3D Graph using Parametric Lines

İfade 15: "s" equals negative 4.8

These two variables help draw the arrow-heads. Can you tell what they do?

İfade 17: "l" equals 0.9 6

İfade 18: "a" Subscript, "l" , Baseline equals 11.2

Now we draw the axes. x_arr and y_arr are helper functions that'll we'll use in equations 23, 24, and 25

İfade 20: "x" Subscript, "a" "r" "r" , Baseline left parenthesis, "t" , "x" Subscript, "s" , Baseline , "y" Subscript, "s" , Baseline , right parenthesis equals left brace, 0 less than "t" less than "l" : "x" Subscript, "s" , Baseline "t" , "l" less than "t" less than StartFraction, "l" plus 1 Over 2 , EndFraction : "l" "x" Subscript, "s" , Baseline plus cos left parenthesis, StartFraction, 5 pi Over 6 , EndFraction , right parenthesis "x" Subscript, "s" , Baseline left parenthesis, "t" minus "l" , right parenthesis minus sin left parenthesis, StartFraction, 5 pi Over 6 , EndFraction , right parenthesis "y" Subscript, "s" , Baseline left parenthesis, "t" minus "l" , right parenthesis , StartFraction, "l" plus 1 Over 2 , EndFraction less than "t" less than 1 : "l" "x" Subscript, "s" , Baseline plus sin left parenthesis, StartFraction, 5 pi Over 6 , EndFraction , right parenthesis "y" Subscript, "s" , Baseline left parenthesis, "t" minus StartFraction, "l" plus 1 Over 2 , EndFraction , right parenthesis plus cos left parenthesis, StartFraction, 5 pi Over 6 , EndFraction , right parenthesis "x" Subscript, "s" , Baseline left parenthesis, "t" minus StartFraction, "l" plus 1 Over 2 , EndFraction , right parenthesis , right brace

İfade 21: "y" Subscript, "a" "r" "r" , Baseline left parenthesis, "t" , "x" Subscript, "s" , Baseline , "y" Subscript, "s" , Baseline , right parenthesis equals left brace, 0 less than "t" less than "l" : "y" Subscript, "s" , Baseline "t" , "l" less than "t" less than StartFraction, "l" plus 1 Over 2 , EndFraction : "l" "y" Subscript, "s" , Baseline plus cos left parenthesis, StartFraction, 5 pi Over 6 , EndFraction , right parenthesis "y" Subscript, "s" , Baseline left parenthesis, "t" minus "l" , right parenthesis plus sin left parenthesis, StartFraction, 5 pi Over 6 , EndFraction , right parenthesis "x" Subscript, "s" , Baseline left parenthesis, "t" minus "l" , right parenthesis , StartFraction, "l" plus 1 Over 2 , EndFraction less than "t" less than 1 : "l" "y" Subscript, "s" , Baseline minus sin left parenthesis, StartFraction, 5 pi Over 6 , EndFraction , right parenthesis "x" Subscript, "s" , Baseline left parenthesis, "t" minus StartFraction, "l" plus 1 Over 2 , EndFraction , right parenthesis plus cos left parenthesis, StartFraction, 5 pi Over 6 , EndFraction , right parenthesis "y" Subscript, "s" , Baseline left parenthesis, "t" minus StartFraction, "l" plus 1 Over 2 , EndFraction , right parenthesis , right brace

İfade 22: left parenthesis, "x" Subscript, "a" "r" "r" , Baseline left parenthesis, "t" , "a" Subscript, "l" , Baseline "x" Subscript, "x" , Baseline , "a" Subscript, "l" , Baseline "x" Subscript, "y" , Baseline , right parenthesis , "y" Subscript, "a" "r" "r" , Baseline left parenthesis, "t" , "a" Subscript, "l" , Baseline "x" Subscript, "x" , Baseline , "a" Subscript, "l" , Baseline "x" Subscript, "y" , Baseline , right parenthesis , right parenthesis

İfade 23: left parenthesis, "x" Subscript, "a" "r" "r" , Baseline left parenthesis, "t" , "a" Subscript, "l" , Baseline "y" Subscript, "x" , Baseline , "a" Subscript, "l" , Baseline "y" Subscript, "y" , Baseline , right parenthesis , "y" Subscript, "a" "r" "r" , Baseline left parenthesis, "t" , "a" Subscript, "l" , Baseline "y" Subscript, "x" , Baseline , "a" Subscript, "l" , Baseline "y" Subscript, "y" , Baseline , right parenthesis , right parenthesis

İfade 24: left parenthesis, "x" Subscript, "a" "r" "r" , Baseline left parenthesis, "t" , "a" Subscript, "l" , Baseline "z" Subscript, "x" , Baseline , "a" Subscript, "l" , Baseline "z" Subscript, "y" , Baseline , right parenthesis , "y" Subscript, "a" "r" "r" , Baseline left parenthesis, "t" , "a" Subscript, "l" , Baseline "z" Subscript, "x" , Baseline , "a" Subscript, "l" , Baseline "z" Subscript, "y" , Baseline , right parenthesis , right parenthesis

And, finally, we draw the graph in blue and green. Can you figure out what g(t) and h(t) do?

İfade 26: "g" left parenthesis, "x" , right parenthesis equals 2 "s" left parenthesis, StartFraction, floor left parenthesis, "x" times left parenthesis, "n" plus 1 , right parenthesis , right parenthesis Over "n" , EndFraction minus .5 , right parenthesis

İfade 27: "h" left parenthesis, "x" , right parenthesis equals 2 "s" left parenthesis, mod left parenthesis, "x" times left parenthesis, "n" plus 1 , right parenthesis , 1 , right parenthesis minus .5 , right parenthesis

İfade 28: left parenthesis, "x" Subscript, "x" , Baseline "g" left parenthesis, "t" , right parenthesis plus "y" Subscript, "x" , Baseline "h" left parenthesis, "t" , right parenthesis plus "z" Subscript, "x" , Baseline "f" left parenthesis, "g" left parenthesis, "t" , right parenthesis , "h" left parenthesis, "t" , right parenthesis , right parenthesis , "x" Subscript, "y" , Baseline "g" left parenthesis, "t" , right parenthesis plus "y" Subscript, "y" , Baseline "h" left parenthesis, "t" , right parenthesis plus "z" Subscript, "y" , Baseline "f" left parenthesis, "g" left parenthesis, "t" , right parenthesis , "h" left parenthesis, "t" , right parenthesis , right parenthesis , right parenthesis

İfade 29: left parenthesis, "x" Subscript, "x" , Baseline "h" left parenthesis, "t" , right parenthesis plus "y" Subscript, "x" , Baseline "g" left parenthesis, "t" , right parenthesis plus "z" Subscript, "x" , Baseline "f" left parenthesis, "h" left parenthesis, "t" , right parenthesis , "g" left parenthesis, "t" , right parenthesis , right parenthesis , "x" Subscript, "y" , Baseline "h" left parenthesis, "t" , right parenthesis plus "y" Subscript, "y" , Baseline "g" left parenthesis, "t" , right parenthesis plus "z" Subscript, "y" , Baseline "f" left parenthesis, "h" left parenthesis, "t" , right parenthesis , "g" left parenthesis, "t" , right parenthesis , right parenthesis , right parenthesis

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