where *R *= length of the radius vector for a point P measured from the pole O; θ = angle

from an arbitrary origin of angle measurement to the radius vector of the point P; *R*0 =

length of radius to arbitrary origin of angle measurement; and α = characteristic constant

angle between the tangent to the curve and radius at any point along the spiral.

A property of the log-spiral curve is that the angle a between the tangent to the curve

and the vector radius at any point along the curve is constant. This leads to the

interesting result that the shape of the log spiral is controlled only by α, with the

parameter *R*o determining the scale of the shape. In fact, the functioning of *R*o is

equivalent to setting a different origin of measurement of the angle θ. In other words,

graphically the log-spiral may be scaled up or down by turning the shape around its pole.

Fig. 2 shows how different values of α alter the shape.

Fig. 1. Definition sketch of the logarithmic spiral shape.

Values of α for headland-bay beaches reported in the literature range from about 45

to 75. In general, as α becomes smaller, the log spiral becomes wider or more open.

There are two singular values or limits for α: if α = 90, the log spiral becomes a circle,

and if α = 0, the log spiral becomes a straight line.

Sensitivity of the log-spiral shape to small changes in α is shown in Fig. 3. Variations

( 1%) in α-values produce large variations in position of the spiral, because the angle

enters the argument of an exponential function. Additionally, the smaller the

characteristic angle α, the larger the difference for the same percentage of angle

variability. The practical consequence is that, because we are interested in fitting this

log-spiral shape to headland-bay beaches especially in design of shore-protection

projects, α has to be accurately defined.

Moreno & Kraus

4

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