Several features of complex numbers make them extremely useful in plane geometry. For example, the simplest way to express a spiral similarity in algebraic terms is by means of multiplication by a complex number. A spiral similarity with center at c, coefficient of dilation r and angle of rotation t is given by a simple formula
This representation of the circle is more convenient in some respects. For example, we may immediately check that the transformation w = f(z) = 1/z maps circles onto circles. Indeed, substituting z = 1/w we get
1/w × 1/w' - (a'/w + a/w') + s = 0
which, if multiplied by ww', leads to
ww' - (wb' + w'b) + t = 0,
where b = a'/s and t = 1/s, an equation in the same form.
Letting a = α + iβ yields yet another form of essentially same equation:
zz' - α(z + z') - iβ(z - z') + s = 0,
where α and β are both real. Yet the most general form of the equation is this
Azz' + Bz + Cz' + D = 0,
which represents a circle if A and D are both real, whilst B and C are complex and conjugate. For A = 0, the equation represents a straight line.
where t a real number. The line is the set {f(t): -∞ < t ≤ ∞} to show that any line contains a point at infinity. (The values at ±∞ are the same, so we chose just one of them, virtually arbitrarily.)
Orthogonality
Given four complex numbers u, v, w, z. Then the following conditions are equivalent and each is satisfied iff the two segments zu and vw are perpendicular:
- (u - v)/(w - z) is purely imaginary,
- (u - v)/(w - z) + (u' - v')/(w' - z') = 0,
- (u - v).(w - z) = 0,
where apostrophe denotes the conjugate of a complex number, and the dot stands for the real product of two numbers.
Collinearity
Given four complex numbers u, v, w, z. Then the following conditions are equivalent and each is satisfied iff the two segments zu and vw are parallel:
- (u - v)/(w - z) is real,
- (u - v)/(w - z) = (u' - v')/(w' - z'),
- (u - v)×(w - z) = 0,
where the cross denotes the complex product of two numbers.
If v = z, we obtain the following condition for the collinearity of three points:
- u, v, w are collinear,
- (u - v)/(w - v) = (u' - v')/(w' - v'),
- (u - v)×(w - v) = 0.
(uvwz) is a common shorthand of the double (cross-) ratio in #2. The latter simply claims that the angles at u and v subtended by wz are either equal or their difference equals π modulo 2π.
In complex analysis, the cross-ratio (uvwz) is more often denoted (u, v; w, z) = (u - w)/(u - z) : (v - w)/(v - z). Collinearity is considered a special case of concyclicity.
As an exercise, you can verify a wonderful property of the cross-ratio. Let f(p), f(q), f(r), f(s) be four points on a line f(t) = (a + tb)/(1 + t). Then
