, however, the curvature is not the same at different points, and it may differ considerably even for two points fairly close to one another. It would not then be accurate to take the amount of bending or deflection between two points as a measure of the curvature of the arc between these points, unless this arc is very small, in fact, unless it is indefinitely small.

If then we consider a very small arc such as (see the next figure ), and if we draw such a circle that an arc of this circle coincides with the arc of the curve more closely than would be the case with any other circle, then the curvature of this circle may be taken as the curvature of the arc of the curve. The smaller the arc , the easier it will be to find a circle an arc of which most nearly coincides with the arc of the curve. When and are very near one another, so that is so small so that the length of the arc is practically negligible, then the coincidence of the two arcs, of circle and of curve, may be considered as being practically perfect, and the curvature of the curve at the point (or ), being then the same as the curvature of the circle, will be expressed by the reciprocal of the radius of this circle, that is, by , according to our way of measuring curvature, explained above.

Now, at first, you may think that, if is very small, then the circle must be very small also. A little thinking will, however, cause you to perceive that it is by no means necessarily so, and that the circle may have any size, according to the amount of bending of the curve along this very small arc . In fact, if the curve is almost flat at that point, the circle will be extremely large. This circle is called the circle of curvature, or the osculating circle at the point considered. Its radius is the radius of curvature of the curve at that particular point.

If the arc is represented by and the angle by , then, if is the radius of curvature,

The secant makes with the -axis the angle , and it will be seen from the small triangle that . When is indefinitely small, so that practically coincides with , the line becomes a tangent to the curve at the point (or ).

Now, depends on the position of the point (or , which is supposed to nearly coincide with it), that is, it depends on , or, in other words, is "a function" of .

Differentiating with regard to to get the slope (see here), we get

or

(see here);

hence

But , and for one may write therefore

but ;¹ hence

and finally,

It is important to note that the radius must always be positive, as a negative radius would have no physical meaning. Therefore, when using the above formula, one must select the sign if the denominator is positive and the sign if the denominator is negative, as the numerator, being a square root, is always positive.²

It has been shown in the chapter on Curvature of Curves that if is positive, the curve is concave upward (also called convex), while if is negative, the curve is concave downward (also simply called concave). If , the radius of curvature is infinitely great, that is, the corresponding portion of the curve is a bit of straight line. This necessarily happens whenever a curve gradually changes from being concave upward to concave downward or vice versa. The point, like in the following figure, where this occurs is called a point of inflection.

The centre of the circle of curvature is called the centre of curvature. If its coordinates are , then the equation of the circle is (see here)

hence

and

Why did we differentiate? To get rid of the constant . This leaves but two unknown constants and ; differentiate again; you shall get rid of one of them. This last differentiation is not quite as easy as it seems; let us do it together; we have:

the numerator of the second term is a product; hence differentiating it gives

so that the result of differentiating (1) is

from this we at once get

Replacing in (1), we get

and give the position of the centre of curvature. The use of these formulae will be best seen by carefully going through a few worked-out examples.

Exercises

1. Depending on , this can be or .↩︎

2. Alternatively, one may write where denotes the absolute value of . For the definition of the absolute value, see page .↩︎

3. Let with coordinates be the centre of this circle whose radius is . Then the equation of this circle is

Since the point lies on this circle, its coordinates must satisfy the equation of this circle. Thus,

Similarly, since and satisfy this equation too, we have

If we subtract Eq. (1) from (2) and (1) from (3) and use , we get or Solving these two equations for two unknowns and , we can find the coordinates of the centre: From Eq. (1), the radius of this circle is ↩︎

4. Notice that . For the definitions of the hyperbolic functions see here.↩︎