Geometry of Complex Numbers (Dover Books on Mathematics)
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Illuminating, widely praised book on analytic geometry of circles, the Moebius transformation, and 2-dimensional non-Euclidean geometries. "This book should be in every library, and every expert in classical function theory should be familiar with this material. The author has performed a distinct service by making this material so conveniently accessible in a single book." — Mathematical Review.
transformations (similarities) form a group. The ratio (z1; z2, z3) is a three-point invariant of this group. Conversely: every transformation Z = f(z) for which (z1; z2, z3) is invariant, is an integral Moebius transformation z = az + b. 5. All displacements (or motions) z = eixz + b (α real) form a subgroup of the group of all integral Moebius transformations. For the subgroup the distance |z1 – z2| is a two-point invariant. 6. All rotations about 0 form a group for which |z| is a one-point
(L.A. § 20). In § 1 it will be shown that every hermitian matrix defines a circle (in a certain sense) and that and λ (λ real ≠ 0) define the same circle. Thus we shall speak of ‘the circle .’ The determinant of a square matrix will be denoted by ׀׀, the modulus of ׀׀, however, by ׀ det ׀. Transformations play an important role in our exposition. A transformation Z = f(z) is given by a complex-valued function f(z) which associates with every point z in the plane of the complex numbers (or in a
well as the anti-homographies are circle-preserving transformations.14 Are there other circle-preserving transformations in the completed plane? The question is similar and in fact closely related to the corresponding problem in projective geometry. It is almost obvious that every invertible linear homogeneous transformation in three homogeneous coordinates represents a collineation in the projective plane, that is, a uniquely invertible transformation carrying points into points and straight
points of the sphere; thus a given pair of points cannot be turned onto every other pair of points. (ii) A similar argument can be applied in the parabolic case. The complex number plane, dotted at infinity, is a domain of simple transitivity for the group of euclidean displacements. A pair of points z1, z2 can be transported into another pair Z1, Z2 if and only if |Z1−Z2| = |z1−z2|. Thus the group is not two-fold transitive. (iii) The hyperbolic group has the interior of the unit circle |z| <
greater we choose b. The angle α* is called the angle of parallelism. 7. All hyperbolic straight lines passing through a point z0 of which are ultra-parallel to a given hyperbolic straight line I not through z0, are situated within the external angle formed by the two hyperbolic parallels to I through z0. Show that any two non-intersecting ultra-parallels to I have a common perpendicular hyperbolic line. 8. Construct an asymptotic triangle having the angle sum zero. Show that four common