Начертательная геометрия

6

& " ) - . @ " " - . @ , D' " . #. ( " . $ -D .

6.1. #"&"' +&" ": %-&+"A" # "$% " %*. "jkk%$% ( %&+ / %*

" , ) " " ( -) D . B -D 1 , -.

D , - " -, , " .

+ -

.

« » ( " . axon metreo D) -« ». M" >

x, y, z , .D D A,

( . 6.1).

# x, y, z D .+ e ( + )" , ex, ey, ez (e=ex=ey=ez).

D QA a, a', a'', - + ex, ey, ez.

:

Q ( -);

102

.

l

– " l -Q ( );

x ! y ! z1 (

)

Zl

= . 6.1

A A a1, a1! a1" A ex, ey, ez + ;

ex1, ey1, ez1 ( ) +-.

H, V, W, -Q l.

# + + " -" !.

# ( ()) :

x	m "	e	x1	, y	n "	ey1		z k "	ez1
							,			.
		ex				ey			ez

+ ()) -D D' :

1. * , m = n = k.

6. $ !

103

2.C , m =k n m = n k.

3.$, m n k.

@ + " : «isos» ( - -()) ) «di» ( ())) «treis» (()) ).

+ DQ -" , " = 90º, - " , #$ .

C , ()) -

:
	"	– m2+n2+k2=2+ctg2 ;
	"	– m2+n2+k2=2.

,, D- ' " " +-.

I , " &. 1853 " " , D: «<D , ', " -- ». C ( ' 1864 ". " " %. ? . ( " - ? .

* " + -

:

- , 5 ! .

6.2. *:">A" ! * # ! * %=": %*

" D D D + D" . "())

104 .

(m=n=k) 0,82 (m2+n2+k2=2; m=n=k= 23 =0,82), -

x1, y1, z1 D " " " 120º ( . 6.2). # D D ' , , - D , ())

m=n=k=1. ( --

1,22 .

# z1 " D , x1 y1

= . 6.2

" 30 "-D.

/ , , -

" ( . 6.3), -( ( . 6.4). C % x1o1ax1, xA A. & xA " ( . 6.3).

= . 6.3

= . 6.4

* ax1 D, D y1, -, yA A, a1;a1 , z1 -zA A. A1 &" -. 6.5. # - . I x y D - + . @ ,x1 O1

6. $ !

105

, xA=a'd'. * - D, D y1. @ ,yA=a'a. CS1 - + . S1 A1, B1, C1, D1, E1, D .

= . 6.5

@ . 6.6 + " .

= . 6.6

106

.

6.3. *:">A" ! * # ! * )%: %*

" ()) x1 z1 D m=k, y1 – + – n=12m.

$" m2+k2+n2=m2+m2+(1 2m)2=2; m=		8	9 =0,94; n=0,47.
	#	z1 ,		x1
	" 7º10´. #			y1
	" 41º25[ "-
	( . 6.7). @
	,		, D
	D		D,
	()) m=k=1,
	n=0,5. (
= . 6.7	1,06 .
	/ " -

A ( . 6.8), ( " ( . 6.9).

# x1 o1ax1,xA A. * ax1 D, Dy1, , yAA, ()) y1 0,5. * a1 a1 1, zA. A1 D D A&

= . 6.8

= . 6.9

" ( . 6.10) . 6.11.

C )1/4 ( " , ' , ). $ ,

6. $ !	107
( O) x,
y, z ( . 6.10).
M D D D' -
.
) " , -
xOz yOz
( . 6.11, ).
	# + " -
( 1, 2, 3, 4) -
2, 3, 4.
. C
( " -
, z1. I -
zH ( .

	6.11, ).
	+ "
= . 6.10	( 5, 6, 7, 8). -
	6, 7, 8. * (

,z1, zB - " .( . 6.11, ).

# " - . + ( . 6.11, ).

= . 6.11

108

.

< + -" , ' D' , -( . 6.12 - , . 6.13 -).

= . 6.12

= . 6.13

6.4. ="' / % "+ >/ "& % % l # *:">A" ! "0+&" ": %%

# ' (- . ( ". ; ( "+ .

, ' -, ( -, D' .

, + ( .

" ! ! *

+ ( ( .D 1,22 . ( + ( 1,22D, -– 0,71D. @ . 6.14 " ) -( . D,AB = 0,71D ( ( ). A B, , AB, " " - . E F . EF=1,22D –+ ( .

x1, y1, z1. x1O1z1D 2. -x1 z1. @ , -

6. $ !

109

. @ , y1 ( -( ), , & ( D (- ). + D ( , D EF ( . 6.15).

= . 6.14

= . 6.15

8 , ( . C ( " .

( " -, + (- .

" ! ! *

+ ( D, -. + ,1,06D. " H ) W( 0,35D, ) V0,94D.

C ( " )" " ( . 6.16).

= . 6.16

110

.

& " 100 35 . % (106 . # + ,D ( AB). # BC 0,35D, - D ( H W.

# AC 1,06D, D + ( . / D " ( AK), -K + " ,AE D 0,94D, ( V.

* -

" -

. 6.17.

@ , -

V

2

x1 z1

-

, -

. @ , -

y1 - = . 6.17 , 0,94D (

( ). + D ( , D 1,06D. .

C

" +D.

+ ", -" + .

+ - ()) : – 1,22 ( . 6.18, ), -– 1,06 ( . 6.18, ).

@ 6.18, , -' D " " ) .