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1、O?A?:? Huang Kun ?fi ?ACM/ICPC? March 14, 2009 ? 1 ? 2 :? 3 ?N 4 ? 5 ;fl K:% 6 ;fl K:? 7 ;fl K:/? O?A?K8?A?fl K double 2:$? ;?.?. ?#IOcmathfabs,cos?|ga.=? 2:?fl K eps = 1e 8 a = b fabs(a b) = b a b eps a = b a b a b + eps a b a /K n?/K AB + BC = AC AB AC = CB ? :fl K ? AB?:C AB?v AC = CB C?I :O?“ OC
2、 OA = ( OB OC) OC = OA + OB 1 + ?:? :(S) a b = | a|b|cos |z b3 a?K ?(?) a b = | a|b|sin |z b3 a?K 5 1 :$?(J?,?$?(J? 2 :?A?,?3? ?K 3 ?A?,?n?/?2?k? ?:? :?A? ?:?A ?A ?mK ?0?du?,dd?n:? AB AC 0 C3 AB? AB AC 0 C3 ABm? ?:?A O?K acos = a b |b| O?Y? = arccos a b | a|b| ?:?A :3?K ?:?:P,?AB ?:P3?AB?KP0 ?: 1:O?
3、 AP3 AB?K? l = AP AB | AB| 2:O? AB? e = AB | AB| 3: OP0= OA + l e ?N O?n?/ S4ABC= ? ? ? ? ? AB AC 2 ? ? ? ? ? = 1 2 111 xAxBxC yAyByC O?/ =z?O?n?/P0,P1,.,Pn1? S = ? ? ? ? ? n1 X i=0 OPi OPi+1 ? ? ? ? ? (P0= Pn) ?N ?“ S4ABC= p p(p a)(p b)(p c),p = a + b + c 2 y S4ABC= 1 2 xB xAyB yA xC xAyC yA S4ABC=
4、 1 2 xB xAxC xA yB yAyC yA S2 4ABC = 1 4 AB AB AB AC AC AB AC AC AC AB = | AC| AB|cosBAC = | AC|2+ | AB|2 | BC|2 2 ?N oNN VABCD= ( AB AC) AD 6 = 1 3! 1111 xAxBxCxD yAyByCyD zAzBzCzD ?N n?“ VABCD= 1 3! xB xAxC xAxD xA yB yAyC yAyD yA zB zAzC zAzD zA VABCD= 1 3! xB xAyB yAzB zA xC xAyC yAzC zA xD xAyD
5、 yAzD zA VABCD=? PKU 2208 ? ? 1 ?5?:?:?: ?:?u,?(“ ?) 2 ?5?:?:?: ? ?: S4ABC S4ABD = | CP| | PD| ddO?P CB?,:“?P ? :?/?X X?:3/S? 4? ? ? :?/?X:? ?:P?/S?X,? lP?,?O? /S?:? ?:?,:3/S? ?:?,:3/? ? :?/?X:? A?/ )?:? ? :?/?X ?/k?A5,? ?:?:?3? ?. dd?I?:3k ?. % n?/?% 4ABC?%P(x,y) x = xA+ xB+ xC 3 y = yA+ yB+ yC 3 ?
6、 OP = OA + OB + OC 3 % /?% /?%Ln?5. r?/y?k?n? /,P1i?n?/?Si,%?Pi,K OP = k1 P i=0 Si OPi k1 P i=0 Si ? ? ?n?:?8? S = P0,P1,.,Pn1 ? ?Sk:?/?”: S? S0=,m n ? ?K STEP 1: ? ?K STEP 2: ? ?K STEP 3: ? ?K STEP 4: ? ?K STEP 5: ? ?K STEP 6: ? ?K,?mE,?O(n2) ?:8S = P0,P1,.,Pn1 ?:?S0=,m n ?: 1:?S0= 2:?Pk0,?vxk0 xi,
7、rPk0?S0,m = 1 3:3k:?Pkm,km6= k0,?vPkm?Pkm1?4? 4:Pkm?S0,m = m + 1 5:if km= k0then 6:?(? 7:else 8:=13? 9:end if ? Graham? STEP 1: ? Graham? STEP 2: ? Graham? STEP 3: ? Graham? STEP 4: ? Graham? STEP 5: ? Graham? STEP 6: ? Graham? STEP 7: ? Graham? STEP 8: ? Graham? STEP 9: ? Graham? ?:8S ?:?S0 1:?A?,?
8、”?,k = 0 2:?P?vxP xi,?S 3:?P?:,U?P?4?S?#?S ?Q1,.,Qn1 4:rQ1?A,k = 2 ? Graham?(Y) ?:8S ?:?S0 1:?P?:,U?P?4?S?#?S ?Q1,.,Qn1 2:rQ1?A,k = 2 3:for i = 1 to n 1 do 4:while Qi3 Ak2Ak1?m?andk 2 do 5:Ak1?,k = k 1 6:end while 7:Qi?,k = k + 1 8:end for 9:A?S?S0?”:S? /? ? ?/S?:?3/S? :?8? ?/? /? /?n)?/S?: /? ? /? ? /? ? /? ? ?q? ?55yfl K?) ?S?3?X? A PKU?SK 1039 1106 1113 1228 1265 1329 1385 1494 1584 1654 1673 1696 1873 1981 2007 2187 2208 2242 2318 3129 3334 3348 3407 The End Thank you!
