This book deals with the geometry of the 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 the 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, it justifies descriptive geometric argumentation, because its embedding in the strict mathematical language is clarified. Geometric terms are of very different kinds; they denote, so to speak, different layers of geometric thinking: some arguments use only concepts such as point, straight line, and incidence, some require angles and distances, and others symmetry. Each of these conceptual fields determines a separate subfield of geometry and a separate chapter of this book, with the exception of the one last-mentioned "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. This book is a translation of the original German 1st 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.

Autorentext

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