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### Equations (39)

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