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B o o l e a n p r o c e d u r e P O I N T I N POLYGON (n, x, y, xO, y0); v a l u e n, x0, y0; i n t e g e r n ; a r r a y x, y; r e a l x0, y0; c o m m e n t if t h e p o i n t s (x[i], y[i]) (i = 1, 2 , . . , n) a r e i n t h i s cyclic o r d e r t h e ve r t i c e s of a s imp le c losed p o l y g o n a n d (x0, y0) is a p o i n t n o t on a n y s ide of t h e po lygon , t h e n t h e procedure d e t e r m i n e s , b y s e t t i n g " p o i n t in p o l y g o n " to t r u e , w h e t h e r (x0, y0) l ies in t h e i n t e r i o r of t h e p o l y g o n ; b e g i n i n t e g e r i ; B o o l e a n b; x [ n + l ] : = x [ 1 ] ; y [ n + l ] : = y [ 1 ] ; b : = t r u e ; f o r i i= 1 s t e p 1 u n t i l n d o i f ( y < y [ i ] ~ y > y [ i + l ] ) h xO -x[i] -(yO -y[i]) X (x[i + 1] -z[i])/(y[i + 1] -y[i]) < 0 t h e n b := --1 b; P O I N T I N POLYGON := ~ b; e n d I~OINT I N P O L Y G O N

Certification of Algorithms 63, 64, 65: Partition, quicksort, find