Three 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