Conformal mapping may be used to map an arbitrary ship’s section to a unit circle centred at the origin. The solution of the potential flow formulation for a unit circle may then be applied to an arbitrary hull form. The conformal mapping has the general form given below in Equation ( 68 ):

Where z is a complex number lying on the unit circle. The coordinate system is shown in Figure 1, and in this case z = ie^-iq.

Figure 1: Mapping Coordinate System

The mapping described in Equation ( 68 ) will map an arbitrary shape in the X-plane to the unit circle in the z-plane. If the entire unit circle is mapped, there is no need for symmetry planes to exist in the y=0 and z=0 axes. This type of mapping is useful for heeled yacht sections or asymmetric catamaran hulls.

A simplified mapping uses only the odd powers of z and is mapped over one quarter of the unit circle. This produces a mapping with symmetry planes in both y=0 and x=0 axes. This mapping equation is given in Equation ( 69 ) and referred to as the symmetric multi-parameter mapping.

It should be noted that several mapping equations may be derived to map ship-like sections to the unit circle. The mapping equations described here follow the work of Ursell (1949), Bishop et al. (1978) and to some extent the work of Sutherland (1987). This is important because of the later derivation and evaluation of the hydrodynamic coefficients.

The symmetric mapping equation described in Equation ( 69 ) may be further simplified so that only the first three terms are used. In this case the mapping is known as a Lewis Mapping and the mapped section is a Lewis Section. The Lewis Mapping is given in Equation ( 70 ).

Where the constants a₀, a₁, and a₂ are calculated from the section water line half beam, section area and section draught. See Equations ( 71 ):

The mapped point in the ship section plane (y, z) of a point on the unit circle, described by its angle, q, from the positive h-axis, for the symmetric, multi-parameter mapping, is given in Equations ( 72 ):