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2024
ɍȾɄ 512.64 ȻȻɄ 22.143
ȼ
Ɋɟɰɟɧɡɟɧɬɵ:
ɞɨɤɬɨɪ ɮɢɡɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɧɚɭɤ ɩɪɨɮɟɫɫɨɪ ɡɚɫɥɭɠɟɧɧɵɣ ɞɟɹɬɟɥɶ ɧɚɭɤɢ ɩɪɨɮɟɫɫɨɪ ɤɚɮɟɞɪɵ ɝɟɨɦɟɬɪɢɢ ɊȽɉɍ ɢɦ Ⱥ ɂ Ƚɟɪɰɟɧɚ
Ⱥɥɟɤɫɟɣ Ʌɟɨɧɢɞɨɜɢɱ ȼɟɪɧɟɪ;
ɤɚɧɞɢɞɚɬ ɬɟɯɧɢɱɟɫɤɢɯ ɧɚɭɤ ɞɨɰɟɧɬ ɡɚɦɟɫɬɢɬɟɥɶ ɞɢɪɟɤɬɨɪɚ ɂɆɈɉ ɋɉɛȽɉɍ
ȼɢɤɬɨɪ ȼɥɚɞɢɦɢɪɨɜɢɱ Ʉɪɚɫɧɨɳɟɤɨɜ
ȼɨɥɤɨɜ Ⱦ ɘ ȼ Ⱥɧɚɥɢɬɢɱɟɫɤɚɹ ɝɟɨɦɟɬɪɢɹ ɢ ɥɢɧɟɣɧɚɹ ɚɥɝɟɛɪɚ ɭɱɟɛɧɨɟ ɩɨɫɨɛɢɟ Ⱦ ɘ ȼɨɥɤɨɜ
Ʉ ȼ Ƚɚɥɭɧɨɜɚ – Ɇɨɫɤɜɚ ȼɨɥɨɝɞɚ ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹ – 116 ɫ ɢɥ ɬɚɛɥ ,6%1 -5- - 94-2
Ɋɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɜɨɩɪɨɫɵ ɤɨɬɨɪɵɟ ɢɡɭɱɚɸɬɫɹ ɜ ɪɚɡɞɟɥɚɯ ©Ⱥɧɚɥɢɬɢɱɟɫɤɚɹ ɝɟɨ ɦɟɬɪɢɹª ɢ ©Ʌɢɧɟɣɧɚɹ ɚɥɝɟɛɪɚª ɤɭɪɫɚ ɜɵɫɲɟɣ ɦɚɬɟɦɚɬɢɤɢ ɉɨɫɨɛɢɟ ɫɨɫɬɨɢɬ ɢɡ ɬɪɟɯ ɱɚɫɬɟɣ ɜ ɤɨɬɨɪɵɯ ɫɨɞɟɪɠɚɬɫɹ ɬɟɨɪɟɬɢɱɟɫɤɢɟ ɫɜɟɞɟɧɢɹ ɩɪɢɦɟɪɵ ɜɵɩɨɥɧɟɧɢɹ ɡɚɞɚɧɢɣ ɢ ɤɨɧɬɪɨɥɶɧɵɟ ɦɚɬɟɪɢɚɥɵ ɩɨ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɦ ɬɟɦɚɦ
Ⱦɥɹ ɫɬɭɞɟɧɬɨɜ ɜɵɫɲɢɯ ɭɱɟɛɧɵɯ ɡɚɜɟɞɟɧɢɣ ɨɛɭɱɚɸɳɢɯɫɹ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɩɨɞ ɝɨɬɨɜɤɢ ɛɚɤɚɥɚɜɪɨɜ ©ɗɤɨɧɨɦɢɤɚª ɢ ©Ɇɟɧɟɞɠɦɟɧɬª ɚ ɬɚɤɠɟ ©Ȼɢɡɧɟɫ-ɢɧɮɨɪɦɚɬɢɤɚª ©Ɍɨɪɝɨɜɨɟɞɟɥɨª ©Ɍɨɜɚɪɨɜɟɞɟɧɢɟª ©Ƚɨɫɭɞɚɪɫɬɜɟɧɧɨɟɢɦɭɧɢɰɢɩɚɥɶɧɨɟ ɭɩɪɚɜɥɟɧɢɟª ©ɍɩɪɚɜɥɟɧɢɟ ɩɟɪɫɨɧɚɥɨɦª
ɍȾɄ 512.64 ȻȻɄ 22.143
ISBN -5- - 94-2 |
ȼɨɥɤɨɜ Ⱦ ɘ Ƚɚɥɭɧɨɜɚ Ʉ ȼ |
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ɂɡɞɚɬɟɥɶɫɬɜɨ ©ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹª |
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Ɉɮɨɪɦɥɟɧɢɟ ɂɡɞɚɬɟɥɶɫɬɜɨ ©ɂɧɮɪɚ-ɂɧɠɟɧɟɪɢɹª |
2
ǵǪDzǧǩDzǬǴǯǬ |
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ȼɜɟɞɟɧɢɟ............................................................................................................... |
4 |
ɑɚɫɬɶ ɉɅɈɋɄɈɋɌɖ......................................................................................... |
5 |
1.1. ɉɨɧɹɬɢɟ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ................................................................. |
5 |
ɍɪɚɜɧɟɧɢɟ ɥɢɧɢɢ ɧɚ ɩɥɨɫɤɨɫɬɢ.......................................................... |
11 |
ɉɪɹɦɚɹ ɧɚ ɩɥɨɫɤɨɫɬɢ........................................................................... |
13 |
ɑɚɫɬɶ ɉɊɈɋɌɊȺɇɋɌȼɈ................................................................................ |
24 |
ɋɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ................................................... |
24 |
ȼɟɤɬɨɪɵ................................................................................................ |
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ɉɥɨɫɤɨɫɬɶ............................................................................................. |
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ɉɪɹɦɚɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ........................................................................ |
45 |
ɑɚɫɬɶ ɅɂɇȿɃɇȺə ȺɅȽȿȻɊȺ....................................................................... |
52 |
Ʉɨɦɩɥɟɤɫɧɵɟ ɱɢɫɥɚ............................................................................. |
52 |
Ʌɢɧɟɣɧɵɟ ɩɪɨɫɬɪɚɧɫɬɜɚ...................................................................... |
64 |
Ɇɚɬɪɢɰɵ ɢ ɨɩɟɪɚɰɢɢ ɧɚɞ ɧɢɦɢ .......................................................... |
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Ɉɩɪɟɞɟɥɢɬɟɥɢ ...................................................................................... |
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Ɋɚɧɝ ɦɚɬɪɢɰɵ....................................................................................... |
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ɋɢɫɬɟɦɵ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ........................................................... |
92 |
Ɉɛɪɚɬɧɚɹ ɦɚɬɪɢɰɚ.............................................................................. |
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ɋɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɢ ɫɨɛɫɬɜɟɧɧɵɟ ɜɟɤɬɨɪɵ ɦɚɬɪɢɰɵ............... |
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Ȼɢɛɥɢɨɝɪɚɮɢɱɟɫɤɢɣ ɫɩɢɫɨɤ............................................................................. |
111 |
ɂɧɬɟɪɧɟɬ-ɪɟɫɭɪɫɵ............................................................................................ |
112 |
3
ǩǩǬǫǬǴǯǬ
ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ ɩɨɫɜɹɳɟɧɨ ɪɚɡɞɟɥɚɦ ɜɭɡɨɜɫɤɨɣ ɞɢɫɰɢɩɥɢɧɵ ©ȼɵɫɲɚɹ ɦɚɬɟɦɚɬɢɤɚª ɞɥɹ ɫɬɭɞɟɧɬɨɜ ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɫɩɟɰɢɚɥɶɧɨɫɬɟɣ ɜ ɤɨɬɨɪɵɯ ɢɡɭɱɚ ɟɬɫɹ ɚɧɚɥɢɬɢɱɟɫɤɚɹ ɝɟɨɦɟɬɪɢɹ ɢ ɥɢɧɟɣɧɚɹ ɚɥɝɟɛɪɚ ɋɨɞɟɪɠɚɧɢɟ ɩɨɫɨɛɢɹ ɨɪɢ ɟɧɬɢɪɨɜɚɧɨ ɧɚ ɩɪɨɝɪɚɦɦɭ ɤɭɪɫɚ ©ȼɵɫɲɚɹ ɦɚɬɟɦɚɬɢɤɚª ɱɢɬɚɟɦɨɝɨ ɨɞɧɢɦ ɢɡ ɚɜɬɨɪɨɜ ɜ ɉɨɥɢɬɟɯɧɢɱɟɫɤɨɦ ɭɧɢɜɟɪɫɢɬɟɬɟ ɉɟɬɪɚ ȼɟɥɢɤɨɝɨ ɉɨɫɨɛɢɟ ɩɪɟɞɧɚ ɡɧɚɱɟɧɨ ɤɚɤ ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɣ ɪɚɛɨɬɵ ɫɬɭɞɟɧɬɨɜ ɩɪɢ ɩɨɞɝɨɬɨɜɤɟ ɤ ɤɨɧ ɬɪɨɥɶɧɵɦ ɦɟɪɨɩɪɢɹɬɢɹɦ ɩɨ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɦ ɬɟɦɚɦ ɬɚɤ ɢ ɞɥɹ ɩɪɟɩɨɞɚɜɚɬɟ ɥɟɣ ɩɨɫɤɨɥɶɤɭ ɫɨɞɟɪɠɢɬ ɜɚɪɢɚɧɬɵ ɤɨɧɬɪɨɥɶɧɵɯ ɦɚɬɟɪɢɚɥɨɜ Ʉɨɧɬɪɨɥɶɧɵɟ ɦɚɬɟɪɢɚɥɵɦɨɝɭɬ ɛɵɬɶɢɫɩɨɥɶɡɨɜɚɧɵɩɪɟɩɨɞɚɜɚɬɟɥɟɦɞɥɹɨɰɟɧɤɢ ɭɪɨɜɧɹɫɮɨɪ ɦɢɪɨɜɚɧɧɨɫɬɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɤɨɦɩɟɬɟɧɰɢɣ ɫɬɭɞɟɧɬɨɜ ɩɨɪɚɡɥɢɱɧɵɦ ɝɪɭɩɩɚɦ ɧɚɩɪɚɜɥɟɧɢɣ ɩɨɞɝɨɬɨɜɤɢ ɛɚɤɚɥɚɜɪɨɜ
ɉɨɫɨɛɢɟ ɫɨɫɬɨɢɬ ɢɡ ɬɪɟɯ ɱɚɫɬɟɣ ȼ ɩɟɪɜɨɣ ɱɚɫɬɢ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɚɧɚɥɢ ɬɢɱɟɫɤɚɹ ɝɟɨɦɟɬɪɢɹ ɩɥɨɫɤɨɫɬɢ ɜɨ ɜɬɨɪɨɣ ɱɚɫɬɢ – ɝɟɨɦɟɬɪɢɹ ɩɪɨɫɬɪɚɧɫɬɜɚ ɡɞɟɫɶ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɬɨɥɶɤɨ ɨɛɴɟɤɬɵ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɜɜɢɞɭ ɨɝɪɚɧɢɱɟɧɧɨ ɫɬɢ ɜɪɟɦɟɧɢ ɜɵɞɟɥɹɟɦɨɝɨɜ ɨɛɳɟɦ ɤɭɪɫɟ ɧɚ ɚɧɚɥɢɬɢɱɟɫɤɭɸ ɝɟɨɦɟɬɪɢɸ ȼ ɬɪɟ ɬɶɟɣ ɱɚɫɬɢ ɩɨɫɨɛɢɹ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɧɟɤɨɬɨɪɵɟ ɜɨɩɪɨɫɵ ɥɢɧɟɣɧɨɣ ɚɥɝɟɛɪɵ ȼ ɬɟɤɫɬɟ ɩɪɢɜɨɞɹɬɫɹ ɤɪɚɬɤɢɟ ɬɟɨɪɟɬɢɱɟɫɤɢɟ ɫɜɟɞɟɧɢɹ ɢ ɞɚɸɬɫɹ ɦɟɬɨɞɢɱɟɫɤɢɟ ɭɤɚɡɚɧɢɹ ɩɨ ɜɵɩɨɥɧɟɧɢɸ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɞɚɧɢɣ
ȼ ɤɨɧɰɟ ɩɨɫɨɛɢɹ ɩɪɢɜɨɞɢɬɫɹ ɫɩɢɫɨɤ ɪɟɤɨɦɟɧɞɭɟɦɨɣ ɥɢɬɟɪɚɬɭɪɵ ɢ ɢɧɬɟɪ ɧɟɬ-ɪɟɫɭɪɫɨɜ ɍɱɟɛɧɢɤɢ > – 9– @ ɦɨɝɭɬ ɩɪɟɞɨɫɬɚɜɢɬɶ ɫɬɭɞɟɧɬɭ ɛɨɥɟɟ ɩɨ ɞɪɨɛɧɵɣ ɦɚɬɟɪɢɚɥ ɩɪɢ ɢɡɭɱɟɧɢɢ ɚɧɚɥɢɬɢɱɟɫɤɨɣ ɝɟɨɦɟɬɪɢɢ ɢ ɥɢɧɟɣɧɨɣ ɚɥ ɝɟɛɪɵ Ⱦɨɩɨɥɧɢɬɟɥɶɧɵɟ ɡɚɞɚɱɢ ɩɨ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɬɟɦɚɬɢɤɟ ɦɨɠɧɨ ɧɚɣɬɢ ɜ ɡɚɞɚɱɧɢɤɚɯ > @
4
ǾȇȘșȣ ǶDzǵǸDZǵǸǹȃ
ǶȕȔȦșȏȌ ȘȏȘșȌȓȢ ȑȕȕȗȋȏȔȇș
Ɉɩɪɟɞɟɥɟɧɢɟ Ƚɨɜɨɪɹɬ ɱɬɨ ɧɚ ɩɪɹɦɨɣ l ɡɚɞɚɧɚ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ ɟɫɥɢ
1)ɜɵɛɪɚɧɚ ɬɨɱɤɚ Ɉ ɤɨɬɨɪɚɹ ɧɚɡɵɜɚɟɬɫɹ ɧɚɱɚɥɨɦ ɤɨɨɪɞɢɧɚɬ
2)ɜɵɛɪɚɧɨ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ
3)ɜɵɛɪɚɧɚ ɟɞɢɧɢɰɚ ɦɚɫɲɬɚɛɚ ɟ.
ǷȏȘ
Ʉɨɨɪɞɢɧɚɬɨɣ ɬɨɱɤɢ Ɇ ɥɟɠɚɳɟɣ ɧɚ ɩɪɹɦɨɣ l ɧɚɡɵɜɚɸɬ ɱɢɫɥɨ x ɤɨɬɨɪɨɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
1)ɚɛɫɨɥɸɬɧɚɹ ɜɟɥɢɱɢɧɚ x ɪɚɜɧɚ ɨɬɧɨɲɟɧɢɸ ɞɥɢɧɵ ɨɬɪɟɡɤɚ >OM @ ɤ ɞɥɢɧɟ ɟɞɢɧɢɱɧɨɝɨ ɨɬɪɟɡɤɚ e:
x
OMe ;
2)ɤɨɨɪɞɢɧɚɬɚ ɯ ɩɨɥɨɠɢɬɟɥɶɧɚ ɟɫɥɢ ɧɚɩɪɚɜɥɟɧɢɟ ɨɬ ɬɨɱɤɢ Ɉ ɤ ɬɨɱɤɟ Ɇ ɫɨɜɩɚɞɚɟɬ ɫ ɩɨɥɨɠɢɬɟɥɶɧɵɦ ɧɚɩɪɚɜɥɟɧɢɟɦ ɧɚ ɩɪɹɦɨɣ ɢ ɨɬɪɢɰɚɬɟɥɶɧɚ ɜ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɦ ɫɥɭɱɚɟ
ɉɭɫɬɶɧɚɩɪɹɦɨɣɡɚɞɚɧɨɞɜɟɬɨɱɤɢ A(x1) ɢ B(x2). Ɋɚɫɫɬɨɹɧɢɟɦɟɠɞɭɷɬɢɦɢ ɬɨɱɤɚɦɢ ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
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AB |
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x1 x2 |
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(1.1.1) |
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ɉɪɢɦɟɪ ɉɭɫɬɶ A B ɬɨɝɞɚ |
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AB |
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ɉɭɫɬɶ ɞɥɹ ɡɚɞɚɧɧɵɯ ɬɨɱɟɤ A(x1) ɢ B(x2) |
x2 x1 ɬɪɟɛɭɟɬɫɹ ɧɚɣɬɢ |
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ɬɨɱɤɭ C ɥɟɠɚɳɭɸ ɦɟɠɞɭ ɧɢɦɢ ɢ ɞɟɥɹɳɭɸ ɨɬɪɟɡɨɤ >A B@ ɜ ɨɬɧɨɲɟɧɢɢ M: |
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AC |
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CB |
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ɝɞɟ M – ɡɚɞɚɧɧɨɟ ɜɟɳɟɫɬɜɟɧɧɨɟ ɱɢɫɥɨ Ɉɛɨɡɧɚɱɢɦ ɧɟɢɡɜɟɫɬɧɭɸ ɤɨɨɪɞɢɧɚɬɭ ɬɨɱɤɢ C ɤɚɤ x ɢ ɡɚɩɢɲɟɦ ɪɚɜɟɧɫɬɜɨ ɱɟɪɟɡ ɤɨɨɪɞɢɧɚɬɵ
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M. |
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x x |
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Ɉɬɫɸɞɚ ɩɨɥɭɱɚɟɦ |
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ɉɪɢɦɟɪ ɉɭɫɬɶ A B ɇɚɣɞɟɦ ɫɟɪɟɞɢɧɭ ɨɬɪɟɡɤɚ >A B@ ɉɨɞ ɫɬɚɜɥɹɹ ɜ ɮɨɪɦɭɥɭ M 1 ɢ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɟɤ A ɢ B ɩɨɥɭɱɚɟɦ ɤɨɨɪɞɢ ɧɚɬɭ ɫɟɪɟɞɢɧɵ ɨɬɪɟɡɤɚ
x 2 6 2. 1 1
Ɉɩɪɟɞɟɥɟɧɢɟ ɇɚ ɩɥɨɫɤɨɫɬɢ ɡɚɞɚɧɚ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ ɟɫɥɢ
1)ɡɚɞɚɧɵ ɞɜɟ ɜɡɚɢɦɧɨ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɟ ɩɪɹɦɵɟ ɬɨɱɤɚ ɢɯ ɩɟɪɟɫɟɱɟ ɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɧɚɱɚɥɨɦ ɤɨɨɪɞɢɧɚɬ Ɉ);
2)ɧɚ ɤɚɠɞɨɣ ɢɡ ɩɪɹɦɵɯ ɜɵɛɪɚɧɨ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ
3)ɜɵɛɪɚɧɚ ɟɞɢɧɢɰɚ ɦɚɫɲɬɚɛɚ e .
ǷȏȘ
ɍɤɚɡɚɧɧɵɟ ɜ ɨɩɪɟɞɟɥɟɧɢɢ ɩɪɹɦɵɟ ɧɚɡɵɜɚɸɬɫɹ ɤɨɨɪɞɢɧɚɬɧɵɦɢ ɨɫɹɦɢ Ɉɞɧɚɢɡ ɧɢɯɧɚɡɵɜɚɟɬɫɹɨɫɶɸɚɛɫɰɢɫɫ ɨɫɶOx ɞɪɭɝɚɹ– ɨɫɶɸ ɨɪɞɢɧɚɬ ɨɫɶ Oy). ɉɭɫɬɶ M – ɬɨɱɤɚ ɩɥɨɫɤɨɫɬɢ Ɉɩɭɫɬɢɦ ɢɡ ɷɬɨɣ ɬɨɱɤɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɵ ɧɚ ɨɫɢ Ox ɢ Oy Ɍɨɱɤɢ P ɢ Q ɩɟɪɟɫɟɱɟɧɢɹ ɩɟɪɩɟɧɞɢɤɭɥɹɪɨɜ ɫ ɤɨɨɪɞɢɧɚɬɧɵɦɢ ɨɫɹɦɢ ɛɭ ɞɟɦ ɧɚɡɵɜɚɬɶ ɩɪɨɟɤɰɢɹɦɢ ɬɨɱɤɢ M ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɫɢ ɉɭɫɬɶ x – ɤɨɨɪ ɞɢɧɚɬɚ ɬɨɱɤɢ P ɧɚ ɨɫɢ Ox ɚ y – ɤɨɨɪɞɢɧɚɬɚ ɬɨɱɤɢ Q ɧɚ ɨɫɢ Oy ɍɩɨɪɹɞɨɱɟɧɧɚɹ ɩɚɪɚ ɱɢɫɟɥ (x y) ɧɚɡɵɜɚɟɬɫɹ ɤɨɨɪɞɢɧɚɬɚɦɢ ɬɨɱɤɢ M ɑɢɫɥɨ x – ɚɛɫɰɢɫɫɚ ɬɨɱɤɢ M y – ɨɪɞɢɧɚɬɚɬɨɱɤɢM Ɇɟɠɞɭɬɨɱɤɚɦɢɩɥɨɫɤɨɫɬɢ ɢɩɚɪɚɦɢ ɤɨɨɪɞɢɧɚɬ ɫɭɳɟɫɬɜɭɟɬ ɜɡɚɢɦɧɨ ɨɞɧɨɡɧɚɱɧɨɟ ɫɨɨɬɜɟɬɫɬɜɢɟ ɤɚɠɞɨɣ ɬɨɱɤɟ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɟɞɢɧɫɬɜɟɧɧɚɹ ɩɚɪɚ ɤɨɨɪɞɢɧɚɬ ɢ ɤɚɠɞɨɣ ɩɚɪɟ ɤɨɨɪɞɢɧɚɬ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɟɞɢɧ ɫɬɜɟɧɧɚɹ ɬɨɱɤɚ ɩɥɨɫɤɨɫɬɢ
6
ɉɭɫɬɶ ɧɚ ɩɥɨɫɤɨɫɬɢ ɡɚɞɚɧɨ ɞɜɟ ɬɨɱɤɢ A x1 y1 ɢ B x2 y2 ɇɚɣɞɟɦ ɪɚɫ ɫɬɨɹɧɢɟ ɦɟɠɞɭ ɷɬɢɦɢ ɬɨɱɤɚɦɢ ɉɨɫɬɪɨɢɦ ɩɪɹɦɭɸ ɩɪɨɯɨɞɹɳɭɸ ɱɟɪɟɡ ɬɨɱɤɭ A ɩɚɪɚɥɥɟɥɶɧɨ ɨɫɢ Ox ɢ ɱɟɪɟɡ ɬɨɱɤɭ B – ɩɚɪɚɥɥɟɥɶɧɨ ɨɫɢ Oy Ɍɨɱɤɭ ɢɯ ɩɟɪɟɫɟ ɱɟɧɢɹ ɨɛɨɡɧɚɱɢɦ ɤɚɤ C.
ǷȏȘ
Ɉɬɪɟɡɨɤ AB – ɝɢɩɨɬɟɧɭɡɚ ɩɪɹɦɨɭɝɨɥɶɧɨɝɨ ɬɪɟɭɝɨɥɶɧɢɤɚ ABC. Ʌɟɝɤɨ ɜɢ ɞɟɬɶ ɱɬɨ ɞɥɢɧɵ ɟɝɨ ɤɚɬɟɬɨɜ AC x2 x1 BC y2 y1 . ɉɨ ɬɟɨɪɟɦɟ ɉɢɮɚ ɝɨɪɚ ɩɨɥɭɱɚɟɦ ɞɥɢɧɭ ɝɢɩɨɬɟɧɭɡɵ
AB |
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ɉɪɢɦɟɪɇɚɣɞɟɦ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɬɨɱɤɚɦɢ A ɢ B :
AB (4 1)2 (5 1)2 5.
ɉɭɫɬɶ ɞɥɹ ɡɚɞɚɧɧɵɯ ɬɨɱɟɤ A x1 y1 ɢ B x2 y2 ɬɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɬɨɱɤɭ C ɥɟɠɚɳɭɸ ɦɟɠɞɭ ɧɢɦɢ ɢ ɞɟɥɹɳɭɸ ɨɬɪɟɡɨɤ >A B@ ɜ ɡɚɞɚɧɧɨɦ ɨɬɧɨɲɟɧɢɢ M:
CBAC
M.
Ɉɛɨɡɧɚɱɢɦ ɧɟɢɡɜɟɫɬɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ C ɤɚɤ (x y). Ɉɩɭɫɬɢɦ ɢɡ ɬɨ ɱɟɤ A B C ɩɟɪɩɟɧɞɢɤɭɥɹɪɵ ɧɚ ɨɫɶ Ox ɢ ɨɛɨɡɧɚɱɢɦ ɩɪɨɟɤɰɢɢ ɷɬɢɯ ɬɨɱɟɤ P1 P2 P ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ
ɉɪɹɦɵɟ AP1 BP2 ɢ CP ɩɚɪɚɥɥɟɥɶɧɵ ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ
AC P1P .
CB PP2
Ɉɬɪɟɡɨɤ >P1 P2@ ɩɪɢɧɚɞɥɟɠɢɬ ɤɨɨɪɞɢɧɚɬɧɨɣ ɩɪɹɦɨɣ ɢ ɤɨɨɪɞɢɧɚɬɭ x ɬɨɱɤɢ P ɞɟɥɹɳɟɣ ɟɝɨ ɜ ɨɬɧɨɲɟɧɢɢ M ɦɨɠɧɨ ɧɚɣɬɢ ɩɨ ɮɨɪɦɭɥɟ ɇɚɣɞɟɧ ɧɚɹ ɜɟɥɢɱɢɧɚ ɹɜɥɹɟɬɫɹ ɢɫɤɨɦɨɣ ɚɛɫɰɢɫɫɨɣ ɬɨɱɤɢ C Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɧɚɣɬɢ
7
ɜɬɨɪɭɸ ɤɨɨɪɞɢɧɚɬɭ ɷɬɨɣ ɬɨɱɤɢ Ɍ ɨ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ C ɧɚɯɨɞɢɦ ɩɨ ɮɨɪ ɦɭɥɚɦ
x |
x1 Mx2 |
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y1 My2 |
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ǷȏȘ |
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ɉɪɢɦɟɪɉɭɫɬɶ |
A B ɇɚɣɞɟɦ ɫɟɪɟɞɢɧɭ ɨɬɪɟɡɤɚ >A B@ |
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ɉɨɞɫɬɚɜɥɹɹ ɜ ɮɨɪɦɭɥɭ (1.1.5) M 1 ɢ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɟɤ A ɢ B ɩɨɥɭɱɚɟɦ ɤɨɨɪ |
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ɞɢɧɚɬɵ ɫɟɪɟɞɢɧɵ ɨɬɪɟɡɤɚ |
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Ɉɩɪɟɞɟɥɟɧɢɟ Ȼɭɞɟɦ ɝɨɜɨɪɢɬɶ ɱɬɨɧɚ ɩɥɨɫɤɨɫɬɢ ɡɚɞɚɧɚ ɩɨɥɹɪɧɚɹ ɫɢ ɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ ɟɫɥɢ
1)ɡɚɞɚɧɚ ɬɨɱɤɚ O ɧɚɡɵɜɚɟɦɚɹ ɩɨɥɸɫɨɦ
2)ɡɚɞɚɧ ɥɭɱ >OP) ɧɚɡɵɜɚɟɦɵɣ ɩɨɥɹɪɧɨɣ ɩɨɥɭɨɫɶɸ
3)ɜɵɛɪɚɧɚ ɟɞɢɧɢɰɚ ɦɚɫɲɬɚɛɚ e.
ǷȏȘ
ɉɨɥɹɪɧɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ɬɨɱɤɢ M ɩɥɨɫɤɨɫɬɢ ɧɚɡɵɜɚɸɬɫɹ ɩɨɥɹɪɧɵɣ ɪɚ ɞɢɭɫ ɪɚɜɧɵɣ ɪɚɫɫɬɨɹɧɢɸ ɨɬ ɬɨɱɤɢ O ɞɨ ɬɨɱɤɢ M: r OM ; ɩɨɥɹɪɧɵɣ ɭɝɨɥ –
ɭɝɨɥ K ɧɚ ɤɨɬɨɪɵɣ ɧɭɠɧɨ ɩɨɜɟɪɧɭɬɶ ɥɭɱ >OP) ɞɥɹ ɬɨɝɨ ɱɬɨɛɵ ɨɧ ɫɨɜɩɚɥ ɫ ɥɭɱɨɦ >OM).
Ʉɚɠɞɨɣ ɬɨɱɤɟ M ɩɥɨɫɤɨɫɬɢ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɩɨɥɹɪ ɧɨɝɨ ɪɚɞɢɭɫɚ r ɉɨɥɹɪɧɵɣ ɭɝɨɥ K ɨɩɪɟɞɟɥɟɧ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɩɟɪɢɨɞɚ Q
8
ɨɛɵɱɧɨ ɜɵɛɢɪɚɸɬ K Q Q@ ɤɪɨɦɟ ɬɨɱɤɢ ɫɨɜɩɚɞɚɸɳɟɣ ɫ ɩɨɥɸɫɨɦ Ⱦɥɹ ɷɬɨɣ ɬɨɱɤɢ r ɚ ɭɝɨɥ K ɧɟ ɨɩɪɟɞɟɥɟɧ ɍɩɨɪɹɞɨɱɟɧɧɨɣ ɩɚɪɟ ɱɢɫɟɥ r K ɫɨ
ɨɬɜɟɬɫɬɜɭɟɬ ɟɞɢɧɫɬɜɟɧɧɚɹ ɬɨɱɤɚ ɩɥɨɫɤɨɫɬɢ M.
ɉɭɫɬɶ ɧɚ ɩɥɨɫɤɨɫɬɢ ɨɞɧɨɜɪɟɦɟɧɧɨ ɡɚɞɚɧɵ ɩɨɥɹɪɧɚɹ ɢ ɞɟɤɚɪɬɨɜɚ ɩɪɹɦɨ ɭɝɨɥɶɧɚɹ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɩɪɢɱɟɦ ɧɚɱɚɥɨ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɫɨɜɩɚɞɚɟɬ ɫ ɩɨɥɸɫɨɦ ɚ ɩɨɥɹɪɧɚɹ ɨɫɶ ɥɟɠɢɬ ɧɚ ɨɫɢ ɚɛɫɰɢɫɫ ɉɭɫɬɶ ɬɨɱɤɚ M ɢɦɟɟɬ ɞɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ (x y) ɢ ɩɨɥɹɪɧɵɟ ɤɨɨɪɞɢɧɚɬɵ r K ɉɨ ɢɡɜɟɫɬ ɧɵɦ ɩɨɥɹɪɧɵɦ ɤɨɨɪɞɢɧɚɬɚɦ r K ɦɨɠɧɨ ɧɚɣɬɢ ɞɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢɫɦ ɪɢɫ
£¦x r cosK
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ǷȏȘ
ȿɫɥɢ ɡɚɞɚɧɵ ɞɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ (x y) ɬɨ ɩɨɥɹɪɧɵɣ ɪɚɞɢɭɫ
ɧɚɯɨɞɢɦ ɩɨ ɮɨɪɦɭɥɟ |
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ɉɪɢɦɟɪɉɭɫɬɶ ɩɨɥɹɪɧɵɟ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ M Q Ⱦɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ ɷɬɨɣ ɬɨɱɤɢ ɨɩɪɟɞɟɥɹɟɦ ɩɨ ɮɨɪɦɭɥɚɦ
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ɉɪɢɦɟɪɉɭɫɬɶ ɞɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ M ɇɚɯɨɞɢɦ |
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ɩɨɥɹɪɧɵɣ ɪɚɞɢɭɫ r ( 2)2 ( 2)2 2 |
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ɉɨɞɫɬɚɜɥɹɹ ɧɚɣɞɟɧɧɭɸ ɜɟɥɢ |
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ɱɢɧɭ ɢ ɞɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ ɜ ɫɢɫɬɟɦɭ ɩɨɥɭɱɚɟɦ |
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ɫɥɟɞɨɜɚɬɟɥɶɧɨ K |
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Ɍɟɤɫɬ ɡɚɞɚɧɢɹ Ɍɨɱɤɢ M1 r1 K1 |
ɢ M2 r2 K2 ɡɚɞɚɧɵ ɫɜɨɢɦɢ ɩɨɥɹɪ |
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ɧɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ɇɚɣɬɢ ɞɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ ɷɬɢɯ ɬɨɱɟɤ ɢ ɫɟɪɟɞɢɧɭ ɨɬ ɪɟɡɤɚ >M1 M2@
ȼɚɪɢɚɧɬɵ ɡɚɞɚɧɢɹ ʋ
ʋ |
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K1 |
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K2 |
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2 |
3 |
S |
4 |
3 |
S |
|
|
|
|
|
|
|
2 |
2 |
2 |
S |
6 |
3 |
S |
3 |
2 |
3 |
S |
4 |
3 |
S |
4 |
4 |
3 |
S |
2 |
3 |
S |
5 |
6 |
3 |
S |
2 |
3 |
S |
6 |
4 |
3 |
S |
2 2 |
S |
|
|
2 |
3 |
S |
4 |
3 |
S |
|
6 |
3 |
S |
4 |
3 |
S |
9 |
2 |
3 |
S |
2 2 |
S |
|
|
|
4 |
S |
|
2 |
S |
11 |
2 |
2 |
S |
4 2 |
S |
|
12 |
|
2 |
S |
2 |
3 |
S |
13 |
4 |
2 |
S |
|
2 |
S |
14 |
|
3 |
S |
2 2 |
S |
|
15 |
4 |
2 |
S |
|
2 |
Q |
16 |
2 |
3 |
S |
|
2 |
S |
|
|
3 |
S |
|
1 |
S |
|
2 |
2 |
S |
|
2 |
S |
19 |
|
2 |
Q |
2 |
3 |
S |
|
|
2 |
S |
|
3 |
S |
21 |
|
2 |
S |
2 |
3 |
S |
22 |
|
2 |
S |
2 2 |
S |
|
23 |
4 |
3 |
S |
|
2 |
Q |
24 |
2 |
2 |
S |
4 |
3 |
S |
10