The area within a maximum range oval of Cassini is developed in this appendix. The development is in two parts: (1) the area within a single oval, where the base-line, L < 2 ?? and ? = bistatic maximum range constant, and (2) the area within two identical ovais, surrounding the transmitter and receiver, where L ? 2 ??. The development uses the polar coordinate System shown in Figure 4.1.

The area within a closed curve, A _B, defined in polar coordinates ( r, ?) is

The equation for an oval of Cassini in polar coordinates is given by (4.5) as

When the oval is a maximum range oval of Cassini, (4.1a) applies:

Substituting (D.3) into (D.2) and solving for r = f( ?) yields:

Substituting (D.4) into (D.1), manipulating terms, and noting that the oval (or ovais) are Symmetrie in each quadrant, and also that

yields:

where only the positive value of the radical is evaluated.

Equation (D.6) can be integrated directly when L ⁴/16 ? ² < 1, i.e., L < 2 ??, the case where a single oval surrounds both transmitter and receiver. In this case the area within the single oval, A _B1, is [152]:

For the two-oval case where L ? 2 ??, (D.6) must be modified by a substitution of variables:

Thus, the area within two ovals, A _B2, becomes

With this substitution of variables, ?