ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ᠊ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ

ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ‍ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ

ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ^{Лобачевский} ᠮᠠᠨ ᠤ ᠶᠢᠷᠲᠢᠨᠴᠦ ᠳ᠋ᠦ ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋  ᠦᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ^geometry᠂ ᠮᠤᠷᠤᠶ᠋ᠢᠯᠲᠠ ᠲᠠᠢ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠬᠣᠶᠠᠷ  ᠤᠨ ᠠᠯᠢ ᠨᠢ ᠪᠣᠳᠠ ᠲᠠᠢ ᠲᠣᠬᠢᠷᠠᠬᠤ ᠪᠤᠢ ᠭᠡᠰᠡᠨ ᠠᠰᠠᠭᠤᠯᠲᠠ ᠳᠡᠪᠰᠢᠭᠦᠯᠵᠡᠢ᠃
ᠡᠭᠦᠪᠡᠷ ‍ᠢᠢᠡᠨ ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋^Euclid  ‍ᠦᠨ ᠪᠤᠰᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ‍ᠢ ᠦᠨᠳᠦᠰᠦᠯᠡᠵᠦᠬᠦᠢ᠃

ᠠᠻᠰᠢᠣᠮ

D ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠫᠠᠷᠠᠯᠯᠧᠯparallel ᠠᠭᠠᠳ M ᠴᠡᠭ ‍ᠢᠢᠡᠷ ᠳᠠᠭᠠᠷᠢᠬᠤ d1᠂ d2᠂ d3 ᠲᠡᠢ ᠠᠳᠠᠯᠢ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠣᠯᠠᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋

ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠪᠠ ᠲᠡᠭᠦᠨ ᠳ᠋ᠤ ᠦᠯᠦ ᠬᠠᠷᠢᠶᠠᠯᠠᠭᠳᠠᠬᠤ ᠴᠡᠭ ‍ᠦᠨ ᠣᠷᠣᠰᠢᠵᠤ ᠪᠠᠢᠢᠭ᠎ ‍ᠠ᠋ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳ᠋ᠤ ᠲᠡᠷᠡ ᠴᠡᠭ ‍ᠢᠢ ᠳᠠᠭᠠᠷᠢᠭᠤᠯᠤᠨ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠦᠯᠦ ᠣᠭᠲᠠᠯᠠᠯᠴᠠᠬᠤ ᠨᠢᠭᠡ ᠡᠴᠡ ᠣᠯᠠᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠲᠠᠵᠤ ᠪᠣᠯᠤᠨ᠎ ‍ᠠ᠋᠃

ᠡᠨᠡ ᠪᠣᠯ ᠯᠣᠪᠠᠴᠧᠣᠰᠻᠢ ‍ᠢᠢᠨ ᠳᠡᠪᠰᠢᠭᠦᠯᠦᠭᠰᠡᠨ ᠫᠣᠰᠲ᠋ᠦ᠋ᠯᠠᠲ^postulate᠃

ᠡᠭᠦᠨ ‍ᠢ ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋ ‍ᠦᠨ 5 ᠳ᠋ᠤᠭᠠᠷ ᠫᠣᠰᠲ᠋ᠦ᠋ᠯᠠᠲ ‍ᠤᠨ ᠣᠷᠤᠨ ᠳ᠋ᠤ ᠰᠠᠨᠠᠯ ᠪᠣᠯᠭᠠᠭᠰᠠᠨ ᠪᠢᠯᠡ᠃

ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋ ‍ᠦᠨ 5 ᠳ᠋ᠤᠭᠠᠷ ᠫᠣᠰᠲ᠋ᠦ᠋ᠯᠠᠲ ᢩᠪᠤᠶᠤ ᠠᠻᠰᠢᠣᠮ ᠨᠢ᠄

ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠪᠠ ᠲᠡᠭᠦᠨ ᠳ᠋ᠤ ᠦᠯᠦ ᠬᠠᠷᠢᠶᠠᠯᠠᠭᠳᠠᠬᠤ ᠴᠡᠭ ‍ᠦᠨ ᠣᠷᠣᠰᠢᠵᠤ ᠪᠠᠢᠢᠭ᠎ ‍ᠠ᠋ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳ᠋ᠤ ᠲᠡᠷᠡ ᠴᠡᠭ ‍ᠢᠢ ᠳᠠᠭᠠᠷᠢᠭᠤᠯᠤᠨ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠫᠠᠷᠠᠯᠯᠧᠯ ᠭᠠᠭᠴᠠ ᠪᠥᠭᠡᠳ ᠵᠥᠪᠬᠡᠨ ᠭᠠᠭᠴᠠ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠲᠠᠵᠤ ᠪᠣᠯᠤᠨ᠎ ‍ᠠ᠋

ᠭᠡᠵᠦ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠳᠠᠳᠠᠭ᠃

ᠯᠣᠪᠠᠴᠧᠡᠰᠻᠢ ‍ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠨᠢ ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋ ‍ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ‍ᠦᠨ ᠡᠨᠡ ᠠᠻᠰᠥᠣᠮ ᠡᠴᠡ ᠪᠤᠰᠤᠳ ᠰᠠᠭᠤᠷᠢ ᠦᠵᠡᠯ ᠪᠠᠷᠢᠮᠲᠠᠯᠠᠯ ‍ᠤᠳ ᠳᠡᠭᠡᠷ ‍ᠡ᠋ ᠦᠨᠳᠦᠰᠦᠯᠡᠭᠰᠡᠨ ᠪᠠᠢᠢᠳᠠᠭ᠃

ᠮᠥᠷᠳᠡᠯᠭᠡ

ᠬᠠᠪᠲᠠᠭᠠᠦ ᠪᠤᠰᠤ ᠭᠠᠳᠠᠷᠭ᠎ ‍ᠠ᠋ ‍ᠳᠠᠬᠢ ᠲᠡᠭᠦᠰ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ

ᠯᠣᠪᠠᠴᠡᠹᠰᠻᠢ ‍ᠢᠢᠨ ᠵᠠᠷᠢᠮ ᠲᠾᠧᠣᠷᠧᠮ ‍ᠦᠳ ᠨᠢ ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋ ‍ᠦᠨ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠬᠤ ᠲᠾᠧᠣᠷᠧᠮ ‍ᠦᠳ ᠡᠴᠡ ᠢᠯᠭᠠᠭ᠎ ‍ᠠ᠋ ᠲᠠᠢ ᠠᠵᠡᠢ᠃

ᠨᠢᠭᠡᠨ ᠲᠾᠧᠣᠷᠧᠮ ᠨᠢ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ‍ᠤ $\alpha$᠂ $\beta$᠂ $\gamma$ ᠥᠨᠴᠦᠭ ‍ᠦᠳ ‍ᠦᠨ ᠨᠡᠢᠢᠯᠡᠪᠦᠷᠢ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠦᠭ ᠡᠴᠡ ᠢᠠᠮᠠᠭᠲᠠ ᠪᠠᠭ᠎ ‍ᠠ᠋ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋ ᠭᠡᠳᠡᠭ᠃

ᠲᠡᠭᠰᠢ ᠥᠨᠴᠦᠭ ᠪᠠ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ‍ᠤ ᠥᠨᠴᠦᠭ ‍ᠦᠳ ‍ᠦᠨ ᠨᠡᠢᠢᠯᠡᠪᠦᠷᠢ ‍ᠢᠢᠨ ᠢᠯᠭᠠᠭ᠎ ‍ᠠ᠋ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ‍ᠤ ᠳ᠋ᠧᠹᠧᠻᠲ ᠭᠡᠨ᠎ ‍ᠡ᠋᠃

ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ‍ᠤ ᠲᠠᠯᠠᠪᠠᠢ ᠨᠢ ᠳᠠᠷᠠᠭᠠᠬᠢ ᠪᠠᠢᠢᠳᠠᠯ ‍ᠢᠢᠠᠷ ᠲᠣᠭᠠᠴᠠᠭᠠᠯᠠᠭᠳᠠᠨ᠎ ‍ᠠ᠋᠄





$S=(\pi -\alpha -\beta -\gamma )({\frac {1}{\sqrt {-K}}})^{2}{}{}\!$



ᠡᠭᠦᠨ ᠳ᠋ᠦ K ᠨᠢ ᠭᠠᠦ᠋ᠰᠰ ‍ᠤᠨ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ ᠭᠡᠳᠡᠭ᠃ ᠡᠨᠡ ᠨᠢ ᠲᠤᠬᠠᠢᠢ ‍ᠢᠢᠨ ᠴᠡᠭ ‍ᠦᠨ ᠣᠷᠴᠢᠮ ᠳ᠋ᠠᠬᠢ ᠭᠠᠳᠠᠷᠭ᠎ ‍ᠢᠢᠨ ᠮᠤᠷᠦᠢᠢᠯᠲᠠ ‍ᠢᠢᠨ ᠬᠡᠮᠵᠢᠶ ‍ᠡ᠋ ᠶᠤᠮ᠃

ᠪᠠᠰᠠ ᠨᠢᠭᠡ ᠮᠥᠷᠳᠡᠯᠭᠡ ᠨᠢ "ᠬᠣᠶᠠᠷ ᠲᠥᠰᠦᠲᠠᠢ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ‍ᠣ ᠲᠠᠯᠠᠪᠠᠢ ᠲᠡᠩᠴᠡᠭᠦᠦ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋" ᠬᠡᠮᠡᠨ᠎ ‍ᠡ᠋᠃ ᠥᠭᠡᠷ ‍ᠡ᠋ ‍ᠢᠢᠡᠷ ᠬᠡᠯᠡᠪᠡᠯ ᠲᠥᠰᠦᠲᠠᠢ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠨᠤᠭᠤᠳ ᠻᠣᠨ᠍ᠭ᠌ᠷᠦ᠋ᠧᠨ᠍ᠲ^congruent ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃

ᠡᠨᠡ ᠮᠡᠲᠦ ᠪᠡᠷ ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ‍ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ‍ᠦᠨ ᠮᠥᠷᠳᠡᠯᠭᠡ ᠨᠦᠭᠦᠳ ᠬᠡᠳᠦᠢ ᠡᠲᠡᠭᠡᠳ ᠭᠠᠵᠢ᠂ ᠫᠠᠷᠠᠳᠣᠻᠰ^paradox ᠶᠤᠮ ᠰᠢᠭ ᠪᠣᠯᠪᠠᠴᠤ ᠤᠯᠠᠮᠵᠢᠯᠠᠯᠲᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠡᠢ ᠪᠦᠷᠢᠨ ᠠᠳᠠᠯᠢ ᠡᠷᠬᠡ ᠲᠡᠢ ᠠᠵᠡᠢ᠃

ᠰᠢᠨ᠎ ‍ᠡ᠋ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠨᠢ ᠣᠷᠤᠨ ᠵᠠᠢ ‍ᠢᠢᠨ ᠮᠥᠨ ᠴᠢᠨᠠᠷ ‍ᠤᠨ ᠲᠤᠬᠠᠢ ᠣᠢᠢᠯᠠᠭᠠᠯᠲᠠ ‍ᠳᠤ ᠰᠠᠭᠤᠷᠢ ᠥᠭᠡᠷᠡᠴᠢᠯᠡᠯᠲᠡ ‍ᠢᠢ ᠠᠪᠴᠢᠷᠠᠭᠰᠠᠨ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃

ᠨᠡᠷ ‍ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ

ᠾᠢᠫᠧᠷᠪᠣᠯ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ - hyperbolic geometry - гиперболическая геометрия, геометрия Лобачевского
ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋  ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ - Euclidean geometry - евклидова геометрия
ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋  ᠦᠨ ᠪᠤᠰᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ - non-Euclidean geometry - неевклидова геометрия
ᠴᠡᠭ - point - точка
ᠰᠢᠯᠠᠭᠤᠨ - line - прямая

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