Construction using conjugation with KB

Dynamic Dynamic geometry on pseudospherical surfaces [Revue .. Volume ... num ... page] 6. Modification in the conjugation with Klein-Beltrami's modele 1. Home \| 2. Lines on PS \| 3. First figures \| 4. Full rolling-up \| 5. Conjugation with KB's model \| 6. New conjugation 7. Hyperbolic PS \| 8. Elliptic PS \| 9. To play on the pseudosphere
	In the last page, the projection was the Beltrami's one, with the centre of the KB's circle Odl image of the point X Now the image of X is the point OExt, moveable by users.
fig 21 Extention of the horocycle as pseudosphere image (incircle and escribed circles) The point OExt (Origine Extented) enlarges the sheets before and after the main sheet. This figure gives a dynamic représentation of an isomorphism : given A, B, C on the pseudosphere, all is constructed on the pseudosphere by the figure in KB. Moving OExt changes all in KB and keeps the final constructions on the pseudosphere invariant. In this file, all lines on speudosphere are monosheet. So it's beter to keep A, B C on the main sheet and modify the latitudes and longitudes to have, for example, one escribed circle on three sheets, as shown in the paper.

fig 22 Golden ration on the pseudosphere The paper explains why we find the golden ration in this situation : three escribed horocycles. With the pseudosphere we can choose that one is centered on the unique point at inifinity of the pseudosphere, so, all the figure is under the associated latitude circle. The triangle MNP can be changed by A and B (latitudes are not important) : these points fixe the two other horocycles's center I and J. We can controle when U and F exists on the pseudosphere and when E and V stay on the main sheet.

fig 23 Circumcycle of a triangle On this file, move C around its latitude circle in the trigonometric direction : the center I leave the pseudosphere, and le circumcircle becomes a circumequidistant and its axis, at its begining, far from the pseudosphere, slowly becomes to exist on the pseudosphere.

fig 24 a Coxeter's dream ? It's an circumequidistant with a meridian (red on the figure) as axis : so this equidistant rolling oneself up on the pseudossphere up to infinite.

1. Home \| 2. Lines on PS \| 3. First figures \| 4. Full rolling-up \| 5. Conjugation with KB's model \| 6. New conjugation 7. Hyperbolic PS \| 8. Elliptic PS \| 9. To play on the pseudosphere