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
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f(z) = r(z - c)(cos(t) + i·sin(t)) + c.
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Circle
A particularly simple equation is that of a circle:
is the circle with radius r and center a. By squaring that equation we obtain
or
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zz' - (za' + z'a) + (aa' - r²) = 0.
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and finally
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zz' - (za' + z'a) + s = 0,
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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 immediatle check that the transformation w = f(z) = 1/z maps circles onto circles. Indeed, substituting z = 1/w we get
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1/w × 1/w' - (a'/w + a/w') + s = 0
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which, if multiplied by ww', leads to
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ww' - (wb' + w'b) + t = 0,
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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:
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zz' - α(z + z') - iβ(z - z') + s = 0,
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where α and β are both real. Yet the most general form of the equation is this
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
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
or,
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| 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), |
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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).
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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 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.
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 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
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(f(p),f(q);f(r),f(s)) = (p,q;r,s).
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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
- 9-point Circle as a locus of concurrency
- A Case of Similarity
- A Property of Cubic Equations
- All About Medians
- Asymmetric Propeller
- Bisector of an imaginary angle may be real
- Bottema's Theorem
- Cantor's Theorem
- Center-circles and Their Chains
- Clifford's Chain
- Clifford's Lemma
- Cycloids
- Equilateral Triangles On Sides of a Parallelogram
- Four Hinged Squares
- Friendly Kiepert's Perspectors
- Harmonic Ratio in Complex Domain
- Hypocycloid Families
- Iterations and the Mandelbrot Set
- J. C. Maxwell's Theorem
- Mandelbrot and Julia sets
- Morley's Miracle: The Original Proof
- Morley's Redux and More
- Napoleon's and Douglas' Theorems
- Napoleon's Propeller
- Napoleon's Relatives
- Napoleon's Theorem
- On Bottema's Shoulders II
- Periodic Points of Quadratic Polynomials
- Product of Diagonals in Regular N-gon
- Remarkable Line in Cyclic Quadrilateral
- Right Isosceles Triangles on Sides of a Quadrilateral
- Three Isosceles Triangles
- Thébault's Problem I
- Thébault's Problem II
- When a Triangle is Equilateral
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, 3rd edition, 1966
- D. Pedoe, Geometry: A Comprehensive Course, Dover, 1988
Complex Numbers
- Algebraic Structure of Complex Numbers
- Division of Complex Numbers
- Useful Identities Among Complex Numbers
- Useful Inequalities Among Complex Numbers
- Trigonometric Form of Complex Numbers
- Real and Complex Products of Complex Numbers
- Complex Numbers and Geometry
- Remarks on the History of Complex Numbers
- Complex Numbers: an Interactive Gizmo
- Cartesian Coordinate System
- Fundamental Theorem of Algebra
- Complex Number To a Complex Power May Be Real
- One can't compare two complex numbers
- Riemann Sphere and Möbius Transformation
- Problems
Copyright © 1996-2009 Alexander Bogomolny
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