Meanders

Year: 2005 Authors: Paul Gailiunas

Core claim

A sine-generated intrinsic curve unifies river meanders, elastic strips, and 3-D ribbon sculptures, and can be explored computationally through turtle geometry.

Topics

river meanders, elastic curves, three-dimensional sculpture, turtle graphics

Domains

intrinsic coordinates, curvature and torsion, sine-generated curves, space curves, sculpture, ribbon forms, visualization, computer-generated art

Methods

intrinsic equation modeling, turtle graphics construction, 3-D VRML extrusion, parameter variation

Media

spring-steel, stainless steel, VRML, computer graphics

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Paul Gailiunas

25 Hedley Terrace, Gosforth

Newcastle, NE3 1DP, England

email: p-gailiunas@argonet.co.uk

Abstract

The shape of a meandering river is approximated by a curve that has a particularly simple equation when expressed in intrinsic coordinates. It is the same shape as the form taken by a strip of spring-steel constrained by bending forces at its ends. These curves are easily generated using computer turtle graphics, as are some related forms. An extension into three dimensions produces a range of sculptural forms, some of which correspond to existing works of art.

The Forms of Rivers

It is well known that a river flowing in a flat landscape does not take a straight path but follows a sinuous curve, e.g. Figure 1, which changes over time. The form of this curve was studied in detail about forty years ago by Luna Leopold [1] ${}^{\text{R}}$ who called it a “sine-generated” curve, since it has the equation

where $a$ is a constant

$\psi$ is the angle to the mean down-valley direction

$s$ is distance measured along the curve.

This is an example of an intrinsic equation, so called because it does not relate to external coordinate axes. In this form it uses a system introduced by Whewell, but it will be more convenient to use Cesáro’s system, which uses radius of curvature, $\rho$ , rather than $\psi$ .

The reciprocal of the radius, $1/\rho$ , is known as the curvature, $\kappa$ .

Figure 1: The river Coquet at Swindon (near Rothbury), Northumberland.

The Elastic Curve

The sine-generated curve is actually an approximation to the “elastic curve”, first described by James Bernoulli, which is the form taken by a flat spring when acted on by forces at its ends (see Figure 2). Eagles [2] gives an approximate method of construction using classical methods, but there is an easier method using computer graphics.

Figure 2: A distorted elastic curve illustrated (Windflow)

Turtle G

Many people will be familiar with turtle geometry domain version, MSW Logo, can be downloaded from a computer, although it can be implemented in almost any language (distance), and RIGHT and LEFT (turn by some angle) do not refer to any external coordinate system, it is the same as the geometry. If, after moving forward by a small step of path can be seen as an arc of a circle with radius, $\rho$ :

Figure 3: Increasing the parameter of a meander ( $a=0.57$ , 0.64, 0.686, 0.715, 0.775).

END

If the $cos$ function takes a value in degrees one cycle will take 360 steps. The additional scaling parameter, $b$ , is needed for other sizes, or if the argument is in radians. In this case a convenient value is $b=20\cdot 0535$ , which gives a cycle of 2 20·0535, which is approximately 126 steps. An equivalent in another language needs a stored variable (the turtle heading) that is changed when the turtle turns. (See Abelson and diSessa [3] for details on implementing turtle geometry in two and three dimensions.) As the parameter, $a$ , is increased the meander curve gets more tightly bent (Figure 3).

An interesting variation occurs if the $\cos (s)$ term is replaced by $\cos (s+\cos (ks))$ , with $k$ close to unity. (Again suitable scaling parameters might be needed.) The whole meander curve itself meanders, suggesting the possibility of self-similar meander curves (Figure 5).

Figure 4: A circular meander of symmetry order 5 as the parameter increases Figure 5: A modified meander suggesting the possibility of self-similarity

Meanders in Three Dimensions

In three dimensions there are three types of turn: roll around the direction in which the turtle is heading, yaw to left or right, and pitch up or down. Since we are not concerned with gravity there is no preferred up direction, so yaw and pitch are equivalent. Most versions of Logo include the possibility of three dimensional geometry but the better display available in VRML makes it worth writing scripts (in Java or JavaScript) to generate an extrusion along a turtle path. Abelson and diSessa [3] pp. 140-144 explains how to implement movement in three dimensions.

A sine-generated curve can be incorporated into a three dimensional space-curve in many ways. One of the most interesting has a constant pitch (so constant curvature), with the roll defined by a cosine function. A ribbon-like extrusion along this curve is a ruled surface of constant curvature.

The intrinsic equations of the curve are

where the curvature, $\kappa$ , gives the pitch, and the torsion, $\tau$ , gives the roll. The constant curvature parameter, $c$ , will be called the bend, and the torsion parameter, $a$ , the twist. As before $b$ is a scaling parameter.

Although things are more complicated than in two dimensions some of the same principles apply. If the curve is to close, the turtle’s intrinsic axes should have an unchanged orientation when it returns to the start (although it might pass through the starting position several times before closing the curve). The roll must complete a whole number of cycles, so, as before, with $b=20\cdot 0535$ a curve with symmetry order 5 closes after approximately 630 steps. The total roll must be zero, so a ribbon along the curve is topologically equivalent to a cylinder.

A meander curve in three dimensions is defined (to similarity) by two parameters, the bend and the twist, and there is a range of possible space-curves as they vary. Considering only those curves with the same symmetry, say order 5, is sufficient to understand the full range of curves, since the same principles apply to curves of any symmetry, and any that does not close is arbitrarily close to one that does.

Begin by thinking about curves with twist $=0$ . They consist of some number of loops around a circle. What happens if the twist is increased? (See Figure 6.) The loops begin to separate, and in Figure 6 begin

to curl inwards. The bend parameter needs to change to take account of this, but eventually the concave parts of the curve cancel the convex parts and any further increase in the twist parameter leads to a curve that cannot close after 630 steps, whatever the bend parameter.

Figure 6: Increasing the twist of a 3-D meander that starts with a 2 loop.

Figure 7: The 3-D meander from an 8 loop moves towards the same limit as twist increases.

The same limiting case can arise starting with a different number of loops (Figure 7), allowing a smooth trajectory between the two starting points.

Figure 8 shows the values of bend and twist that give curves with 5-symmetry. Notice that a circle with five loops is equivalent to a single circuit, since each loop is identical, and it is a limiting case. It does not close for any non-zero twist. The dotted lines correspond to space-curves where the concave and convex parts exactly cancel, so they go to infinity (Figure 9). Curves on opposite sides of these lines have centres

of symmetry on opposite sides

. Figure 8: Values of bend and twist that produce 3-D meanders with 5-symmetry.

Figure 9: Infinite 3-D meanders.

These looping curves have an obvious aesthetic appeal, and many variations are possible. Figure 10 illustrates how changing the width of the ribbon can significantly alter their appearance. It is not surprising that some sculptors have produced works with similar features.

Figure 10: Changing the ribbon width of two 3-D meanders (not to scale).

Figure 11: Pax Mundi, by Brent Collins.

Many pieces by Brent Collins [4] have the overall shape of 3-D meanders, but usually the twist maintains the same sign rather than oscillating, so they are not topological cylinders. His ribbon sculptures are exceptions [5], and

Jan Zach’s work [6] with stainless steel in the late 1960s inevitably expresses elastic curves, since the steel automatically takes on these shapes, although

other forces are also at work, in particular gravity. Windflower No.1 (Figure 12) is a good example.

In Flower of Freedom No.6 (Figure 13) the ribbon it is obviously constrained near the centre. This il curves provide believable images of ribbons, but in spring back to a circular cylinder.

kind permission of Hallie Ford Museum of Art.

Figure 12: Windflower No.1 (1967). All the images of Jan Zach’s work are reproduced by

References

[1] Leopold, L. B. and Langbein, W.B., “River Meanders”, Scientific American, 214, (Jun 1966), pp.60-70. [2] Eagles, T.H., Constructive Geometry of Plane Curves, Macmillan, 1885, pp.348-354. [4] Collins, B., Visualization: From Biology to Culture, Bridges Proceedings 2000, pp.309-315. [6] Hall, R., Intersections, The Life and Art of Jan Zach, Hallie Ford Museum of Art, Willamette University, 2002.