This book deals with the geometry of visual space in all its aspects. As in any branch of mathematics, the aim is to trace the hidden to the obvious; the peculiarity of geometry is that the obvious is sometimes literally before one's eyes.

Starting from intuition, spatial concepts are embedded in the pre-existing mathematical framework of linear algebra and calculus. The path from visualization to mathematically exact language is itself the learning content of this book. This is intended to close an often lamented gap in understanding between descriptive preschool and school geometry and the abstract concepts of linear algebra and calculus. At the same time, descriptive geometric modes of argumentation are justified because their embedding in the strict mathematical language has been clarified.

The concepts of geometry are of a very different nature; they denote, so to speak, different layers of geometric thinking: some arguments use only concepts such as point, straight line, and incidence, others require angles and distances, still others symmetry considerations. Each of these conceptual fields determines a separate subfield of geometry and a separate chapter of this book, with the exception of the last-mentioned conceptual field "symmetry", which runs through all the others:

- Incidence: Projective geometry

- Parallelism: Affine geometry

- Angle: Conformal Geometry

- Distance: Metric Geometry

- Curvature: Differential Geometry

- Angle as distance measure: Spherical and Hyperbolic Geometry

- Symmetry: Mapping Geometry.

The mathematical experience acquired in the visual space can be easily transferred to much more abstract situations with the help of the vector space notion. The generalizations beyond the visual dimension point in two directions: Extension of the number concept and transcending the three illustrative dimensions.

This book is a translation of the original German 1^st edition Geometrie ¿ Anschauung und Begriffe by Jost-Hinrich Eschenburg, published by Springer Fachmedien Wiesbaden GmbH, part of Springer Nature in 2020. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors.

Prof. Dr. Jost-Hinrich Eschenburg, University of Augsburg, Institute of Mathematics

What is geometry.- Parallelism: affine geometry.- From affine geometry to linear algebra.- Definition of affine space.- Parallelism and semiaffine mappings.- Parallel projections.- Affine coordinates and center of gravity.- Incidence: projective geometry.- Central perspective.- Far points and straight lines of projection.- Projective and affine space.-Semi-projective mappings and collineations.- Conic sections and quadrics; homogenization.- The theorems of Desargues and Brianchon.- Duality and polarity; Pascal's theorem.- The double ratio.- Distance: Euclidean geometry.- The Pythagorean theorem.-  Isometries of Euclidean space.- Classification of isometries.- Platonic solids.- Symmetry groups of Platonic solids.- Finite rotation groups and crystal groups.- Metric properties of conic sections.- Curvature: differential geometry.- Smoothness.- Fundamental forms and curvatures.- Characterization of spheres and hyperplanes.- Orthogonal hyperface systems.- Angles: conformal geometry.- Conformal mappings.- Inversions.- Conformal and spherical mappings.- The stereographic projection.- The space of spheres.- Angular distance: spherical and hyperbolic geometry. The hyperbolic space. Distance on the sphere and in hyperbolic space. Models of hyperbolic geometry.- Exercises.- Solutions