Representing Negotiation Space - Regional Science Matters - page 277

Mathematical Appendix

Elliptic Area

The ellipse has characteristic equation:

ax 2 by 2

2

þ ¼ ˁ ð 14 : 27 Þ

where x and y are the Cartesian coordinates and a given constant; a and b, inverses

of the squares of the semi-axes, are here strictly positive parameters, other cases
being however possible (hyperbolas and parabolas). Equation (14.27) is written as
follows in polar coordinates:

ˁ

¼ ˁ

r 2 a sin 2 b cos 2

2

ˆ þ ˆ ð 14 : 28 Þ

and where r is now the radius, the angle expressed in radians.

From (14.28) one computes the elliptic area as:

ˆ

ð 14 : 29 Þ

ð 14 : 30 Þ

The value of the definite integral (14.30) is known (C.R.C., p.313, No 419) and
equal to:

ðÞ

1 = 2

2

s ¼ ˀˁ ab ð 14 : 31 Þ

Reduction of an Oblique Quadratic to its Canonical Form

Starting from:

ax 2 by 2 r 2 þ þ cxy ¼ ð 14 : 32 Þ

one applies the following transformation (Fisher and Ziebur 1967 , p. 421, Theorem
95-1):

a cos 2 b sin 2 a ¼ α þ α þ c sin α cos α ð 14 : 33 Þ

a sin 2 b cos 2 b ¼ α þ α c sin α cos α ð 14 : 34 Þ

c ¼ ð b a Þ sin 2 α þ c cos 2 α ¼ 0 ð 14 : 35 Þ

from where it follows that:

cotg 2 α ¼ ð a b Þ= c, 0 < α < ˀ= 2 ð 14 : 36 Þ