Dynamic Visualization of Hyperbolic Geometry in the

Dynamic Visualization of Hyperbolic Geometry in the Poincaré Disk Model Lajos Szilassi1 1 Faculty 2 The

Zoltán Kovács2

of Education, University of Szeged, Hungary

Private University College of Education of the Diocese of Linz, Austria

Aplimat conference 5–7 February 2019 Bratislava, Slovakia

Second author is partially supported by the grant MTM2017-88796-P from the Spanish MINECO and the ERDF.

Abstract

In a letter to his father, János Bolyai wrote in 1823: “out of nothing I have created a strange new universe.” This new universe is demonstrated in an electronic GeoGebra book that makes the basics of hyperbolic geometry easy to understand and visually clear. Even more important aim is to sketch up a new perspective for some well-known Euclidean notions and constructions. Most of the problems discussed in this talk do not use the Euclidean axiom of parallels.

A system of axioms as introduced in the geometry class A classification of the most commonly used axioms

1. incidence axioms 2. order axioms 3. congruence axioms 4. continuity (measuring) axioms 5. axiom of parallels

Absolute geometry No axiom of parallels is present


Absolute geometry No axiom of parallels is present


Three important statements They refer to three different kinds of geometry

1. Lines perpendicular to a given line of the plane are not intersecting, thus there are non-intersecting lines, i.e. parallel lines in the plane. A theorem that follows from axioms 1–4.


1. Lines perpendicular to a given line of the plane are not intersecting, thus there are non-intersecting lines, i.e. parallel lines in the plane. A theorem that follows from axioms 1–4. 2. Given a line and a point outside it, there is at most one line through the given point which lies in the plane of the given line and point so that the two lines are parallel. Axiom of parallels in Euclidean geometry.


1. Lines perpendicular to a given line of the plane are not intersecting, thus there are non-intersecting lines, i.e. parallel lines in the plane. A theorem that follows from axioms 1–4. 2. Given a line and a point outside it, there is at most one line through the given point which lies in the plane of the given line and point so that the two lines are parallel. Axiom of parallels in Euclidean geometry. 3. Given a line and a point outside it, there are at least two lines through the given point which lie in the plane of the given line and point so that the two lines are parallel. Axiom of parallels in hyperbolic geometry.


1. Lines perpendicular to a given line of the plane are not intersecting, thus there are non-intersecting lines, i.e. parallel lines in the plane. A theorem that follows from axioms 1–4.


1. Lines perpendicular to a given line of the plane are not intersecting, thus there are non-intersecting lines, i.e. parallel lines in the plane. A theorem that follows from axioms 1–4. 2. Given a line and a point outside it, there is at most one line through the given point which lies in the plane of the given line and point so that the two lines are parallel. Axiom of parallels in Euclidean geometry.





“At least two. . .”

Bolyai (1802-1860) and Lobachevsky (1792-1856)

The Poincaré model of hyperbolic geometry H-plane, H-point, H-line, H-segment, H-reflection

Perpendicular bisectors Are they always concurrent?

Perpendicular bisectors A theorem that is valid in absolute geometry

Theorem If any two perpendicular bisectors of a triangle intersect each other, then all the three perpendicular bisectors are concurrent. The concurrency point is the center of the circumcircle of the triangle.

Congruency A definition of congruent objects without the concept of distance

Definition Two geometric objects are congruent if there exists a finite number of consecutive reflections that transform one object into the other.

Further theorems that are valid in absolute geometry

Theorem The three medians of a triangle are concurrent.

Further theorems that are valid in absolute geometry

Theorem The three medians of a triangle are concurrent.

Theorem In cyclic quadrilaterals the sums of opposite angles are equal. (In Euclidean geometry in cyclic quadrilaterals the sums of opposite angles are equal to 180o .)

α+γ =β+δ

What would Thales ask from Bolyai?

Constructing tangents to a circle through an external point

Problem: Given the circle c (with its center O and a circumpoint of it) and a point P outside it. The tangents to c through P are to be constructed.


Problem: Given the circle c (with its center O and a circumpoint of it) and a point P outside it. The tangents to c through P are to be constructed. Solution: Use Thales’ circle theorem.


Problem: Given the circle c (with its center O and a circumpoint of it) and a point P outside it. The tangents to c through P are to be constructed. Solution: Use Thales’ circle theorem.


Problem: Given the circle c (with its center O and a circumpoint of it) and a point P outside it. The tangents to c through P are to be constructed. Solution: Use. . .


Problem: Given the circle c (with its center O and a circumpoint of it) and a point P outside it. The tangents to c through P are to be constructed. Solution: Use Euclid’s idea from Elements III/17.

Constructing tangents to a circle through an external point Using Euclid’s idea from Elements Book III Proposition 17

Euclidean plane

Hyperbolic plane

Conclusion I A set of tools provided as a framework in the dynamic geometry system GeoGebra I 10 basic tools (hyperbolic line, segment, ray, reflection, circle, perpendicular bisector, perpendicular line, angle bisector, intersection of lines, intersection of circle segments) I 4 additional tools (hyperbolic angle, distance unit setting, distance, relationship of lines)

I A GeoGebra book with 11 chapters and 70+ applets I Full text available only in Hungarian at the moment I Parts are also available in English (still a work in progress)

ˇ Dakujem za pozornosˇt! Thank you!