21  Concavity and curvature

It is an easy visual task to discern the slope of a line segment. A glance shows whether the slope at that point is positive or negative. Comparing the slopes at two locales is also an automatic visual task: most people have little difficulty saying which slope is steeper. One consequence of this visual ability: it is easy to recognize whether a line that touches the graph at a point is tangent to the graph.

There are other aspects of functions, introduced in Section 6.2, that are also readily discerned from a glance at the function graph.

21.1 Quantifying concavity and curvature

It often happens in building models that the modeler (you!) knows something about the concavity or the curvature of a function. For example, concavity is essential in classical economics; the curve for supply as a function of price is concave down while the curve for demand as a function of price is concave up. For a train, car, or plane, sideways forces depend on the curvature of the track, road, or trajectory. Road designers need to calculate the curvature to know if the road is safe at the indicated speed.

It turns out that quantifying these properties of functions or shapes is naturally done by calculating second derivatives.

Application area 21.1 Designing a highway curve

Imagine designing a highway. Due to the terrain, part of the road is oriented east-west and another part north-south. Those two parts need to be connected together for vehicles to use the road! (In math-speak, we might say that the road has to be continuous, but this is just common sense.)

Experience with highways shows that the connection will be a smooth curve. If the curve is part of a circle, the design needs to specify the radius of curvature. Too tight a radius and the traffic will not be able to handle the centrifugal force; vehicles will drift or skid off the road. A big radius provides safety, but making the radius bigger than required adds road construction costs.

Real-world highway on- and off-ramps are usually not precisely sections of a circle, so specifying the shape of the ramp is not as simple as setting the radius of the curve. Instead, the radius changes at the entry and exit of the curve. The American Association of State Highway and Transportation Officials Policy on Geometric Design of Highways and Streets (1994) explains why:

Any motor vehicle follows a transition path as it enters or leaves a circular horizontal curve. The steering change and the consequent gain or loss of centrifugal force cannot be effected instantly. For most curves the average driver can effect a suitable transition path within the limits of normal lane width. However, with combinations of high speed and sharp curvature the resultant longer transition can result in crowding and sometimes actual occupation of adjoining lanes. In such instances transition curves would be appropriate because they make it easier for a driver to confine the vehicle to his or her own lane. The employment of transition curves between tangents and sharp circular curves and between circular curves of substantially different radii warrants consideration.

21.2 Concavity

Recall that to find the slope of a function \(f(x)\) at any input \(x\), you compute the derivative of that function, which we’ve been writing \(\partial_x\,f(x)\). Plug in some value for the input \(x\) and the output of \(\partial_x\, f(x)\) will be the slope of \(f(x)\) at that input. (Chapter 20 introduced some techniques for computing the derivative of any given function.)

Now we want to show how differentiation can quantify the concavity of a function. First, remember that when we speak of the “derivative” of a function, we mean the first derivative of the function. That full name naturally suggests that there will be a second derivative, a third derivative, and higher-order derivatives.

As introduced in Section 6.2, the concavity of a function describes not the slope but the change in the slope. Figure 21.1 depicts the slope of the function in each of the three boxes labelled A, B, and C. In the subdomain marked A, the function slope is positive, while in the subdomain B, the function slope is negative. This transition from the slope at A to the slope at B corresponds to the negative concavity of the function between A and B.

Figure 21.1: Concavity is about how the slope changes from one place in the domain to another.

Similarly, the function’s concavity in the interval B to C reflects the transition in the instantaneous slope at B to the different instantaneous slope at C.

Let’s look at this using symbolic notation. Keep in mind that the function graphed is \(f(x)\) while the slope is the function \(\partial_x\,f(x)\). We’ve seen that the concavity is indicated by the change in slope of \(f()\), that is, the change in \(\partial_x\, f(x)\). We will go back to our standard way of describing the rate of change near an input \(x\):

\[\text{concavity.of.f}(x) \equiv\ \text{rate of change in}\ \partial_x\, f(x) = \] \[= \partial_x \left(\strut\partial_x f(x)\right) = \lim_{h\rightarrow 0}\frac{\partial_x f(x+h) - \partial_x f(x)}{h}\] We are defining the concavity of a function \(f()\) at any input \(x\) to be \(\partial_x \left(\strut\partial_x f(x)\right)\). We create the concavity_of_f(x) function by applying differentiation twice to the function \(f()\).

Such a double differentiation of a function \(f(x)\) is called the second derivative of \(f(x)\). The second derivative is so important in applications that it has its own compact notation: \[\text{second derivative of}\ f()\ \text{is written}\ \partial_{xx} f(x)\] Look carefully to see the difference between the first derivative \(\partial_x f(x)\) and the second derivative \(\partial_{xx} f(x)\): it is all in the double subscript \(_{xx}\).

Computing the second derivative is merely a matter of computing the first derivative \(\partial_x f(x)\) and then computing the (first) derivative of \(\partial_x f(x)\). In R this process looks like:

dx_f  <- D(   f(x) ~ x)   # First deriv. of f()
dxx_f <- D(dx_f(x) ~ x)   # Second deriv. of f()
A shortcut for second derivatives

A notation shortcut for the two-step process above: double up on the x on the right-hand side of the tilde:

dxx_f <- D(f(x) ~ x & x)

Read the ~ x & x as “differentiate by \(x\) and then once more by \(x\) again.”

21.3 Curvature

As you see from Section 21.2, it is easy to quantify the concavity of a function \(f(x)\): just evaluate the second derivative \(\partial_{xx} f(x)\). However, it turns out that people cannot do a good job of estimating the quantitative value of concavity by eye.

To illustrate, consider the square function, \(f(x) \equiv x^2\). (See Figure 21.2.)

Figure 21.2: Does the concavity of the square function vary with \(x\)?

The square function is concave up. Now a test: Looking at the graph of the square function, where is the concavity the largest? Don’t read on until you’ve pointed where you think the concavity is largest.

 

Have you decided where the concavity is largest?

 

With your answer to the test question in mind, we can calculate the concavity of the square function using derivatives.

\[f(x) \equiv x^2\ \text{ so }\ \partial_x f(x) = 2 x\ \text{ and therefore }\ \partial_{xx} f(x) = 2\]

The second derivative of \(f(x)\) is positive, as expected for a function that is concave up. Surprisingly, however, the second derivative is constant.

The concavity-related property that the human eye reads from the function graph is not the concavity itself but the curvature of the function. The curvature of \(f(x)\) at \(x_0\) is defined to be the radius of the circle tangent to the function at \(x_0\).

Figure 21.3 illustrates the changing curvature of \(f(x) \equiv x^2\) by inscribing tangent circles at several points on the function graph, marked with dots. That the function’s thin black line goes right down the middle of the broader lines used to draw the circles shows the tangency of the circle to the function graph.

Figure 21.3: At any point on the graph of a smooth function, there is a circle tangent to the graph. The radius of this circle is \(1/{\cal K}\).

Black dots along the graph at the points indicate where the function graph is tangent to the inscribed circle. The visual sign of tangency is that the function graph goes right down the circle’s center.

The inscribed circle at \(x=0\) is tightest, the circle at \(x=1\) larger and the radius of the circle at \(x=-1.5\) is the largest of all. Whereas the concavity is the same at all points on the graph, the visual impression that the function is most highly curved near \(x=0\) is better captured by the radius of the inscribed circle. The radius of the inscribed circle at any point is the reciprocal of a quantity \({\cal K}\) called the curvature.

The curvature \({\cal K}\) of a function \(f(x)\) depends on both the first and second derivative. The formula for curvature \(K\) is somewhat off-putting; you are not expected to memorize it. But you can see where \(\partial x f()\) and \(\partial_{xx}f()\) come into play.

\[{\cal K}_f(x) \equiv \frac{\left|\partial_{xx} f(x)\right|}{\ \ \ \ \left|1 + \left[\strut\partial_x f(x)\right]^2\right|^{3/2}}\]

Mathematically, the curvature function \(\cal K()\) corresponds to the reciprocal of the radius of the tangent circle. When the tangent circle is tight, the output of \(\cal K()\) is large. When radius of the tangent circle is large, that is, when the function is very close to approximating a straight line, the output of \(\cal K()\) is very small.

Try it! 21.1 Calculate the curvature

Calculate the curvature of the square function: \(f(x) \equiv x^2\). Active R chunk 21.1 will get you started by defining some of the functions needed:

Active R chunk 21.1

Is the curvature constant as a function of \(x\)?

The formula for \(K()\) is

abs(dx_f(x))/abs(1 + (dxx_f(x)^2))^(2/3).

Application area 21.2 Back to the highway

Returning to the highway design example in Application area 21.1 … The Policy on geometric design of highways and streets called for the curvature of a road to change gently, giving the driver time to adjust the steering and accommodate the centrifugal force of the car going around the curve.

Changing curvature implies that \(\partial_x {\cal K}\) is non-zero. Since \({\cal K}\) depends on the first and second derivatives of \(f(x)\), the Policy on gradual change means that the third derivative of \(f(x)\) is non-zero.