Expression 28: "x" equals "x" Subscript, "p" "t" 1 , Baseline left brace, "y" greater than or equal to "p" left parenthesis, "x" Subscript, "p" "t" 1 , Baseline , right parenthesis , right bracex=xpt1y≥pxpt1
28
The line from a point on the parabola to the focus. The incoming ray and the reflected ray form equal angles about the line perpendicular to the parabola.
29
Expression 30: "l" Subscript, "t" "o" "F" "o" "c" "u" "s" , Baseline left parenthesis, "x" , right parenthesis equals "l" Subscript, "p" "t" "s" , Baseline left parenthesis, 0 , "y" Subscript, "f" "o" "c" "u" "s" , Baseline , "x" Subscript, "p" "t" 1 , Baseline , "p" left parenthesis, "x" Subscript, "p" "t" 1 , Baseline , right parenthesis , "x" , right parenthesisltoFocusx=lpts0,yfocus,xpt1,pxpt1,x
30
Expression 31: "l" Subscript, "t" "o" "F" "o" "c" "u" "s" 1 , Baseline left parenthesis, "x" , right parenthesis equals left brace, 0 less than or equal to "x" less than or equal to "x" Subscript, "p" "t" 1 , Baseline : "l" Subscript, "t" "o" "F" "o" "c" "u" "s" , Baseline left parenthesis, "x" , right parenthesis , right braceltoFocus1x=0≤x≤xpt1:ltoFocusx
31
Expression 32: "l" Subscript, "t" "o" "F" "o" "c" "u" "s" 2 , Baseline left parenthesis, "x" , right parenthesis equals left brace, "x" Subscript, "p" "t" 1 , Baseline less than or equal to "x" less than or equal to 0 : "l" Subscript, "t" "o" "F" "o" "c" "u" "s" , Baseline left parenthesis, "x" , right parenthesis , right braceltoFocus2x=xpt1≤x≤0:ltoFocusx
32
The angle of the line perpendicular to the parabola. p_angle is computed from the slope of the perpendicular line. p_upAngle is angle of the upper part of the perpendicular.
33
Expression 34: "p" Subscript, "a" "n" "g" "l" "e" , Baseline equals arc tangent left parenthesis, negative StartFraction, 1 Over "p" Subscript, "s" "l" "o" "p" "e" , Baseline left parenthesis, "x" Subscript, "p" "t" 1 , Baseline , right parenthesis , EndFraction , right parenthesispangle=arctan−1pslopexpt1
equals=
negative 0.9 0 6 9 3 3 6 1 6 5 4 4−0.906933616544
34
Expression 35: "p" Subscript, "u" "p" "A" "n" "g" "l" "e" , Baseline equals left brace, "x" Subscript, "p" "t" 1 , Baseline greater than or equal to 0 : "p" Subscript, "a" "n" "g" "l" "e" , Baseline plus pi , "p" Subscript, "a" "n" "g" "l" "e" , Baseline , right bracepupAngle=xpt1≥0:pangle+π,pangle
equals=
2.2 3 4 6 5 9 0 3 7 0 52.23465903705
35
f_upAngle is the angle of the reflected ray
36
Expression 37: "f" Subscript, "u" "p" "A" "n" "g" "l" "e" , Baseline equals left brace, "x" Subscript, "p" "t" 1 , Baseline greater than or equal to 0 : "f" Subscript, "a" "n" "g" "l" "e" , Baseline left parenthesis, "x" Subscript, "p" "t" 1 , Baseline , "p" left parenthesis, "x" Subscript, "p" "t" 1 , Baseline , right parenthesis , 0 , "y" Subscript, "f" "o" "c" "u" "s" , Baseline , right parenthesis plus pi , "f" Subscript, "a" "n" "g" "l" "e" , Baseline left parenthesis, "x" Subscript, "p" "t" 1 , Baseline , "p" left parenthesis, "x" Subscript, "p" "t" 1 , Baseline , right parenthesis , 0 , "y" Subscript, "f" "o" "c" "u" "s" , Baseline , right parenthesis , right bracefupAngle=xpt1≥0:fanglexpt1,pxpt1,0,yfocus+π,fanglexpt1,pxpt1,0,yfocus
equals=
2.8 9 8 5 2 1 7 4 7 32.8985217473
37
The angle of incidence and the angle of reflection, both with respect to the angle of the line perpendicular to the parabola.
38
Expression 39: "I" Subscript, "a" "n" "g" "l" "e" , Baseline equals StartFraction, pi Over 2 , EndFraction minus "p" Subscript, "u" "p" "A" "n" "g" "l" "e" , BaselineIangle=π2−pupAngle
equals=
negative 0.6 6 3 8 6 2 7 1 0 2 5 1−0.663862710251
39
Expression 40: "R" Subscript, "a" "n" "g" "l" "e" , Baseline equals "p" Subscript, "u" "p" "A" "n" "g" "l" "e" , Baseline minus "f" Subscript, "u" "p" "A" "n" "g" "l" "e" , BaselineRangle=pupAngle−fupAngle
equals=
negative 0.6 6 3 8 6 2 7 1 0 2 5 1−0.663862710251
40
The distance from the point on the parabola to the focus and to the directrix. The directrix is y=0, so the distance to the directrix is simply p(x).
41
Expression 42: "d" left parenthesis, "x" Subscript, "p" "t" 1 , Baseline , "p" left parenthesis, "x" Subscript, "p" "t" 1 , Baseline , right parenthesis , 0 , "y" Subscript, "f" "o" "c" "u" "s" , Baseline , right parenthesisdxpt1,pxpt1,0,yfocus
equals=
2.3 6 9 6 5 9 8 6 3 9 52.36965986395
42
Expression 43: "p" left parenthesis, "x" Subscript, "p" "t" 1 , Baseline , right parenthesispxpt1
equals=
2.3 6 9 6 5 9 8 6 3 9 52.36965986395
43
Expression 44: "x" equals "x" Subscript, "p" "t" 1 , Baseline left brace, 0 less than "y" less than "p" left parenthesis, "x" Subscript, "p" "t" 1 , Baseline , right parenthesis , right bracex=xpt10<y<pxpt1
44
Expression 45: left parenthesis, "x" minus 0 , right parenthesis left parenthesis, 0 minus "y" Subscript, "f" "o" "c" "u" "s" , Baseline , right parenthesis equals left parenthesis, "y" minus "y" Subscript, "f" "o" "c" "u" "s" , Baseline , right parenthesis left parenthesis, "x" Subscript, "p" "t" 1 , Baseline minus 0 , right parenthesis left brace, "x" Subscript, "p" "t" 1 , Baseline less than "x" less than 0 , 0 less than "x" less than "x" Subscript, "p" "t" 1 , Baseline , right bracex−00−yfocus=y−yfocusxpt1−0xpt1<x<0,0<x<xpt1
