Complex Numbers and Geometry

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

f(z) = r(z - c)(cos(t) + i·sin(t)) + c.

Circle

A particularly simple equation is that of a circle:

{z: |z - a| = r},

is the circle with radius r and center a. By squaring that equation we obtain

(z - a)(z' - a') = r²

zz' - (za' + z'a) + (aa' - r²) = 0.

and finally

zz' - (za' + z'a) + s = 0,

where s is a real number. The circle is centered at a and has the radius r = √aa' - s, provided the root is real.

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.

Straight Line

A straight line through point (complex number) a and parallel to the vector (another complex number) v is defined by

(1)

f(t) = a + tv,

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.)

From (1) we can derive the equation of a line through two points, a and b say. Indeed, if the line contains both a and b, then it is parallel to the number b-a. Thus the equation becomes

f(t) = a + t(b - a),

or,

f(t) = (1 - t)a + tb

= (1 - t)a + tb

= sa + tb, where s = 1 - t,

= (sa + tb) / (s + t), since s + t = 1,

= (a + rb) / (1 + r),

where r = t/s = t / (1 - t). The latter defines a hyperbola in the (t, r) plane so that r takes exactly the same values as t. In terms if thus defined r the straight line through a and b has the equation

(2)

f(r) = (a + rb) / (1 + r).

The point at infinity is now obtained for r = -1. a = f(0), b = f(∞), (a + b)/2 = f(1).

Orthogonality

Given four complex numbers u, v, w, z. Then the following conditions are equivalent and each is satisfied iff the two segments uv and wz 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 uv and wz 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.

Concyclicity

Given four complex numbers u, v, w, z. Then the following conditions are equivalent:

u, v, w, z are concyclic (or collinear),
(u - w)/(u - z) : (v - w)/(v - z) is real,
(u - w)/(u - z) : (v - w)/(v - z) = (u' - w')/(u' - z') : (v' - w')/(v' - z')
(uvwz) is real.

(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

(f(p), f(q); f(r), f(s)) = (p, q; r, s).

Similarity

Given two triangles A(a)B(b)C(c) and A₁(a₁)B₁(b₁)C₁(c₁). Then the following are equivalent"

The triangles are similar and have the same orientation,
(b₁ - a₁)/(c₁ - a₁) = (b - a)/(c - a).

Also,

The triangles are similar and have different orientations,
(b₁ - a₁)/(c₁ - a₁) = (b' - a')/(c' - a').

Equilateral Triangles

For a positively oriented triangle A(a)B(b)C(c), the following conditions are equivalent

ABC is equilateral.
|a - b| = |b - c| = |c - a|.
a² + b² + c² = ab + bc + ca.
(b - a)/(c - b) = (c - b)/(a - b).
(z - a)^-1 + (z - b)^-1 + (z - c)^-1 = 0, where z = (a + b + c)/3.
(a + eb + e²c)(a + ec + e²b) = 0, where e = cos(2p/3) + i·sin(2p/3).

The following links point to a variety of applications of complex numbers in geometry:

Problems

References

T. Andreescu, D. Andrica, Complex Numbers From A to ... Z, Birkhäuser, 2006
C. W. Dodge, Euclidean Geometry and Transformations, Dover, 2004 (reprint of 1972 edition)
Liang-shin Hahn, Complex Numbers & Geometry, MAA, 1994
E. Landau, Foundations of Analisys, Chelsea Publ, 3^rd edition, 1966
D. Pedoe, Geometry: A Comprehensive Course, Dover, 1988

Complex Numbers

71931301

f(t)	= (1 - t)a + tb
	= (1 - t)a + tb
	= sa + tb, where s = 1 - t,
	= (sa + tb) / (s + t), since s + t = 1,
	= (a + rb) / (1 + r),