Lobachevskian Plane Models

A unique base, which is an image of all three mod­els on a Lobachevsky plane, is a trans­par­ent semi­sphere with semi­cir­cles, drawn on it, which are per­pen­dic­u­lar to its bor­der-an equa­tor. So if an equa­tor of a semi­sphere lies in a hor­i­zon­tal plane, a cir­cle should be drawn in ver­ti­cal planes.

A model of Poin­caré in a cir­cle can be received, if one places a semi­sphere on a table by a pole, and a point source of light ‘in an oppo­site, North­ern pole of a sphere. Tak­ing into con­sid­er­a­tion that it is a point source, namely rays dis­perse from it on straight lines to dif­fer­ent sides, one receives a stere­o­graphic pro­jec­tion of the sphere (in our case semi­sphere) on the plane. A stere­o­graphic pro­jec­tion saves angles between lines, and turns any cir­cles on the sphere into cir­cles on a plane. (To be more pre­cise, cir­cles, which don’t pass through the cen­tre of a pro­jec­tion, pass to cir­cles on the plane, and those, that pass through it-to straight lines).

An equa­tor of a semi­sphere turns into an absolute on a Lobachevsky plane (its points don’t belong to it), which is a bor­der of a Poin­caré’s model in a cir­cle. Cir­cles on a semi­sphere, per­pen­dic­u­lar to an equa­tor, turn into cir­cles in a cir­cle, per­pen­dic­u­lar to absolute. Namely they are “straight lines” on Lobachevsky plane.

A model of Bel­trami-Klein in a cir­cle (a pro­jec­tive model) will be cre­ated, if in case of the same posi­tion of a semi­sphere a source of light will be “ran into“, namely will lighten a semi­sphere by ver­ti­cal par­al­lel rays. An absolute will be a pro­jec­tion of an equa­tor, and straight planes of Lobachevsky are hordes of a cir­cle. A pro­jec­tion of equa­tor will be an absolute, and straight planes of Lobachevsky are hordes of a cir­cle.

A model of Poin­caré on a semi­plane can be received, if a semi­sphere is set­tled on a table so that its equa­tor is in a ver­ti­cal plane, and source of light-in the North Pole. steno­graphic pro­jec­tion, where a semi­sphere is pro­jected in a plane and an equa­tor of a semi­sphere goes into a straight line, arouses again, - an absolute of Lobachevsky’s plane in this model. Straight lines in Poin­caré’s model in a semi­plane will be cir­cles and straight lines, per­pen­dic­u­lar to an absolute,-pro­jec­tions from a semi­sphere of cir­cles, which pass through the North Pole.

If one rotates a semi­sphere, one can observe trans­for­ma­tions (move­ments) of Lobachevsky plane. In case of Poin­caré model on a semi­plane a semi­sphere should be rotated around the axis (so that the equa­tor is always located in the same plane). Trans­for­ma­tions (for read­ers-ellip­tic) planes of Lobachevsky in case the Poin­caré model can be observed, if we rotate a semi­sphere around the axis, bend­ing it to a ver­ti­cal line, which join the source of light and tan­gency point of semi­sphere and plane of pro­jec­tion. (In case of this posi­tion of a semi­sphere a stere­o­graphic pro­jec­tion is received again. Its listed qual­i­ties guar­an­tee, that in this case a model of Poin­caré in a cir­cle is also received).

Straight planes of Lobachevsky, which pass through one point and are par­al­lel to the given line, are con­structed accord­ing to the model of Poin­caré in a cir­cle. For the fixed cir­cle on a semi­sphere, for the cho­sen points two cir­cles that arise from dif­fer­ent ends of the fixed one, can be made. As only cir­cles, per­pen­dic­u­lar to an equa­tor, are drawn, their pro­jec­tions-straight planes of Lobachevsky-go out of “com­mon” points on the absolute, per­pen­dic­u­lar to it (remem­ber that points of an absolute are not involved in tha Lobachevsky plane). It means that any of newly con­structed straight lines is par­al­lel to the fixed one. And any straight line, which passes through the cho­sen point and lies “be­tween” the con­structed ones, is called “di­ver­gent” with the given one and also doesn’t have com­mon points with it