Klein model in a circle (and cross ratio)You need plugins for Cabri2/Cabri3D to see
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 Basics of Klein model
 Relations with upper sphere model;
line&circle, horicircle&equi distant line
 Relations with Poincare disk;
part1, part2
 construction of circle, equi distant line
& horicircle
 construction of inversion
 construction of circle,equi distant line
& horicircle
 distance and angle（analytic)
 angle of 'sphere model' & 'Klein model'
 distance is direction dependant
 cross ratio(double ratio) of 4 points in
space (R^{3})
 cross ratio and inversion (with respect to
a sphere)
 cross rario & distance (geometric)
 Poincare upper planeH^{+}: cross ratio&distance
 upper sphere: cross ratio &distance,
inversion which transforms H+
onto upper
sphere
 Poincare Disk: cross ratio&distance,
relations among 3 models
 Klein model: cross ratio&distance,
relations with upper sphere model
 cross ratio&angle
 Poincare upper planeH+
 upper sphere
 Poincare Disk
 Klein model
 cross ratios of various kinds
 cross ratio of 4 points on a line
 cross ratio of pencil (a set of lines which
meet together)
 cross ratio of 4 points on a circle:
when points move on a circle, when a pole moves
 'distance & angle' of Klein model in
a conic
 Klein model in a circle: angle&cross ratio of a pencil
 Klein model in a conic: distance, angle
 PDF files (in Japanese)
Please use 'back button', when you come back
from links!
 Basics of Klein model
 construction of circle, equi distant line
& horicircle
 distance and angle（analytic)
 cross ratio(double ratio) of 4 points in
space (R^{3})
 cross rario & distance (geometric) （H^{+,}Poincare Disk, sphere model)
 cross ratio&angle (same as above)
 cross ratios of various kinds
 'distance & angle' of Klein model in
a conic
 Original Cabri files
 Macros of Cabri2

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