These two variables help draw the arrow-heads. Can you tell what they do?
21
Expression 22: "l" equals 0.7l=0.7
negative 10−10
1010
22
Expression 23: "a" Subscript, "l" , Baseline equals 9al=9
negative 10−10
2020
23
Now we draw the axes. x_arr and y_arr are helper functions that'll we'll use in equations 23, 24, and 25
24
Expression 25: "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 bracexarrt,xs,ys=0<t<l:xst,l<t<l+12:lxs+cos5π6xst−l−sin5π6yst−l,l+12<t<1:lxs+sin5π6yst−l+12+cos5π6xst−l+12
25
Expression 26: "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 braceyarrt,xs,ys=0<t<l:yst,l<t<l+12:lys+cos5π6yst−l+sin5π6xst−l,l+12<t<1:lys−sin5π6xst−l+12+cos5π6yst−l+12
26
Expression 27: 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 parenthesisxarrt,alxx,alxy,yarrt,alxx,alxy
domain t Minimum: 00
less than or equal to "t" less than or equal to≤t≤
domain t Maximum: 11
27
Expression 28: 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 parenthesisxarrt,alyx,alyy,yarrt,alyx,alyy
domain t Minimum: 00
less than or equal to "t" less than or equal to≤t≤
domain t Maximum: 11
28
Expression 29: 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 parenthesisxarrt,alzx,alzy,yarrt,alzx,alzy
domain t Minimum: 00
less than or equal to "t" less than or equal to≤t≤
domain t Maximum: 11
29
And, finally, we draw the graph in blue and green. Can you figure out what g(t) and h(t) do?
30
Expression 31: "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 parenthesisgx=2sfloorx·n+1n−.5
31
Expression 32: "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 parenthesishx=2smodx·n+1,1−.5
32
Expression 33: 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 parenthesisxxgt+yxht+zxfgt,ht,xygt+yyht+zyfgt,ht
domain t Minimum: 00
less than or equal to "t" less than or equal to≤t≤
domain t Maximum: 11
33
Expression 34: 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 parenthesisxxht+yxgt+zxfht,gt,xyht+yygt+zyfht,gt
domain t Minimum: 00
less than or equal to "t" less than or equal to≤t≤
domain t Maximum: 11
34
Hidden Label: left parenthesis, "a" , "b" , right parenthesisa,b