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Sine-generated curves belong to a class of intrinsic functions which describe a curve by specifying its “direction angle”. The curve is determined by ω, the maximum angle which the curve makes with the horizontal, and the fact that the direction angle changes in a sinusoidal fashion along the path. Sine-generated curves are shown to be excellent approximations to the path of minimal average curvature, and expressions for radius of curvature and curve sinuosity are derived.

This paper will examine, both theoretically and empirically, various aspects of a family of parametric curves known as sine-generated curves. These curves arise most naturally in geophysics in the study of meandering structures such as rivers [

the horizontal. Specifically,

fines the direction of the curve at each point,

where

These equations are satisfied by each value l in 0 < l < L. We can see that when the value of l changes, the values of x, y, and θ will change in response and so we can think of x, y, and θ as functions of l. To find functions for x and y, we can integrate the two above equations with respect to l.

The sine generated curve is then described parametrically by:

where t varies across the curve length from 0 to L.

Sine generated curves are of interest in that they provide a convenient closed-form approximation to the curve which has the least average curvature per unit length. In other words, the curve which, as it meanders between two points, A and B, provides the minimum changes in direction for a particle traveling along the curve. This would, for example, minimize the energy needed to accelerate the particle (by changing its direction) through a curved path. Of course, the absolute minimum change in direction would be a straight line segment between points A and B. We assume that our curves deviate from this straight-line course, and take instead a winding, curved path. The question is then to specify the shape of this curve according to the optimality criterion presented above. As noted above, these curves have been analyzed in the setting of river meanders, where the curvature of rivers seems to follow the trajectory of sine-generated curves. This analysis has been undertaken mainly by Leopold and Langbein [

This paper will present a more direct derivation of sine-generated curves as curves of minimal average curvature, similar to that offered by Adam [_{x}, defined as in a cosine or sine wave as horizontal distance along the x-axis from trough to trough, and the radius of curvature R, which defines the radius of a circle which has as its arc the peak of the curve. To normalize for the effect of curve length, and give dimensionless numbers which can be compared between curves, these parameters are combined into two ratios,

After empirically examining these parameters of interest, theoretical derivations of simplified expressions for_{, }can be represented in terms of a Bessel function. This derivation was apparently not available in the early analyses of sine-generated curves, where various authors such as Leoplod and Langbein [

Let the curvature of a curve be defined as

From the above, we are looking for the curve with minimal changes in direction, and so we will require the mean squared derivative of the curvature to be a minimum, as this would be one way to minimize the energy

required for turning. Therefore, we are looking for curves which, for a given average curvature

To find curves which minimize Equation (4) subject to the constraint of Equation (3), we will follow lines similar to Adam [

Now, the integrals are with respect to the normalized path length

The problem now becomes to minimize the integral of Equation (6) subject to the constraint of Equation (5). This is an isoperimetric problem, and can be solved by introducing a Lagrange multiplier λ into a calculus of variations formulation, and form a new functional

Here, we know the form of

respect to normalized motion along the curve

To find the desired function, we introduce a theorem from the calculus of variations that

Given the centrality of this equation to the exposition, it is derived in Appendix A.

Upon substituting

The general solution to this equation is

By direct integration, we then get

Here,

us

A less direct, but perhaps more interesting, derivation of the sine-generated curve begins with a different definition of average curvature [

In this case, the problem becomes to minimize the integral:

Similar to the above, the problem can be tackled through the calculus of variations [

where α is a constant, ω is the maximal direction angle as defined previously, and where θ implies

Although this integral has no closed form solution, it can be shown that

The reason this derivation is of interest is that Equation (13) results naturally from a probabilistic approach to the mathematical modeling of particle paths. If a particle takes a random walk from point A to point B in a fixed number of steps, able to change its direction at random at each step, it is possible to search for the most probable resulting curve, which leads to a path defined by minimizing Equation (13) ( [

As noted above, the intrinsic form of the sine-generated curve,

of curves, each specified by the value of ω. Each different value of ω produces a unique sine-generated curve. Any curve of this family can be transformed into a set of parametric equations which represent the curve conventionally on Cartesian coordinates (Equations (2a), (2b))

These curves are rendered using numerical integration, assuming a path length (L) of 10. We see that the shape of the sine-generated curve will be essentially uniquely determined by ω.

Because of the well-known approximation that

One interesting result of the sine-generated curve model is that it generates loops for values of ω above approximately 126˚ (see

The sine-generated curve should be the curve of minimum average curvature (i.e., minimum changes of direction). It is worth empirically verifying this proposition in comparison to some “reasonable” curves which have a similar appearance. Returning to Equation (12), average curvature per unit length for the sine-generated curve can now be rewritten as:

Average curvature is calculated using numerical integration of Equation (15).

In

A more difficult challenge is to test the sine-generated curve against a curve specifically designed to approxi-

mate its shape. The sine-generated curve is specified intrinsically as per Equation (2), with

We note that

l = L, and l = L/2. We construct a similar function,

zeros as

mum as

This has the same maximum and the same zeros as

The curves are extremely similar in appearance, as they must be, given the similarity of their direction functions. The average curvature per unit length of the sine-generated curve is 0.468, while that of the spline curve is 0.489. This helps support the argument that the sine-generated curve represents a curve of minimal average curvature.

It is also instructive to investigate

By studying these parameters across multiple values of ω, it is possible to obtain an empirical equation which provides a good fit for the relationship of s to ω. With ω expressed in radians, we note that

Having established a valid relationship, we can solve for s in Equation (18) to produce the following:

It is noted that

These empirical results motivate a further theoretical exploration of parameters of interest for sine-generated curves.

As specified above in Equation (2), the sine generated curve is described parametrically by:

According to a well-known result from calculus, for a parametric curve, the radius of curvature R(t) along any point on the curve is given by:

Now, we let ω be specified in radians as

Thus,

Similarly,

From here, it is straightforward to calculate

Plugging these expressions into Equation (20) for curvature above, we get (after some algebra):

Our interest lies in the radius of curvature of the apex of the sine generated curve, i.e., where t = L/2. Plugging this into the above equation and taking the absolute value, we get:

Since ω is specified in radians as

We are interested in the parameter L/R, which has the remarkably simple and elegant expression:

Next, we can turn our attention to investigating

again, starting with

Now, recalling Equation (1a), that

have

Now, substituting Equation (23) into Equation (24), we can state that

Now, we can integrate both sides. The differential on the left is a differential of a normalized length, like

Now, if we let

This integral is quite significant, since it is essentially a Bessel function. It is shown in a variety of sources [

Therefore, from Equation (26), it follows directly that

This theoretical result is valid up to the first zero of the Bessel function at

Dean Hathout, (2015) Sine-Generated Curves: Theoretical and Empirical Notes. Advances in Pure Mathematics,05,689-702. doi: 10.4236/apm.2015.511063

The Euler-Lagrange equation is one of the fundamental theorems in the calculus of variations and is used above to derive the differential equation which can then be solved to yield the sine-generated curve. We follow the derivation presented by Nahin [

if x_{1} and x_{2} are known, F is given, and

Let us define a function

where _{1} and x_{2}. Second, u(x) must vanish at the endpoints of the integral. Therefore,

Now, J will, in general, depend of the value of

where

Because

We can now continue by using Leibniz’s rule for differentiating an integral. In the simplest case, where the limits are not functions of

We know

So,

Using previous results,

Now, we can proceed to integrate the second term of this integral using the technique of integration by parts. This gives:

If we now recall our initial condition that

Plugging this result back in, we get that

We are now only one step away from completing the derivation of the Euler-Lagrange equation. The final step requires the use of a fundamental lemma of the calculus of variations that states:

If, for arbitrary u(x), _{1} < x < x_{2})

Applying this result to our integral, we have that:

and thus,

This is the well known Euler-Lagrange differential equation.

We will now show that a probability based approach for finding the most probable random path in the plane during a random walk leads to Equation (13). This derivation follows the approach taken by Von Schelling in studying particle paths in the plane [

We normalize the particle’s speed (i.e., call it 1) in the plane, and let its direction change randomly with each step, understanding that the particle will move from A to B in a fixed number of steps. The particle direction is measured by the direction angle

Since the particle’s speed is 1, the particle traverses the same distance between any two direction changes. This distance is given by

Using the additional assumption that the values

Now, if we ask for which set of direction changes

does the expression

Now, we can rewrite this expression as a sum in the following way:

Now, let us take steps at smaller and smaller time intervals, i.e., assume that the time interval between successive steps,

Once again,