H+ model

Poincare upper plane model(H⁺)

You need plugins for Cabri2/Cabri3D to see files embedded in htmls below, even if you have those softwares. To download plugins,please check these: cabri2 , cabri3D ．Once you install the plugins, you can manuplate figures on the files with mouse ． To move the whole figure, please drag with Ctrl key down . For reconstraction, please click the red inverted triangle. At the bottom, there are also original cabri files which you can directly manipulate with Cabri/Cabri3D SOFTWARE (not plugins)．

There are also Geogebra(3.2) files. To see these files, you need Java(JDK/JVM) which you can download from here.

If you can read Japanese , you can get detailed explanations of these files and mathematical proves from the PDF section below. There are one-to-one links to Cabri/Geogebra files. Note that , however, you should use 'back button' of your browser when you come back from these links, if you close the brouser, it also closes original PDFs.

Cabri/Geogebra(embedded in html files)
-You need Cabri2 plugins and Java environment-
1. Definition of upper plane(H⁺).
2. Examples of hyperbolic congruent transformation
  - inverse with respect to a circle centered on the real axis
  - resolution of congruent transformations down to atoms
3. hyperbolic straight line
  - circle centered on the real axis
4. parallel & super parallel
  - definition
5. 'atoms & molecules' of hyperbolic congruent transformation
  - rotation and symmetric transformation on Euclid plane
  - rotation & symmetric transformation on H⁺- ex１, ex2
  - mobius transformation on H⁺- hyperbolic, palabolic ,elliptic
6. hyperbolic angle
  - definition
  - construction of 90 degree
7. hyperbolic circle
  - construction of hyperbolic circle
  - proves
8. parallel angle
  - definition
  - construction of parallel angle-by Geogebra, by Cabri
9. horicircle
  - horicircle parallel to the real axis
  - horicircle touching the real axis
  - all horicircles are congruent
  - horicircle is a circle with infinite radius-type1, type2
  - construction of the axis of 2 hyperbolic lines- by Geogebra, Cabri
  - construction of horicircle by Bolyai (using the construction above)
10. equi-distant line
  - 2 types of equi distant line
  - how to move from one type to the other
  - equi distant line and symmetric transformation-type1, type2
  - construction of equi distant line using symmetric transformations
    -type1,type2
11. mobius transformation
  1. standerd form-elliptic, hyperbolic, parabolic
  2. shteiner circles (cordinates by circles)-hyperbolic, elliptic, parabolic
  3. breaking down to 2 inversions-hyperbolic, elliptic, parabolic
  4. composition of 2 inversions makes a mobius transformation
12. geometric theories of hyperbolic geometry
  - sum of angles of a quadrangle is less than 360 degrees
  - sum of angles of a triangle is less than 180 gegrees-part1, part2
  - The bigger a triangle is, the smaller the sum of the angles
  - there are no similar triangles-part1, part2
  - medians meet at one point (center of gravity)
  - angle bisectors meet at one point(inner center)
  - outer center doesn't necessarily exists
  - orthocenter doesn't necessarily exists
PDF files（in Japanese)
-Please use 'back button' when you return from links!-
Cabri/Geogebra original files (*.fig,*.bbg,*.mac)
- program.zip（fig/ggb files)
- macro.zip (macros for cabri2)

Non euclidian geometry

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