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The principle aim of this unique text is to illuminate the beauty of the subject both with abstractions like proofs and mathematical text, and with visuals, such as abundant illustrations and diagrams. With few mathematical prerequisites, geometry is presented through the lens of linear fractional transformations. The exposition is motivational and the well-placed examples and exercises give students ample opportunity to pause and digest the material. The subject builds from the fundamentals of Euclidean geometry, to inversive geometry, and, finally, to hyperbolic geometry at the end. Throughout, the author aims to express the underlying philosophy behind the definitions and mathematical reasoning. This text may be used as primary for an undergraduate geometry course or a freshman seminar in geometry, or as supplemental to instructors in their undergraduate courses in complex analysis, algebra, and number theory. There are elective courses that bring together seemingly disparate topics and this text would be a welcome accompaniment.

Group theory --- Differential geometry. Global analysis --- Geometry --- differentiaal geometrie --- wiskunde --- geometrie --- Transformations (Mathematics) --- Transformacions (Matemàtica)

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Convolutions (Mathematics) --- Convolucions (Matemàtica) --- Convolution transforms --- Transformations, Convolution --- Distribution (Probability theory) --- Functions --- Integrals --- Transformations (Mathematics) --- Transformacions de convolució --- Transformades de convolució --- Distribució (Teoria de la probabilitat) --- Funcions --- Transformacions (Matemàtica)

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Geometria --- Matemàtica --- Congruències (Geometria) --- Dibuix lineal --- Envolupants (Geometria) --- Esfera --- Geometria algebraica --- Geometria computacional --- Geometria conforme --- Geometria convexa --- Geometria de l'espai --- Geometria diferencial --- Geometria euclidiana --- Porismes --- Programació geomètrica --- Ràtio i proporció --- Similitud (Geometria) --- Teorema de Pitàgores --- Transformacions (Matemàtica) --- Trigonometria --- Geometria en l'art --- Geometry. --- Mathematics --- Euclid's Elements

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Politops --- Teoria dels reticles --- Anàlisi estructural (Matemàtica) --- Reticles (Matemàtica) --- Teoria de reticles --- Teoria reticular --- Àlgebra abstracta --- Àlgebra de Boole --- Teoria de conjunts --- Teoria de grups --- Topologia --- Transformacions (Matemàtica) --- Percolació (Física estadística) --- Reticles cristal·lins --- Cristal·lografia matemàtica --- Hiperespai --- Politops convexos --- Polytopes. --- Discrete mathematics. --- Lattice theory --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis --- Hyperspace --- Topology

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Teoria dels reticles --- Anàlisi estructural (Matemàtica) --- Reticles (Matemàtica) --- Teoria de reticles --- Teoria reticular --- Àlgebra abstracta --- Àlgebra de Boole --- Teoria de conjunts --- Teoria de grups --- Topologia --- Transformacions (Matemàtica) --- Percolació (Física estadística) --- Reticles cristal·lins --- Cristal·lografia matemàtica --- Lattice theory. --- Lattices (Mathematics) --- Space lattice (Mathematics) --- Structural analysis (Mathematics) --- Algebra, Abstract --- Algebra, Boolean --- Group theory --- Set theory --- Topology --- Transformations (Mathematics) --- Crystallography, Mathematical