ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ

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

ᠠᠻᠰᠢᠣᠮ

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᠃ ^[2]

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᠃^[3]

ᠪᠠᠰᠠ᠄

ᠲᠤᠬᠠᠢ ‍ᠢᠢᠨ ᠰᠢᠯᠠᠭᠤᠨ ᠳ᠋ᠤ ᠫᠧᠷᠫᠧᠨ᠍ᠡᠳᠢᠻᠦ᠋ᠯᠠᠷ 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 ᠪᠢᠲᠠᠭᠦ ᠪᠠᠢᠢᠳᠠᠭ᠃^[4]

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 ᠣᠯᠤᠨᠯᠢᠭ 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)᠃

ᠧᠯᠯᠢᠫᠰ ᠭᠧᠣᠮᠧᠲ᠋ᠷ ᠲᠦ ᠭᠤᠷᠪᠠᠯᠵᠢᠨ ‍ᠤ ᠥᠨᠴᠦᠭ ‍ᠦᠳ ‍ᠦᠨ ᠨᠡᠢᠢᠯᠡᠪᠦᠷᠢ ᠢᠮᠠᠭᠲᠠ 180° ᠡᠴᠡ ᠶᠡᠬᠡ ᠪᠠᠢᠢᠨ᠎ ‍ᠠ᠋᠃^[5]

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

ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ

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

ᠷᠢᠮᠠᠨ ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠷ ᠧᠦ᠋ᠻ᠊ᠯᠢᠳ᠋᠂ ᠭᠠᠦ᠋ᠰᠰ᠂ ᠪᠣᠯᠢᠶᠠᠢ᠂ 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 ᠴᠣ ᠰᠢᠨ ᠎ᠡ ᠪᠣᠯᠤᠮᠵᠢ ᠶᠢ ᠬᠦᠮᠦᠨ ᠲᠥᠷᠥᠯᠬᠢᠲᠡᠨ  ᠦ ᠣᠶᠤᠨ ᠤᠬᠠᠭᠠᠨ  ᠳᠤ ᠨᠡᠭᠡᠭᠡᠵᠦ ᠥᠭᠴᠡᠢ᠃

ᠨᠡᠷ ‍ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ

ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ

↑ 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
↑ https://bigenc.ru/mathematics/text/2652171 Н. В. Ефимов. Неевклидовы геометрии. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/02/23
↑ Benno Klotzek: . 1. Auflage. Harri Deutsch, Frankfurt am Main 2001, ISBN 3-8171-1583-0
↑ О.А.Старова. Геометрия Римана. 2012/12. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02]
↑ WolframMathWorld. Elliptic Geometry. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02

[1] ttps://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

[2] ttps://bigenc.ru/mathematics/text/2652171 Н. В. Ефимов. Неевклидовы геометрии. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/02/23

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

[4] О.А.Старова. Геометрия Римана. 2012/12. ᠬᠠᠨᠳᠤᠭᠰᠠᠨ 2021/03/02]

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

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ᠷᠢᠮᠠᠨ ᠊ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ

Гарчиг

ᠷᠢᠮᠠᠨ ‍ᠤ ᠭᠧᠣᠮᠧᠲ᠋ᠧᠷ

ᠠᠻᠰᠢᠣᠮ

ᠠᠴᠢ ᠬᠣᠯᠪᠤᠭᠳᠠᠯ

ᠨᠡᠷ ‍ᠡ᠋ ᠲᠣᠮᠢᠶᠠᠯᠠᠯ

ᠡᠬᠢ ᠰᠤᠷᠪᠤᠯᠵᠢ