3000

2000

1000

60

50

70

0

Pole

Log Spiral

-1000

-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

1000

Distance Alongshore, m

Fig. 2. Variation of logarithmic-spiral shape with α (*R*0 = 500 m).

2000

1500

1000

500

0

Pole

Log-spiral α = 60.6

-500

Log-spiral α = 60.0

Log-spiral α = 59.4

-1000

-3500

-3000

-2500

-2000 -1500 -1000 -500

0

500

1000

Distance Alongshore, m

Fig. 3. Sensitivity of log-spiral shape to small variations of the parameter α (*R*0 = 500 m).

To analyze the validity of the assumption that headland-bay beaches can be

adequately described by a log-spiral function, the pole location, radius at origin of angle

measurement (scaling factor), and the spiral characteristic angle α must be identified,

giving four unknowns. If the pole position is known, a solution procedure for the best-fit

shape, that is, to find the best-fit value of α, follows. Eq. 1 can be rewritten as

ln (*R*) = ln (*R*0 ) + *θ *cot *α*

(2)

where *R *= radius to point P, θ = angle between polar axis and radius vector to point *P*.

This linear relationship is convenient for fitting a log-spiral curve to data. The slope of

the straight line is cot α, from which α can be obtained.

A computer program was written that searches for the pole position while minimizing

the fitting error. It also determines the characteristic angle α that best fits the shoreline

Moreno & Kraus

5

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