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l 1 2 + m 1 2 + n 1 2 = 1
cos θ = l 1 l 2 + m 1 m 2 + n 1 n 2
l 1 l 2 + m 1 m 2 + n 1 n 2 = 0
A 1 A 2 + B 1 B 2 + C 1 C 2 = 0
x - x 0 : y - y 0 : z - z 0
x - x 0 ( x - x 0 ) 2 + ( y - y 0 ) 2 + ( z - z 0 ) 2 ,             y - y 0 ( x - x 0 ) 2 + ( y - y 0 ) 2 + ( z - z 0 ) 2 , z - z 0 ( x - x 0 ) 2 + ( y - y 0 ) 2 + ( z - z 0 ) 2
x = 1 4 f ( y 2 + z 2 )
dx = 1 2 f ( ydy + zdz )
1 2 f ( ydy + zdz ) : dy : dz
l 2 + m 2 + n 2 = 1
l 2 f ( ydy + zdz ) + m dy + n dz = 0
l = - 2 f 4 f 2 + y 2 + z 2 , m = y 4 f 2 + y 2 + z 2 , n = z 4 f 2 + y 2 + z 2
- 2 f : y : z
l ( x - x 0 ) + m ( y - y 0 ) + n ( z - z 0 ) = 0
l ( - 2 f ) + m y + n z = 0
m = | 2 l f z - l ( x - x 0 ) z - z 0 | | y z y - y 0 z - z 0 | = l [ 2 f ( z - z o ) + z ( x - x 0 ) ] y 0 z - y z 0
n = | y 2 l f y - y 0 - l ( x - x 0 ) | | y z y - y 0 z - z 0 | = - l [ 2 f ( y - y 0 ) + y ( x - x 0 ) ] y 0 z - y z 0
cos P 0 PN = 2 f ( x - x 0 ) - y ( y - y 0 ) - z ( z - z 0 ) 4 f 2 + y 2 + z 2             ( x - x 0 ) 2 + ( y - y 0 ) 2 + ( z - z 0 ) 2
cos NPR = - 2 fl + ym + zn 4 f 2 + y 2 + z 2
2 f ( x - x 0 ) - y ( y - y 0 ) - z ( z - z 0 ) 4 f 2 + y 2 + z 2             ( x - x 0 ) 2 + ( y - y 0 ) 2 + ( z - z 0 ) = - 2 fl + ym + zn 4 f 2 + y 2 + z 2
l l + m l 2 f ( z - z 0 ) + z ( x - x 0 ) y 0 z - yz 0 - n l [ 2 f ( y - y 0 ) + y ( x - x 0 ) ] y 0 z - yz 0 = 0
m [ 2 f ( z - z 0 ) + z ( x - x 0 ) ] - n [ 2 f ( y - y 0 ) + y ( x - x 0 ) ] = - l ( y 0 z - yz 0 )
my + nz = 2 fl + 2 f ( x - x 0 ) - y ( y - y 0 ) - z ( z - z 0 ) ( x - x 0 ) 2 + ( y - y 0 ) 2 + ( z - z 0 ) 2
m = | - l ( y 0 z - yz 0 ) - [ 2 f ( y - y 0 + y ( x - x 0 ) ] 2 fl + 2 f ( x - x 0 ) - y ( y - y 0 ) - 2 ( z - z 0 ) ( x - x 0 ) 2 + ( y - y 0 ) 2 + ( z - z 0 ) 2 z | z [ 2 f ( z - z 0 ) + z ( x - x 0 ) ] + y [ 2 f ( y - y 0 ) + y ( x - x 0 ) ]
n = | 2 f ( z - z 0 ) + z ( x - x 0 ) - 1 ( y 0 z - yz 0 ) y 2 fl + 2 f ( x - x 0 ) - y ( y - y 0 ) - z ( z - z 0 ) ( x - x 0 ) 2 + ( y - y 0 ) 2 + ( z - z 0 ) 2 | z [ 2 f ( z - z 0 ) + z ( x - x 0 ) ] + y [ 2 f ( y - y 0 ) + y ( x - x 0 ) ]
l = 1 - m 2 - n 2
l = cos O ° = 1 ,             m = cos 90 ° = O ,             n = cos 90 ° = O .
m = | yz 0 - y 0 z - y ( f + x ) 4 fx ( f - x 0 ) - ( f - x ) ( yy 0 + zz 0 ) ( f + x ) 2 z | 4 fx ( f + x )
m = y ( f - x 0 ) ( f + x ) 2 + [ - 4 f 2 + y 2 - z 2 ] y 0 4 f ( f + x ) 2 + y z z 0 2 f ( f + x ) 2
n = z ( f - x 0 ) ( f + x ) 2 + [ - 4 f 2 + z 2 - y 2 ] z 0 4 f ( f + x ) 2 + y z y 0 2 f ( f + x ) 2
l = 1 - ½ ( m 2 + n 2 )
m = n = 0.01767 ,             l = 0.99969
m = n = 0.01846 ,             l = 0.99964
y = 4 fx cos θ             z = 4 fx sin θ
y = 2.12 x cos θ             z = 2.12 x sin θ
FP = ( f - x ) 2 + y 2 = ( f - x ) 2 + 4 fx = f + x = 1.125 + x
1.15 1.125 + x
m = 0.11 y z 2 f ( f + x ) 2             and             n = 0.11 [ - 4 f 2 + z 2 - y 2 ] 4 f ( f + x )
m = 6.30 y z 2 f ( f + x ) 2             and             n = 6.30 [ - 4 f 2 + z 2 - y 2 ] 4 f ( f + x ) 2