San José State University
Department of Economics
Thayer Watkins
Silicon Valley
& Tornado Alley

Simple, Yet Amazing Theorems Concerning
the Cumulative Curvature of a Curve Between Two Points

A Simple, Yet Amazing Theorem
Concerning Average Curvature of a Curve

A special case of the theorem will be presented first to acquaint readers with the nature of the subject matter.

Let f(x) be a twice differentiable function. Then the
average curvature between critical points of f(x) is equal to zero.

Critical points may be relative maximums, relative minimums or inflections points. They are the points such that the first derivative is zero, f'(x)=0. The theorem applies not just to adjacent critical points; it applies to any two critical points of the function be they maximums, minimums or inflection points.

The theorem can be easily generalized and the proof of the general theorem is so simple it is not worth bothering with the proof of the specialized theorem. For the general theorem let us define two points as being parallel tangency points if the slopes of the curve at those two points are equal; i.e., if f'(a)=f'(b) the f(a) and f(b) are parallel tangency points.

The Theorem Concerning
Cumulative Curvature of a Line

Theorem: If f(x) is a twice differentiable function
on some interval then the cumulative curvature
between two parallel tangency points is equal to zero.


Let a and b be any two values of x in the interval of definition for f(x). Curvature is the second derivative of the function, f"(x). The cumulative curvature between a and b, C(a,b), is given by
which reduces to

When a and b are parallel tangency points of the function, f'(a)=f'(b). Therefore the cumulative curvature between parallel tangency points is 0.

End of proof.

Critical points are simply parallel tangency points for which the slopes are zero; i.e., f'(a)=f'(b)=0. It does not matter that there may be other values of x between a and b, say c, such that f'(c)=f'(a)=f'(b). The cumulative curvature between a and c is zero and between c and b is zero, but so is it zero between a and b.

If the cumulative curvature is zero then obviously the average curvature is also zero.

The Effect of Statistical Noise
on the Perception of Curvature

Generally random variation of a function smudges out the perception of curvature so stochastic functions have an appearance of piecewise linearity. Consider the following display in which random amounts of expected value zero are added to a function.

Each time the REFRESH button is pressed a new set of random values are added to the function. The general impression is of linear episodes with some flatness at the transitions.

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