A Method for Determining the Curvature of Natural Forms and Its Application to Certain Tail Feathers of the Superb Lyrebird Menura novaehollandiae

Abstract

A method for determining the curvature pattem of a two-dimensional curve is described and demonstrated by its application to three types of tail feathers of the Superb Lyrebird. Assuming that the curve is part of a circle, the straight line AB joining the ends A and B of the curve is a chord of that circle. The perpendicular bisector of the chord AB passes through the centre of the circle 0 and meets the curve (circle) at C, and at D, a distance of 2R from C, where R is the radius of the circle. Because DOC is a diameter of a circle, angle DAC (= angle DBC) equals 90 degrees Guclidean theorem) and AC (= BC = L)I2R = Cos θ, where θ is the angle between AC and DOC. Thus, R = L/2 Cos θ. The paper describes how L and θ are measured. Extension of the method to the shorter elements resulting from the first step enables a curvature pattem for the entire curve (feather) to be determined. The plain median rectrix of a Superb Lyrebird, and the median and lyrate rectrices of a mature male Superb Lyrebird, were all more highly curved at the distal end.