You are here

Euclidean Plane and Its Relatives

Anton Petrunin
Publication Date: 
Number of Pages: 
[Reviewed by
Joel Haack
, on

Anton Petrunin’s Euclidean Plane and its Relatives: A minimalistic introduction is indeed a lean textbook for a foundations of geometry course with a calculus prerequisite. The text is based on the experience of the author teaching such a course at Penn State University. This is a review of version 0.9 of the text, so it is possible that a later version will have been edited to clear up some issues with nonidiomatic language that make the text somewhat awkward to read.

The author has provided an axiomatic presentation of Euclidean geometry based on Birkhoff’s metric approach, hence providing an introduction to metric spaces as well. Euclidean geometry is developed up through a chapter on triangles. Introductions to inversive geometry, hyperbolic geometry, incidence geometry, spherical geometry, and geometric constructions conclude the text. I found the presentation of inversions and the development of the Poincaré disk model for hyperbolic geometry appealing.

In what way is the presentation minimalist?  It is a minimal introduction to Euclidean and non-Euclidean geometry which is still rigorous.

Unlike some of the competitive texts, there is almost no coverage of history, which enriches, for example, Marvin Greenberg’s Euclidean and Non-Euclidean Geometries: Development and History. Instead, Petrunin’s text maintains its focus on providing a rigorous introduction to geometry. The concept of “area” is not developed; as the author notes, “The formal definition of area is quite long and tedious.” [96])

Instructors seeking a text for future secondary geometry teachers may prefer a text that covers a different selection of topics. If the audience of the geometry course is instead sophomore mathematics majors, this approach would be of interest. In any case, this book would be a useful reference for someone teaching such a course.

The text of this book is also available on

Joel Haack is Professor of Mathematics at the University of Northern Iowa.


Prerequisite. Overview.

1 Preliminaries

What is axiomatic approach? What is model? Metric spaces. Examples. Shortcut for distance. Isometries, motions and lines. Half-lines and segments. Angles. Reals modulo 2π. Continuity. Congruent triangles.

Euclidean geometry

2 The Axioms

The Axioms. Lines and half-lines. Zero angle. Straight angle. Vertical angles.

3 Half-planes

Sign of angle. Intermediate value theorem. Same sign lemmas. Half-planes. Triangle with the given sides.

4 Congruent triangles

Side-angle-side condition. Angle-side-angle condition. Isosceles triangles. Side-side-side condition.

5 Perpendicular lines

Right, acute and obtuse angles. Perpendicular bisector. Uniqueness of perpendicular. Reflection. Perpendicular is shortest. Angle bisectors. Circles. Geometric constructions.

6 Parallel lines and similar triangles

Parallel lines. Similar triangles. Pythagorean theorem. Angles of triangle. Transversal property. Parallelograms. Method of coordinates.

7 Triangle geometry

Circumcircle and circumcenter. Altitudes and orthocenter. Medians and centroid. Bisector of triangle. Incenter.

Inversive geometry

8 Inscribed angles

Angle between a tangent line and a chord. Inscribed angle. Inscribed quadrilaterals. Arcs.

9 Inversion

Cross-ratio. Inversive plane and circlines. Ptolemy’s identity. Perpendicular circles. Angles after inversion.

Non-Euclidean geometry

10 Absolute plane

Two angles of triangle. Three angles of triangle. How to prove that something can not be proved? Curvature.

11 Hyperbolic plane

Poincaré disk model. The plan. Auxiliary statements. Axioms: I, II, III, IV, h-V.

12 Geometry of h-plane

Angle of parallelism. Inradius of triangle. Circles, horocycles and equidistants. Hyperbolic triangles. Conformal interpretation.

Incidence geometry

13 Affine geometry

Affine transformations. Constructions with parallel tool and ruler. Matrix form. On inversive transformations.

14 Projective geometry

Real projective plane. Euclidean space. Perspective projection. Projective transformations. Desargues’ theorem.Duality. Axioms.

Additional Topics

15 Spherical geometry

Spheres in the space. Pythagorean theorem. Inversion of the space. Stereographic projection. Central projection.

16 Klein model

Special bijection of h-plane to itself. Klein model. Hyperbolic Pythagorean theorem. Bolyai’s construction.

17 Complex coordinates

Complex numbers. Complex coordinates. Conjugation and absolute value. Euler’s formula. Argument and polar coordinates. Möbius transformations. Elementary transformations. Complex cross-ratio. Schwarz–Pick theorem.

18 Geometric constructions

Classical problems. Constructable numbers. Construction with set square. More impossible constructions.




Used resources