10:37, 20 Гуравдугаар сар 2021-ий байдлаарх одоогийн засвар

ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ[засварлах | кодоор засварлах]

ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ ᠲᠠᠢ ᠣᠭᠲᠠᠷᠭᠤᠢ ‍ᠢᠢᠨ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠨᠢ ᠷᠢᠮᠠᠨ ‍ᠤ ᠵᠤᠤᠪᠠᠩ ᠭᠠᠳᠠᠷᠭᠤ ‍ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠶᠤᠮ᠃

ᠷᠢᠮᠠᠨ^Riemann  ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠷ^geometry ᠨᠢ ᠲᠣᠭᠲᠠᠮᠠᠯ ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠶ᠋ᠢᠯᠲᠠ ᠲᠠᠢ ᠭᠠᠳᠠᠷᠭᠤ ᠪᠤᠶᠤ ᠵᠤᠤᠪᠠᠩ ᠭᠠᠳᠠᠷᠭᠤ ᠳᠡᠭᠡᠷ᠎ᠡ ᠬᠡᠷᠡᠭᠵᠢᠳᠡᠭ᠃^[1]

ᠤᠯᠠᠮᠵᠢᠯᠠᠯᠲᠤ ᠪᠤᠰᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ‍ᠦᠨ ᠡᠨᠡ ᠬᠡᠯᠪᠡᠷᠢ ‍ᠢᠢ ᠮᠥᠨ ᠧᠯᠯᠢᠫᠰ ‍ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ^{elliptic geometry} ᠭᠡᠳᠡᠭ᠃^[2] ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋^Euclid ‍ᠦᠨ ᠪᠤᠶᠤ ᠤᠯᠠᠮᠵᠢᠯᠠᠲᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠨᠢ ᠲᠡᠭ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ ᠲᠠᠢ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠣᠷᠤᠨ ᠵᠠᠢ ‍ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠡᠷ ᠪᠠᠢᠢᠳᠠᠭ ᠪᠣᠯ ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ‍ᠢᠢᠨ^{Лобачевский} ᠬᠢ ᠨᠢ ᠰᠥᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ ᠲᠠᠢ ᠣᠷᠤᠨ ᠵᠠᠢ ‍ᠢᠢ ᠳᠦᠷᠰᠦᠯᠡᠳᠡᠭ ᠪᠢᠯᠡ᠃

ᠬᠠᠷᠢᠨ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠨᠢ ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ ᠲᠠᠢ ᠭᠦᠳᠦᠭᠦᠷ ᠣᠷᠤᠨ ᠵᠠᠢ ‍ᠢᠢᠨ ᠵᠠᠭᠪᠤᠷ ᠪᠣᠯᠤᠨ᠎ ‍ᠠ᠋᠃

ᠷᠢᠮᠠᠨ ᠥᠪᠡᠷ ‍ᠦᠨ ᠪᠠᠢᠢᠭᠤᠯᠤᠭᠰᠠᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ‍ᠢ ᠬᠣᠶᠠᠷ ᠬᠡᠮᠵᠢᠭᠳᠠᠬᠦᠨᠲᠦ ᠬᠠᠪᠲᠠᠭᠠᠢ᠂ ᠭᠤᠷᠪᠠᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠬᠦᠨᠲᠦ ᠬᠢᠭᠡᠳ ᠲᠡᠭᠦᠨ ᠡᠴᠡ ᠳᠡᠭᠡᠭᠰᠢ ᠬᠡᠮᠵᠢᠭᠳᠡᠬᠦᠨᠲᠦ ᠣᠭᠲᠠᠷᠭᠤᠢ ᠳ᠋ᠤ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠪᠠᠢᠢᠳᠠᠭ᠃

ᠠᠻᠰᠢᠣᠮ[засварлах | кодоор засварлах]

ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠲᠡᠭᠰᠢ ᠥᠨᠴᠦᠭ ‍ᠢᠢᠡᠷ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠵᠤ ᠪᠤᠢ ᠬᠣᠶᠠᠷ ᠰᠢᠯᠠᠭᠤᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠪᠠᠨ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠨ᠎ ‍ᠠ᠋

ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠨᠢᠭᠡ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳ᠋ᠤ ᠣᠷᠣᠰᠢᠵᠤ ᠪᠠᠢᠢᠭ᠎ ‍ᠠ᠋ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠤᠰ ᠪᠦᠷᠢ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠨ᠎ ‍ᠠ᠋᠃ ^[3]

ᠪᠠᠰᠠ᠄

ᠬᠡᠷᠪᠡ $g$ ᠰᠢᠯᠠᠭᠤᠨ ᠱᠤᠭᠤᠮ ᠪᠥᠭᠡᠳ $P$ ᠴᠡᠭ ᠨᠢ ᠲᠡᠭᠦᠨ ‍ᠦ ᠭᠠᠳᠠᠨ᠎ ᠎ᠠ ᠣᠷᠤᠰᠢᠵᠤ ᠪᠠᠢᠢᠭ᠎ ‍ᠠ᠋ ᠪᠣᠯ $g$ ᠰᠢᠯᠠᠭᠤᠨ ᠪᠠ $P$ ᠴᠡᠭ ‍ᠢ ᠳᠠᠮᠨᠠᠭᠰᠠᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳ᠋ᠤ $g$ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠦᠯᠦ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠬᠤ ᠨᠢᠭᠡ ᠴᠦ ᠰᠢᠯᠠᠭᠤᠨ $h$ ᠪᠠᠢᠢᠬᠤ ᠦᠭᠡᠢ᠃^[4]

ᠪᠠᠰᠠ᠄

ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠳ᠋ᠤ ᠫᠧᠷᠫᠧᠨ᠍ᠡᠳᠢᠻᠦ᠋ᠯᠠᠷ^{perpendicular} ᠬᠣᠶᠠᠷ ᠰᠢᠯᠠᠭᠤᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠪᠠᠨ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠨ᠎ ‍ᠠ᠋᠃

ᠶᠡᠷᠦ ᠨᠢ᠂ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ ᠨᠢᠭᠡ ᠲᠠᠯ ‍ᠠ᠋ ᠳᠠᠬᠢ ᠪᠦᠬᠢ ᠫᠧᠷᠫᠧᠨ᠍ᠳᠢᠻᠦ᠋ᠯᠠᠷ ‍ᠤᠳ ᠲᠡᠷᠡ ᠬᠦ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ "ᠦᠨᠡᠮᠯᠡᠬᠦᠢ ᠲᠤᠢᠢᠯ" ᠬᠡᠮᠡᠬᠦ ᠭᠠᠭᠴᠠ ᠴᠡᠭ ᠲᠦ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠨ᠎ ‍ᠠ᠋᠃

ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ ᠨᠥᠭᠦᠭᠡ ᠲᠠᠯ ‍ᠠ᠋ ᠳ᠋ᠠᠬᠢ ᠫᠧᠷᠫᠧᠨ᠍ᠳᠢᠻᠦ᠋ᠯᠠᠷ ‍ᠤᠳ ᠴᠤ ᠮᠥᠨ ᠨᠢᠭᠡ ᠴᠡᠭ ᠲᠦ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠨ᠎ ‍ᠠ᠋᠃

ᠬᠣᠶᠠᠷ ᠡᠰᠡᠷᠭᠦ ᠲᠠᠯ ‍ᠠ᠋ ᠳ᠋ᠠᠬᠢ ᠲᠤᠢᠢᠯ ‍ᠤᠳ ᠨᠢᠭᠡ ᠯᠡ ᠲᠤᠢᠢᠯ ᠪᠠᠢᠢᠨ᠌ ᠊ᠠ᠋᠃

ᠤᠯᠠᠮᠵᠢᠯᠠᠯᠲᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠰᠢᠯᠠᠭᠤᠨ ᠨᠢ ᠬᠣᠶᠠᠷ ᠵᠦᠭ ᠲᠦ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠢᠢᠳᠠᠭ ᠪᠣᠯ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠰᠢᠯᠠᠭᠤᠨ ᠨᠢ ᠪᠢᠲᠠᠭᠦᠦ ᠪᠠᠢᠢᠳᠠᠭ᠂ ᠥᠭᠡᠷ ‍ᠡ᠋ ᠪᠠᠡ ᠬᠡᠯᠡᠪᠡᠯ ᠬᠣᠶᠠᠷ ᠦᠵᠦᠭᠦᠷ ᠨᠢ ᠠᠭᠤᠯᠵᠠᠨ ᠨᠡᠢᠢᠯᠡᠳᠡᠭ᠃^[5]

ᠰᠢᠯᠠᠭᠤᠨ ᠭᠡᠵᠦ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠪᠥᠮᠪᠦᠷᠴᠡᠭ ‍ᠦᠨ ᠪᠦᠰᠡᠯᠡᠭᠦᠷ ᠲᠣᠭᠤᠷᠢᠭ ‍ᠤᠳ ‍ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠴᠢᠩᠬ᠎ᠠ ᠡᠰᠡᠷᠭᠦ ᠲᠠᠯ ‍᠋ᠠ᠋ ᠳ᠋ᠤ ᠪᠠᠢᠢᠷᠢᠯᠠᠬᠤ ᠴᠡᠭ ‍ᠦᠳ ‍ᠦᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯᠵᠤ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃

ᠰᠢᠯᠠᠭᠤᠨ ‍ᠢ ᠢᠩᠭᠢᠵᠦ ᠲᠣᠳᠤᠷᠬᠠᠶ᠋ᠢᠯᠠᠭᠰᠠᠨ ‍ᠢᠢᠠᠷ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ ᠤᠷᠲᠤ ᠨᠢ ᠳᠠᠷᠠᠭᠠᠬᠢ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ‍ᠢᠢᠡᠷ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠳᠠᠨ᠎ ‍ᠠ᠋᠄

$L=\pi r$

ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄
L ᠨᠢ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ ᠤᠷᠲᠤ᠂
r ᠨᠢ ᠬᠡᠭᠡᠰᠦ (ᠷᠠᠳᠢᠦ᠋ᠰ^radius)᠃

ᠧᠯᠯᠢᠫᠰ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ‍ᠤ ᠥᠨᠴᠦᠭ ‍ᠦᠳ ‍ᠦᠨ ᠨᠡᠢᠢᠯᠡᠪᠦᠷᠢ ᠢᠮᠠᠭᠲᠠ 180° ᠡᠴᠡ ᠶᠡᠬᠡ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃^[6]

ᠮᠥᠨ ᠲᠥᠰᠦᠲᠠᠢ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠨᠤᠭᠤᠳ ᠵᠠᠪᠠᠯ ᠻᠣᠨ᠍ᠭ᠌ᠷᠦ᠋ᠧᠨ᠍ᠲ^congruent ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃

ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ[засварлах | кодоор засварлах]

1. ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠳᠤ ᠦᠯᠦ ᠬᠠᠮᠢᠶᠠᠷᠬᠤ ᠴᠡᠭ ‍ᠢ ᠳᠠᠭᠠᠷᠢᠬᠤ ᠨᠢᠭᠡ ᠯᠡ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠳ᠋ᠤ ᠫᠠᠷᠠᠯᠯᠧᠯ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃ (ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋ ‍ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ)
2. ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠳᠤ ᠦᠯᠦ ᠬᠠᠮᠢᠶᠠᠷᠬᠤ ᠴᠡᠭ ‍ᠢ ᠳᠠᠭᠠᠷᠢᠬᠤ ᢩᠪᠥᠭᠡᠳ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠳ᠋ᠤ ᠫᠠᠷᠠᠯᠯᠧᠯ ᠣᠷᠤᠰᠢᠬᠤ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠣᠯᠠᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃ (ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ)
3. ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠳᠤ ᠦᠯᠦ ᠬᠠᠮᠢᠶᠠᠷᠬᠤ ᠴᠡᠭ ‍ᠢ ᠳᠠᠭᠠᠷᠢᠬᠤ ᠬᠣᠶᠠᠷ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ ᠠᠯᠢ ᠨᠢ ᠴᠤ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠬᠤ ᠦᠭᠡᠢ᠃ (ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ‍ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ)

ᠷᠢᠮᠠᠨ ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠷ ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋᠂ ᠭᠠᠦ᠋ᠰᠰ^Gauss᠂ ᠪᠣᠯᠢᠶᠠᠢ^Boliay᠂ ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ᠨᠠᠷ ‍ᠤᠨ ᠣᠨᠣᠯ  ᠢ ᠪᠦᠭᠦᠳᠡ ᠶᠢ ᠨᠢᠭᠡ ᠳᠡᠭᠡᠪᠦᠷ ᠳᠣᠣᠷ᠎ᠠ ᠣᠷᠣᠭᠤᠯᠵᠤ ᠥᠭᠭᠦᠭᠰᠡᠨ ᠪᠠᠶ᠋ᠢᠨ᠎ᠠ᠃
ᠲᠡᠭᠦᠨᠴᠢᠯᠡᠨ᠂ ᠲᠡᠳᠡᠭᠡᠷ  ᠢ ᠦᠯᠡᠮᠵᠢ ᠥᠷᠭᠡᠳᠬᠡᠵᠦ᠂ ᠡᠮᠦᠨ᠎ᠡ ᠨᠢ ᠬᠡᠨ  ᠦ ᠴᠤ ᠲᠥᠰᠦᠭᠡᠯᠡᠵᠦ ᠪᠠᠶ᠋ᠢᠭᠠᠳᠦᠢ ᠨᠡᠩ ᠰᠢᠨ ᠎ᠡ ᠪᠣᠯᠤᠮᠵᠢ ᠶᠢ ᠬᠦᠮᠦᠨ ᠲᠥᠷᠥᠯᠬᠢᠲᠡᠨ  ᠦ ᠣᠶᠤᠨ ᠤᠬᠠᠭᠠᠨ  ᠳᠤ ᠨᠡᠭᠡᠭᠡᠵᠦ ᠥᠭᠴᠡᠢ᠃
ᠪᠣᠳᠠᠲᠤ ᠹᠢᠽᠢᠻ^physic ᠣᠷᠤᠨ ᠵᠠᠢ᠂ ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠴᠢᠨᠠᠷ ‍ᠤᠳ ᠨᠢ ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋ ‍ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠪᠢᠰᠢ ᠬᠠᠷᠢᠨ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠳᠠᠭ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃^[7]

ᠲᠡᠦᠬᠡᠨ ᠳᠠᠷᠠᠭᠠᠯᠠᠯ  ᠤᠨ ᠬᠤᠪᠢ  ᠳᠤ ᠷᠢᠮᠠᠨ  ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠨᠢ ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ‍ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠡᠴᠡ ᠬᠣᠵᠢᠭᠤ᠂ 1854 ᠣᠨ  ᠳᠤ ᠭᠠᠷᠴᠤ ᠢᠷᠡᠭᠰᠡᠨ᠃

ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋  ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ  ᠦᠨ ᠰᠠᠭᠤᠷᠢ  ᠶᠢ ᠷᠢᠮᠠᠨ ᠪᠥᠷᠢᠨ ᠬᠢᠨᠠᠵᠤ ᠦᠵᠠᠭᠡᠳ᠂ ᠲᠡᠭᠦᠨ ᠦ ᠣᠷᠣᠨ ᠳᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ ᠥᠪᠡᠷ  ᠦᠨ ᠵᠠᠷᠴᠢᠮ  ᠤᠳ  ᠢ ᠰᠠᠨᠠᠯ ᠪᠣᠯᠭᠠᠵᠠᠢ᠃

ᠡᠭᠦᠪᠡᠷ  ‍ᠢᠶ᠋ᠡᠨ ᠧᠦᠻ᠊ᠯᠢᠳ᠋  ᠦᠨ ᠪᠤᠰᠤ ᠭᠧᠣᠮᠧᠡᠲ᠋ᠧᠷ  ᠦᠨ ᠣᠯᠠᠨ ᠬᠡᠯᠪᠡᠷᠢ ᠪᠠᠶ᠋ᠢᠵᠤ᠂ ᠣᠭᠲᠠᠷᠭᠤᠢ ᠶᠢᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠣᠯᠠᠨ ᠥᠭᠡᠷ ‍ᠠ᠋ ᠪᠦᠲᠦᠴᠡ ᠶᠢ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠵᠤ ᠪᠣᠯᠬᠤ ᠶᠢ ᠲᠤᠩᠬᠠᠭᠯᠠᠭᠰᠠᠨ ᠪᠠᠶ᠋ᠢᠨ᠎ᠠ᠃

ᠷᠢᠮᠠᠨ ᠡᠳᠡᠭᠡᠷ ᠪᠣᠯᠤᠮᠵᠢᠲᠤ ᠬᠤᠪᠢᠯᠪᠤᠷᠢ ᠨᠤᠭᠤᠳ ᠠᠴᠠ ᠡᠩ  ᠦᠨ ᠨᠢᠭᠡᠨ ᠲᠣᠬᠢᠶᠠᠯᠳᠤᠯ ᠪᠣᠯᠬᠤ ᠲᠣᠭᠲᠠᠮᠠᠯ ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠶ᠋ᠢᠯᠲᠠ ᠲᠠᠢ ᠣᠷᠣᠨ ᠵᠠᠢ ᠶᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ  ᠢ ᠲᠠᠨᠢᠯᠴᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠪᠢᠯᠡ᠃᠎^[5]

ᠬᠠᠷᠢᠴᠠᠩᠭᠤᠢ ‍ᠢᠢᠨ ᠶᠡᠷᠦᠩᠬᠡᠢ ᠣᠨᠤᠯ ‍ᠢ ᠪᠠᠢᠢᠭᠤᠯᠤᠭᠰᠠᠨ ᠧᠢᠢᠨ᠍ᠱᠲ᠋ᠠᠢᠢᠨ^Einstein ᠥᠪᠡᠷ ᠲᠦ ᠪᠡᠨ ᠬᠡᠷᠡᠭ ᠲᠡᠢ ᠦᠨᠡᠨ ᠵᠥᠪ ᠮᠠᠲᠾᠧᠮᠠᠲᠢᠻ^mathematic ᠬᠡᠷᠡᠭᠰᠡᠯ ‍ᠢ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠡᠴᠡ ᠣᠯᠵᠤ^[8]᠂ ᠲᠡᠭᠦᠨ ‍ᠳᠤ ᠲᠦᠰᠢᠭᠯᠡᠭᠰᠡᠨ ᠪᠢᠯᠡ᠃

ᠨᠡᠷ ‍ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ[засварлах | кодоор засварлах]

ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ - Riemannian geometry - геометрия Римана

ᠧᠯᠯᠢᠫᠰ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ - elliptic geometry - эллиптическая геометрия

ᠪᠥᠮᠪᠦᠷᠴᠡᠭ ‍ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ - spherical geometry - сферическая геометрия

ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ - positive curvature - положительная кривизна

ᠰᠥᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ - negative curvature - отрицательная кривизна

ᠫᠧᠷᠫᠧᠨ᠍ᠳᠢᠻᠦ᠋ᠯᠠᠷ - perpendicular - перпендикуляр

ᠫᠠᠷᠠᠯᠯᠧᠯ - parallel - параллель

ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠬᠤ - intersect - пересекаться

ᠬᠠᠪᠲᠠᠭᠠᠢ - plane - плоскость

ᠣᠭᠲᠠᠷᠭᠤᠢ᠂ ᠣᠷᠤᠨ ᠵᠠᠢ - space - пространство
ᠦᠨᠡᠮᠯᠡᠬᠦᠢ ᠲᠤᠢᠢᠯ - absolute pole - абсолютный полюс

ᠪᠥᠮᠪᠦᠷᠴᠡᠭ ‍- sphere - сфера

ᠲᠣᠭᠤᠷᠢᠭ - circle - окружность

ᠬᠡᠮᠵᠢᠭᠳᠡᠬᠦᠨ - dimension - измерение

ᠤᠶᠠᠯᠳᠤᠭ᠎ ‍ᠠ᠋ ᠨᠡᠢᠢᠲᠡᠯᠡᠯ[засварлах | кодоор засварлах]

ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋ ᠊ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ
ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ᠊ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ

ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ[засварлах | кодоор засварлах]

↑ Наука. Геометрия Римана. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/02/28
↑ Н. В. Ефимов. Неевклидовы геометрии. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02
↑ Н. В. Ефимов. Неевклидовы геометрии. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/02/23
↑ Benno Klotzek: . 1. Auflage. Harri Deutsch, Frankfurt am Main 2001, ISBN 3-8171-1583-0
↑ ^5.0 ^5.1 О.А.Старова. Геометрия Римана. 2012/12. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02]
↑ WolframMathWorld. Elliptic Geometry. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02
↑ В.А.Фок. Теория пространства, времени и тяготения. ГИТТЛ. Москва 1955. с. 253
↑ Max Born. Physics in My Generation (Springer, 1969), ISBN 0-387-90008-X. p. 329

[1] Наука. Геометрия Римана. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/02/28

[2] Н. В. Ефимов. Неевклидовы геометрии. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02

[3] Н. В. Ефимов. Неевклидовы геометрии. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/02/23

[4] Benno Klotzek: . 1. Auflage. Harri Deutsch, Frankfurt am Main 2001, ISBN 3-8171-1583-0

[:0-5] 5.0 ^5.1 О.А.Старова. Геометрия Римана. 2012/12. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02]

[6] WolframMathWorld. Elliptic Geometry. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02

[7] В.А.Фок. Теория пространства, времени и тяготения. ГИТТЛ. Москва 1955. с. 253

[8] Max Born. Physics in My Generation (Springer, 1969), ISBN 0-387-90008-X. p. 329

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@@ 1-р мөр: / 1-р мөр: @@
 <div style="max-height: 80vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'">
 = ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ =
-ᠷᠢᠮᠠᠨ ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠷ ᠨᠢ ᠲᠣᠭᠲᠠᠮᠠᠯ ᠡᠶᠡᠷᠡᠭ ᠮᠤᠷᠤᠶ᠋ᠢᠯᠲᠠ ᠲᠠᠢ ᠭᠠᠳᠠᠷᠭᠤ ᠪᠤᠶᠤ ᠵᠤᠤᠪᠠᠩ ᠭᠠᠳᠠᠷᠭᠤ ᠳᠡᠭᠡᠷ᠎ᠡ ᠬᠡᠷᠡᠭᠵᠢᠳᠡᠭ᠃<ref>https://science.wikia.org/ru/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F_%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD%D0%B0 Наука. Геометрия Римана. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/02/28</ref>
+<br>
-<br><br>
+{{ImageX
+|Filename=Comparison_of_geometries.svg
+|Text=ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ ᠲᠠᠢ ᠣᠭᠲᠠᠷᠭᠤᠢ ‍ᠢᠢᠨ ᠳᠦᠷᠰᠦᠯᠡᠯ ᠨᠢ ᠷᠢᠮᠠᠨ ‍ᠤ ᠵᠤᠤᠪᠠᠩ ᠭᠠᠳᠠᠷᠭᠤ ‍ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠶᠤᠮ᠃
+|Height=50
+}}
+<br>
+ᠷᠢᠮᠠᠨ<sup>Riemann</sup>  ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠷ<sup>geometry</sup> ᠨᠢ ᠲᠣᠭᠲᠠᠮᠠᠯ ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠶ᠋ᠢᠯᠲᠠ ᠲᠠᠢ ᠭᠠᠳᠠᠷᠭᠤ ᠪᠤᠶᠤ ᠵᠤᠤᠪᠠᠩ ᠭᠠᠳᠠᠷᠭᠤ ᠳᠡᠭᠡᠷ᠎ᠡ ᠬᠡᠷᠡᠭᠵᠢᠳᠡᠭ᠃<ref>[https://science.wikia.org/ru/wiki/%D0%93%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F_%D0%A0%D0%B8%D0%BC%D0%B0%D0%BD%D0%B0 Наука. Геометрия Римана]. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/02/28</ref>
+ᠤᠯᠠᠮᠵᠢᠯᠠᠯᠲᠤ ᠪᠤᠰᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ‍ᠦᠨ ᠡᠨᠡ ᠬᠡᠯᠪᠡᠷᠢ ‍ᠢᠢ ᠮᠥᠨ ᠧᠯᠯᠢᠫᠰ ‍ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ<sup>elliptic geometry</sup> ᠭᠡᠳᠡᠭ᠃<ref>[https://www.booksite.ru/fulltext/1/001/008/080/804.htm Н. В. Ефимов. Неевклидовы геометрии]. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02 </ref> ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋<sup>Euclid</sup> ‍ᠦᠨ ᠪᠤᠶᠤ ᠤᠯᠠᠮᠵᠢᠯᠠᠲᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠨᠢ ᠲᠡᠭ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ ᠲᠠᠢ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠣᠷᠤᠨ ᠵᠠᠢ ‍ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠡᠷ ᠪᠠᠢᠢᠳᠠᠭ ᠪᠣᠯ [[ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ᠊ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ|ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ]] ‍ᠢᠢᠨ<sup>Лобачевский</sup> ᠬᠢ ᠨᠢ ᠰᠥᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ ᠲᠠᠢ ᠣᠷᠤᠨ ᠵᠠᠢ ‍ᠢᠢ ᠳᠦᠷᠰᠦᠯᠡᠳᠡᠭ ᠪᠢᠯᠡ᠃
+ᠬᠠᠷᠢᠨ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠨᠢ ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ ᠲᠠᠢ ᠭᠦᠳᠦᠭᠦᠷ ᠣᠷᠤᠨ ᠵᠠᠢ ‍ᠢᠢᠨ ᠵᠠᠭᠪᠤᠷ ᠪᠣᠯᠤᠨ᠎ ‍ᠠ᠋᠃
+ᠷᠢᠮᠠᠨ ᠥᠪᠡᠷ ‍ᠦᠨ ᠪᠠᠢᠢᠭᠤᠯᠤᠭᠰᠠᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ‍ᠢ ᠬᠣᠶᠠᠷ ᠬᠡᠮᠵᠢᠭᠳᠠᠬᠦᠨᠲᠦ ᠬᠠᠪᠲᠠᠭᠠᠢ᠂ ᠭᠤᠷᠪᠠᠨ ᠬᠡᠮᠵᠢᠭᠳᠡᠬᠦᠨᠲᠦ ᠬᠢᠭᠡᠳ ᠲᠡᠭᠦᠨ ᠡᠴᠡ ᠳᠡᠭᠡᠭᠰᠢ ᠬᠡᠮᠵᠢᠭᠳᠡᠬᠦᠨᠲᠦ ᠣᠭᠲᠠᠷᠭᠤᠢ ᠳ᠋ᠤ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠪᠠᠢᠢᠳᠠᠭ᠃  <br><br>
 == ᠠᠻᠰᠢᠣᠮ ==
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 }}
 <br>
-'''ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠨᠢᠭᠡ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳ᠋ᠤ ᠣᠷᠣᠰᠢᠵᠤ ᠪᠠᠢᠢᠭ᠎ ‍ᠠ᠋ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠤᠰ ᠪᠦᠷᠢ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠨ᠎ ‍ᠠ᠋'''᠃ <ref>https://bigenc.ru/mathematics/text/2652171 Н. В. Ефимов. Неевклидовы геометрии. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/02/23</ref>
+'''ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠨᠢᠭᠡ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳ᠋ᠤ ᠣᠷᠣᠰᠢᠵᠤ ᠪᠠᠢᠢᠭ᠎ ‍ᠠ᠋ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠤᠰ ᠪᠦᠷᠢ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠨ᠎ ‍ᠠ᠋'''᠃ <ref>[https://bigenc.ru/mathematics/text/2652171 Н. В. Ефимов. Неевклидовы геометрии]. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/02/23</ref>
 ᠪᠠᠰᠠ᠄
-ᠬᠡᠷᠪᠡ g ᠰᠢᠯᠠᠭᠤᠨ ᠱᠤᠭᠤᠮ ᠪᠥᠭᠡᠳ P ᠴᠡᠭ ᠨᠢ ᠲᠡᠭᠦᠨ ‍ᠦ ᠭᠠᠳᠠᠨ᠎ ᠎ᠠ ᠣᠷᠤᠰᠢᠵᠤ ᠪᠠᠢᠢᠭ᠎ ‍ᠠ᠋ ᠪᠣᠯ g ᠰᠢᠯᠠᠭᠤᠨ ᠪᠠ P ᠴᠡᠭ ‍ᠢ ᠳᠠᠮᠨᠠᠭᠰᠠᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳ᠋ᠤ g ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠦᠯᠦ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠬᠤ ᠨᠢᠭᠡ ᠴᠦ ᠰᠢᠯᠠᠭᠤᠨ h ᠪᠠᠢᠢᠬᠤ ᠦᠭᠡᠢ᠃<ref>Benno Klotzek: . 1. Auflage. Harri Deutsch, Frankfurt am Main 2001, ISBN 3-8171-1583-0</ref>
+ᠬᠡᠷᠪᠡ <math>g</math> ᠰᠢᠯᠠᠭᠤᠨ ᠱᠤᠭᠤᠮ ᠪᠥᠭᠡᠳ <math>P</math> ᠴᠡᠭ ᠨᠢ ᠲᠡᠭᠦᠨ ‍ᠦ ᠭᠠᠳᠠᠨ᠎ ᠎ᠠ ᠣᠷᠤᠰᠢᠵᠤ ᠪᠠᠢᠢᠭ᠎ ‍ᠠ᠋ ᠪᠣᠯ <math>g</math> ᠰᠢᠯᠠᠭᠤᠨ ᠪᠠ <math>P</math> ᠴᠡᠭ ‍ᠢ ᠳᠠᠮᠨᠠᠭᠰᠠᠨ ᠬᠠᠪᠲᠠᠭᠠᠢ ᠳ᠋ᠤ <math>g</math> ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠦᠯᠦ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠬᠤ ᠨᠢᠭᠡ ᠴᠦ ᠰᠢᠯᠠᠭᠤᠨ <math>h</math> ᠪᠠᠢᠢᠬᠤ ᠦᠭᠡᠢ᠃<ref>Benno Klotzek: . 1. Auflage. Harri Deutsch, Frankfurt am Main 2001, ISBN 3-8171-1583-0</ref>
 ᠪᠠᠰᠠ᠄
-ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠳ᠋ᠤ ᠫᠧᠷᠫᠧᠨ᠍ᠡᠳᠢᠻᠦ᠋ᠯᠠᠷ ᠬᠣᠶᠠᠷ ᠰᠢᠯᠠᠭᠤᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠪᠠᠨ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠨ᠎ ‍ᠠ᠋᠃
+ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠳ᠋ᠤ ᠫᠧᠷᠫᠧᠨ᠍ᠡᠳᠢᠻᠦ᠋ᠯᠠᠷ<sup>perpendicular</sup> ᠬᠣᠶᠠᠷ ᠰᠢᠯᠠᠭᠤᠨ ᠬᠣᠭᠤᠷᠤᠨᠳᠤ ᠪᠠᠨ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠨ᠎ ‍ᠠ᠋᠃
 ᠶᠡᠷᠦ ᠨᠢ᠂ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ ᠨᠢᠭᠡ ᠲᠠᠯ ‍ᠠ᠋ ᠳᠠᠬᠢ ᠪᠦᠬᠢ ᠫᠧᠷᠫᠧᠨ᠍ᠳᠢᠻᠦ᠋ᠯᠠᠷ ‍ᠤᠳ ᠲᠡᠷᠡ ᠬᠦ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ "ᠦᠨᠡᠮᠯᠡᠬᠦᠢ ᠲᠤᠢᠢᠯ" ᠬᠡᠮᠡᠬᠦ ᠭᠠᠭᠴᠠ ᠴᠡᠭ ᠲᠦ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠨ᠎ ‍ᠠ᠋᠃
@@ 26-р мөр: / 39-р мөр: @@
 ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ ᠨᠥᠭᠦᠭᠡ ᠲᠠᠯ ‍ᠠ᠋ ᠳ᠋ᠠᠬᠢ ᠫᠧᠷᠫᠧᠨ᠍ᠳᠢᠻᠦ᠋ᠯᠠᠷ ‍ᠤᠳ ᠴᠤ ᠮᠥᠨ ᠨᠢᠭᠡ ᠴᠡᠭ ᠲᠦ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠨ᠎ ‍ᠠ᠋᠃<br>
-ᠬᠣᠶᠠᠷ ᠡᠰᠡᠷᠭᠦ ᠲᠠᠯ ‍ᠠ᠋ᠳ᠋ᠠᠬᠢ ᠲᠤᠢᠢᠯ ‍ᠤᠳ ᠨᠢᠭᠡ ᠯᠡ ᠲᠤᠢᠢᠯ ᠪᠠᠢᠢᠨ᠌ ᠊ᠠ᠋᠃
+ᠬᠣᠶᠠᠷ ᠡᠰᠡᠷᠭᠦ ᠲᠠᠯ ‍ᠠ᠋ ᠳ᠋ᠠᠬᠢ ᠲᠤᠢᠢᠯ ‍ᠤᠳ ᠨᠢᠭᠡ ᠯᠡ ᠲᠤᠢᠢᠯ ᠪᠠᠢᠢᠨ᠌ ᠊ᠠ᠋᠃
+ᠤᠯᠠᠮᠵᠢᠯᠠᠯᠲᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠰᠢᠯᠠᠭᠤᠨ ᠨᠢ ᠬᠣᠶᠠᠷ ᠵᠦᠭ ᠲᠦ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠪᠠᠢᠢᠳᠠᠭ ᠪᠣᠯ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠰᠢᠯᠠᠭᠤᠨ ᠨᠢ ᠪᠢᠲᠠᠭᠦᠦ ᠪᠠᠢᠢᠳᠠᠭ᠂ ᠥᠭᠡᠷ ‍ᠡ᠋ ᠪᠠᠡ ᠬᠡᠯᠡᠪᠡᠯ ᠬᠣᠶᠠᠷ ᠦᠵᠦᠭᠦᠷ ᠨᠢ ᠠᠭᠤᠯᠵᠠᠨ ᠨᠡᠢᠢᠯᠡᠳᠡᠭ᠃<ref name=":0">[http://www.e-osnova.ru/PDF/osnova_3_24_3845.pdf О.А.Старова. Геометрия Римана. 2012/12]. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02]</ref>
+ᠰᠢᠯᠠᠭᠤᠨ ᠭᠡᠵᠦ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠪᠥᠮᠪᠦᠷᠴᠡᠭ ‍ᠦᠨ ᠪᠦᠰᠡᠯᠡᠭᠦᠷ ᠲᠣᠭᠤᠷᠢᠭ ‍ᠤᠳ ‍ᠢ ᠬᠡᠯᠡᠳᠡᠭ ᠪᠥᠭᠡᠳ ᠡᠨᠡ ᠨᠢ ᠴᠢᠩᠬ᠎ᠠ ᠡᠰᠡᠷᠭᠦ ᠲᠠᠯ ‍᠋ᠠ᠋ ᠳ᠋ᠤ ᠪᠠᠢᠢᠷᠢᠯᠠᠬᠤ ᠴᠡᠭ ‍ᠦᠳ ‍ᠦᠨ ᠣᠯᠠᠨᠯᠢᠭ ᠪᠣᠯᠵᠤ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃
+ᠰᠢᠯᠠᠭᠤᠨ ‍ᠢ ᠢᠩᠭᠢᠵᠦ ᠲᠣᠳᠤᠷᠬᠠᠶ᠋ᠢᠯᠠᠭᠰᠠᠨ ‍ᠢᠢᠠᠷ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ ᠤᠷᠲᠤ ᠨᠢ ᠳᠠᠷᠠᠭᠠᠬᠢ ᠲᠡᠭᠰᠢᠳᠬᠡᠯ ‍ᠢᠢᠡᠷ ᠲᠣᠮᠢᠶᠠᠯᠠᠭᠳᠠᠨ᠎ ‍ᠠ᠋᠄
+<br><br>
+<span style="writing-mode: horizontal-tb;">
+<br><br><br><br>
+<math>L = \pi r</math>
+<br><br>
+<span style="max-height:50vh; writing-mode: vertical-lr; font-family: 'Noto Sans Mongolian'">
+ᠡᠭᠦᠨ ᠳ᠋ᠦ᠄ <br><span style="writing-mode: horizontal-tb;">L</span> ᠨᠢ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ ᠤᠷᠲᠤ᠂ <br><span style="writing-mode: horizontal-tb;">r</span> ᠨᠢ ᠬᠡᠭᠡᠰᠦ (ᠷᠠᠳᠢᠦ᠋ᠰ<sup>radius</sup>)᠃
+</span>
+</span><br><br>
+ᠧᠯᠯᠢᠫᠰ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ‍ᠤ ᠥᠨᠴᠦᠭ ‍ᠦᠳ ‍ᠦᠨ ᠨᠡᠢᠢᠯᠡᠪᠦᠷᠢ ᠢᠮᠠᠭᠲᠠ <span style="writing-mode: horizontal-tb;">180°</span> ᠡᠴᠡ ᠶᠡᠬᠡ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃<ref>[https://mathworld.wolfram.com/EllipticGeometry.html WolframMathWorld. Elliptic Geometry]. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02</ref>
+ᠮᠥᠨ ᠲᠥᠰᠦᠲᠠᠢ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ᠨᠤᠭᠤᠳ ᠵᠠᠪᠠᠯ ᠻᠣᠨ᠍ᠭ᠌ᠷᠦ᠋ᠧᠨ᠍ᠲ<sup>congruent</sup> ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃
 <br><br>
 ==ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ==
@@ 32-р мөр: / 63-р мөр: @@
 {{ImageX
 |Filename=Euclidian_and_non_euclidian_geometry.png
-|Text=<span style="writing-mode: horizontal-tb; "><b>1</b>.</span> ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠳᠤ ᠦᠯᠦ ᠬᠠᠮᠢᠶᠠᠷᠠᠬᠤ ᠴᠡᠭ ‍ᠢ ᠳᠠᠭᠠᠷᠢᠬᠤ ᠨᠢᠭᠡ ᠯᠡ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠳ᠋ᠤ ᠫᠠᠷᠠᠯᠯᠧᠯ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃ (ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋ ‍ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ) <br><span style="writing-mode: horizontal-tb; "><b>2</b>.</span> ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠳᠤ ᠦᠯᠦ ᠬᠠᠮᠢᠶᠠᠷᠠᠬᠤ ᠴᠡᠭ ‍ᠢ ᠳᠠᠭᠠᠷᠢᠬᠤ ᢩᠪᠥᠭᠡᠳ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠳ᠋ᠤ ᠫᠠᠷᠠᠯᠯᠧᠯ ᠣᠷᠤᠰᠢᠬᠤ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠣᠯᠠᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃  (ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ) <br><span style="writing-mode: horizontal-tb; "><b>3</b>.</span> ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠳᠤ ᠦᠯᠦ ᠬᠠᠮᠢᠶᠠᠷᠠᠬᠤ ᠴᠡᠭ ‍ᠢ ᠳᠠᠭᠠᠷᠢᠬᠤ ᠬᠣᠶᠠᠷ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ ᠠᠯᠢ ᠨᠢ ᠴᠤ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠳ᠋ᠤ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠬᠤ ᠦᠭᠡᠢ᠃ (ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ‍ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ)
+|Text=<span style="writing-mode: horizontal-tb; "><b>1</b>.</span> ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠳᠤ ᠦᠯᠦ ᠬᠠᠮᠢᠶᠠᠷᠬᠤ ᠴᠡᠭ ‍ᠢ ᠳᠠᠭᠠᠷᠢᠬᠤ ᠨᠢᠭᠡ ᠯᠡ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠳ᠋ᠤ ᠫᠠᠷᠠᠯᠯᠧᠯ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃ (ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋ ‍ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ) <br><span style="writing-mode: horizontal-tb; "><b>2</b>.</span> ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠳᠤ ᠦᠯᠦ ᠬᠠᠮᠢᠶᠠᠷᠬᠤ ᠴᠡᠭ ‍ᠢ ᠳᠠᠭᠠᠷᠢᠬᠤ ᢩᠪᠥᠭᠡᠳ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠳ᠋ᠤ ᠫᠠᠷᠠᠯᠯᠧᠯ ᠣᠷᠤᠰᠢᠬᠤ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠣᠯᠠᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃  (ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ) <br><span style="writing-mode: horizontal-tb; "><b>3</b>.</span> ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠳᠤ ᠦᠯᠦ ᠬᠠᠮᠢᠶᠠᠷᠬᠤ ᠴᠡᠭ ‍ᠢ ᠳᠠᠭᠠᠷᠢᠬᠤ ᠬᠣᠶᠠᠷ ᠰᠢᠯᠠᠭᠤᠨ ‍ᠤ ᠠᠯᠢ ᠨᠢ ᠴᠤ ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠲᠠᠢ ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠬᠤ ᠦᠭᠡᠢ᠃ (ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ‍ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ)
 |Height=50
 }}
 <br>
-ᠷᠢᠮᠠᠨ ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠷ ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋᠂ ᠭᠠᠦ᠋ᠰᠰ᠂ ᠪᠣᠯᠢᠶᠠᠢ᠂ ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ᠨᠠᠷ ‍ᠤᠨ ᠣᠨᠣᠯ  ᠢ ᠪᠦᠭᠦᠳᠡ ᠶᠢ ᠨᠢᠭᠡ ᠳᠡᠭᠡᠪᠦᠷ ᠳᠣᠣᠷ᠎ᠠ ᠣᠷᠣᠭᠤᠯᠵᠤ ᠥᠭᠭᠦᠭᠰᠡᠨ ᠪᠠᠶ᠋ᠢᠨ᠎ᠠ᠃ <br>
+ᠷᠢᠮᠠᠨ ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠷ ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋᠂ ᠭᠠᠦ᠋ᠰᠰ<sup>Gauss</sup>᠂ ᠪᠣᠯᠢᠶᠠᠢ<sup>Boliay</sup>᠂ ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ᠨᠠᠷ ‍ᠤᠨ ᠣᠨᠣᠯ  ᠢ ᠪᠦᠭᠦᠳᠡ ᠶᠢ ᠨᠢᠭᠡ ᠳᠡᠭᠡᠪᠦᠷ ᠳᠣᠣᠷ᠎ᠠ ᠣᠷᠣᠭᠤᠯᠵᠤ ᠥᠭᠭᠦᠭᠰᠡᠨ ᠪᠠᠶ᠋ᠢᠨ᠎ᠠ᠃ <br>
-ᠲᠡᠭᠦᠨᠴᠢᠯᠡᠨ᠂ ᠲᠡᠳᠡᠭᠡᠷ  ᠢ ᠦᠯᠡᠮᠵᠢ ᠥᠷᠭᠡᠳᠬᠡᠵᠦ᠂ ᠡᠮᠦᠨ᠎ᠡ ᠨᠢ ᠬᠡᠨ  ᠦ ᠴᠤ ᠲᠥᠰᠦᠭᠡᠯᠡᠵᠦ ᠪᠠᠶ᠋ᠢᠭᠠᠳᠦᠢ ᠴᠣ ᠰᠢᠨ ᠎ᠡ ᠪᠣᠯᠤᠮᠵᠢ ᠶᠢ ᠬᠦᠮᠦᠨ ᠲᠥᠷᠥᠯᠬᠢᠲᠡᠨ  ᠦ ᠣᠶᠤᠨ ᠤᠬᠠᠭᠠᠨ  ᠳᠤ ᠨᠡᠭᠡᠭᠡᠵᠦ ᠥᠭᠴᠡᠢ᠃ <br>
+ᠲᠡᠭᠦᠨᠴᠢᠯᠡᠨ᠂ ᠲᠡᠳᠡᠭᠡᠷ  ᠢ ᠦᠯᠡᠮᠵᠢ ᠥᠷᠭᠡᠳᠬᠡᠵᠦ᠂ ᠡᠮᠦᠨ᠎ᠡ ᠨᠢ ᠬᠡᠨ  ᠦ ᠴᠤ ᠲᠥᠰᠦᠭᠡᠯᠡᠵᠦ ᠪᠠᠶ᠋ᠢᠭᠠᠳᠦᠢ ᠨᠡᠩ ᠰᠢᠨ ᠎ᠡ ᠪᠣᠯᠤᠮᠵᠢ ᠶᠢ ᠬᠦᠮᠦᠨ ᠲᠥᠷᠥᠯᠬᠢᠲᠡᠨ  ᠦ ᠣᠶᠤᠨ ᠤᠬᠠᠭᠠᠨ  ᠳᠤ ᠨᠡᠭᠡᠭᠡᠵᠦ ᠥᠭᠴᠡᠢ᠃ <br>
-<br>
+ᠪᠣᠳᠠᠲᠤ ᠹᠢᠽᠢᠻ<sup>physic</sup> ᠣᠷᠤᠨ ᠵᠠᠢ᠂ ᠴᠠᠭ ᠬᠤᠭᠤᠴᠠᠭᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠴᠢᠨᠠᠷ ‍ᠤᠳ ᠨᠢ [[ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋ ᠊ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ|ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋ ‍ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ]] ᠪᠢᠰᠢ ᠬᠠᠷᠢᠨ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠲᠦ ᠬᠠᠷᠠᠭᠠᠯᠵᠠᠳᠠᠭ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃<ref>В.А.Фок. Теория пространства, времени и тяготения. ГИТТЛ. Москва 1955. с. 253  </ref>
+ᠲᠡᠦᠬᠡᠨ ᠳᠠᠷᠠᠭᠠᠯᠠᠯ  ᠤᠨ ᠬᠤᠪᠢ  ᠳᠤ ᠷᠢᠮᠠᠨ  ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠨᠢ ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ‍ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠡᠴᠡ ᠬᠣᠵᠢᠭᠤ᠂ 1854 ᠣᠨ  ᠳᠤ ᠭᠠᠷᠴᠤ ᠢᠷᠡᠭᠰᠡᠨ᠃
+ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋  ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ  ᠦᠨ ᠰᠠᠭᠤᠷᠢ  ᠶᠢ ᠷᠢᠮᠠᠨ ᠪᠥᠷᠢᠨ ᠬᠢᠨᠠᠵᠤ ᠦᠵᠠᠭᠡᠳ᠂ ᠲᠡᠭᠦᠨ ᠦ ᠣᠷᠣᠨ ᠳᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠪᠠᠢᠢᠭᠤᠯᠬᠤ ᠥᠪᠡᠷ  ᠦᠨ ᠵᠠᠷᠴᠢᠮ  ᠤᠳ  ᠢ ᠰᠠᠨᠠᠯ ᠪᠣᠯᠭᠠᠵᠠᠢ᠃
+ᠡᠭᠦᠪᠡᠷ  ‍ᠢᠶ᠋ᠡᠨ ᠧᠦᠻ᠊ᠯᠢᠳ᠋  ᠦᠨ ᠪᠤᠰᠤ ᠭᠧᠣᠮᠧᠡᠲ᠋ᠧᠷ  ᠦᠨ ᠣᠯᠠᠨ ᠬᠡᠯᠪᠡᠷᠢ ᠪᠠᠶ᠋ᠢᠵᠤ᠂ ᠣᠭᠲᠠᠷᠭᠤᠢ ᠶᠢᠨ ᠬᠢᠵᠠᠭᠠᠷ ᠦᠭᠡᠢ ᠣᠯᠠᠨ ᠥᠭᠡᠷ ‍ᠠ᠋ ᠪᠦᠲᠦᠴᠡ ᠶᠢ ᠲᠣᠳᠣᠷᠬᠠᠶ᠋ᠢᠯᠠᠵᠤ ᠪᠣᠯᠬᠤ ᠶᠢ ᠲᠤᠩᠬᠠᠭᠯᠠᠭᠰᠠᠨ ᠪᠠᠶ᠋ᠢᠨ᠎ᠠ᠃
+ᠷᠢᠮᠠᠨ ᠡᠳᠡᠭᠡᠷ ᠪᠣᠯᠤᠮᠵᠢᠲᠤ ᠬᠤᠪᠢᠯᠪᠤᠷᠢ ᠨᠤᠭᠤᠳ ᠠᠴᠠ ᠡᠩ  ᠦᠨ ᠨᠢᠭᠡᠨ ᠲᠣᠬᠢᠶᠠᠯᠳᠤᠯ ᠪᠣᠯᠬᠤ ᠲᠣᠭᠲᠠᠮᠠᠯ ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠶ᠋ᠢᠯᠲᠠ ᠲᠠᠢ ᠣᠷᠣᠨ ᠵᠠᠢ ᠶᠢ ᠳᠦᠷᠰᠦᠯᠡᠭᠰᠡᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ  ᠢ ᠲᠠᠨᠢᠯᠴᠠᠭᠤᠯᠤᠭᠰᠠᠨ ᠪᠢᠯᠡ᠃᠎<ref name=":0" />
+[[ᠬᠠᠷᠢᠴᠠᠩᠭᠤᠢ ᠊ᠢᠢᠨ ᠶᠡᠷᠦᠩᠬᠡᠢ ᠣᠨᠤᠯ|ᠬᠠᠷᠢᠴᠠᠩᠭᠤᠢ ‍ᠢᠢᠨ ᠶᠡᠷᠦᠩᠬᠡᠢ ᠣᠨᠤᠯ]] ‍ᠢ ᠪᠠᠢᠢᠭᠤᠯᠤᠭᠰᠠᠨ ᠧᠢᠢᠨ᠍ᠱᠲ᠋ᠠᠢᠢᠨ<sup>Einstein</sup> ᠥᠪᠡᠷ ᠲᠦ ᠪᠡᠨ ᠬᠡᠷᠡᠭ ᠲᠡᠢ ᠦᠨᠡᠨ ᠵᠥᠪ ᠮᠠᠲᠾᠧᠮᠠᠲᠢᠻ<sup>mathematic</sup> ᠬᠡᠷᠡᠭᠰᠡᠯ ‍ᠢ ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ ᠡᠴᠡ ᠣᠯᠵᠤ<ref>Max Born. Physics in My Generation (Springer, 1969), <nowiki>ISBN 0-387-90008-X</nowiki>. p. 329</ref>᠂ ᠲᠡᠭᠦᠨ ‍ᠳᠤ ᠲᠦᠰᠢᠭᠯᠡᠭᠰᠡᠨ ᠪᠢᠯᠡ᠃<br><br>
 ==ᠨᠡᠷ ‍ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ==
+ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ - Riemannian geometry - геометрия Римана
+ᠧᠯᠯᠢᠫᠰ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ - elliptic geometry - эллиптическая геометрия
+ᠪᠥᠮᠪᠦᠷᠴᠡᠭ ‍ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ - spherical geometry - сферическая геометрия
+ᠡᠶᠡᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ - positive curvature - положительная кривизна
+ᠰᠥᠷᠭᠦ ᠮᠤᠷᠤᠢᠢᠯᠲᠠ - negative curvature - отрицательная кривизна
+ᠫᠧᠷᠫᠧᠨ᠍ᠳᠢᠻᠦ᠋ᠯᠠᠷ - perpendicular - перпендикуляр
+ᠫᠠᠷᠠᠯᠯᠧᠯ - parallel - параллель
+ᠣᠭᠲᠤᠯᠤᠯᠴᠠᠬᠤ - intersect - пересекаться
+ᠬᠠᠪᠲᠠᠭᠠᠢ - plane - плоскость
+ᠣᠭᠲᠠᠷᠭᠤᠢ᠂ ᠣᠷᠤᠨ ᠵᠠᠢ - space - пространство<br>ᠦᠨᠡᠮᠯᠡᠬᠦᠢ ᠲᠤᠢᠢᠯ - absolute pole - абсолютный полюс
+ᠪᠥᠮᠪᠦᠷᠴᠡᠭ ‍- sphere - сфера
+ᠲᠣᠭᠤᠷᠢᠭ - circle - окружность
+ᠬᠡᠮᠵᠢᠭᠳᠡᠬᠦᠨ - dimension - измерение
 <br><br>
+==ᠤᠶᠠᠯᠳᠤᠭ᠎ ‍ᠠ᠋ ᠨᠡᠢᠢᠲᠡᠯᠡᠯ==
+[[ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋ ᠊ᠦᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ]]<br>
+[[ᠯᠣᠪᠠᠴᠧᠹᠰᠻᠢ ᠊ᠢᠢᠨ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ]]
+<br>
 == ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ ==
+<references />
+[[Ангилал:ᠮᠠᠲ᠋ᠾᠧᠮᠠᠲᠢᠭ]]
+[[Ангилал:ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ]]