Animations

Axiom 1 of Euclidian and Hyperbolic geometry

Given two points, there is a unique line through them

Line through two points

Axiom 2 of Euclidian and hyperbolic geometry

This is an animation about the second axiom of geometry which states that a line can be prolonged infinitely

Lines have no end

Axiom 3 of Euclidian and hyperbolic geometry

This animation exhibits the third axiom that states the uniquness of a circle with a given center and a given radius

Circle through a center with given radius

Axiom 4 of Euclidian and hyperbolic geometry

This is an animation about the fourth axiom of geometry. This axiom states that all right angles are congruent

All right angles are congruent

Axiom 5 of Euclidian geometry


Animations on Euclidean Geometry

  1. Construction of equilateral triangle from a given segment (to play the animation download the pdf file and click on play).

Equilateral triangle construction.

2. Circle trhoug three points Circle trhough three given points

3. No lines int the projective are parallel Projective Plane

4. Hyperbolic lines in the Poincare disk model Hyperbolic lines

5. In the file Geometry yo can play the following animations (you must download and click at the bottom of the images to play the animations

  • Postulate of Euclidean geometry. A segment could be prolongued forever.

  • Postulate of Euclidean geometry. One could drow a circle with any center and any radius.

  • Postulate of Euclidean geometry. All right angles are congruent.

  • Fifth postulate. If a line crosses two other lines forming internal angles whose sum is less than 180 degrees, these two lines intersect on the same side where these angles are located.

  • Axiom by Gauss. Given any triangle, one could construct any triangle with larger area.

  • Construction of hyperbolic lines

  • "Short" hyperbolic lines "look" like euclidean lines.

  • Paralelisms on the plane introduced by connections.

Animations on Differential affine Geometry

6. The group of affine transformations of a group preserving a flat affine connection has as linear basis of its Lie algebra the set of complete infinitesimal affine transformations, and these can be determined as the vector fields whose images under the developing map preserve the orbit. This can be observed in the link Flows preserving an orbit