วกำล, งสอง, บทความน, ไม, การอ, างอ, งจากแหล, งท, มาใดกร, ณาช, วยปร, บปร, งบทความน, โดยเพ, มการอ, างอ, งแหล, งท, มาท, าเช, อถ, เน, อความท, ไม, แหล, งท, มาอาจถ, กค, ดค, านหร, อลบออก, เร, ยนร, าจะนำสารแม, แบบน, ออกได, อย, างไรและเม, อไร, หร, ควอดร, งกฤษ, quadric,. bthkhwamniimmikarxangxingcakaehlngthimaidkrunachwyprbprungbthkhwamni odyephimkarxangxingaehlngthimathinaechuxthux enuxkhwamthiimmiaehlngthimaxacthukkhdkhanhruxlbxxk eriynruwacanasaraemaebbnixxkidxyangiraelaemuxir phiwkalngsxng hrux khwxdrik xngkvs quadric surface inthangkhnitsastr hmaythung phiw hypersurface in D miti sungkahndodykhatxbhruxthangedinrakkhxngsmkarphhunamkalngsxng quadratic polynomial thaeraphicarnaphikd x 0 x 1 x 2 x D displaystyle x 0 x 1 x 2 ldots x D phiwkalngsxngthukkahnddwysmkarphichkhnitdngtxipni i j 0 D Q i j x i x j i 0 D P i x i R 0 displaystyle sum i j 0 D Q i j x i x j sum i 0 D P i x i R 0 ody Q khux emthriks miti D 1 aela P khux ewketxr miti D 1 aela R khux khakhngthi khakhxng Q P aela R 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2 a 2 frac y 2 a 2 frac z 2 a 2 1 thrngpharaoblaechingwngri x 2 a 2 y 2 b 2 z 0 displaystyle frac x 2 a 2 frac y 2 b 2 z 0 thrngpharaoblaechingwngklm x 2 a 2 y 2 a 2 z 0 displaystyle frac x 2 a 2 frac y 2 a 2 z 0 thrngpharaoblaechingihephxrobla x 2 a 2 y 2 b 2 z 0 displaystyle frac x 2 a 2 frac y 2 b 2 z 0 thrngihephxroblachinediyw x 2 a 2 y 2 b 2 z 2 c 2 1 displaystyle frac x 2 a 2 frac y 2 b 2 frac z 2 c 2 1 thrngihephxroblasxngchin x 2 a 2 y 2 b 2 z 2 c 2 1 displaystyle frac x 2 a 2 frac y 2 b 2 frac z 2 c 2 1 thrngkrwy x 2 a 2 y 2 b 2 z 2 c 2 0 displaystyle frac x 2 a 2 frac y 2 b 2 frac z 2 c 2 0 thrngkrabxkechingwngri x 2 a 2 y 2 b 2 1 displaystyle frac x 2 a 2 frac y 2 b 2 1 thrngkrabxkechingwngklm x 2 a 2 y 2 a 2 1 displaystyle frac x 2 a 2 frac y 2 a 2 1 thrngkrabxkechingihephxrobla x 2 a 2 y 2 b 2 1 displaystyle frac x 2 a 2 frac y 2 b 2 1 thrngkrabxkechingihpharaobla x 2 2 y 0 displaystyle x 2 2y 0 phakhkhyaykhxngphiwkalngsxng aekikhnxkehnuxcakrupaebbphiwkalngsxngmatrthanthiidklawthungipaelw yngmikarddaeplngrupaebbkhxngsmkarphunphiwdngklawephuxichinkaraethnrupthrngerkhakhnitthisbsxnkhun echn supepxrkhwxdrik aelaihepxrkhwxdrik supepxrkhwxdrik aekikh smkarbrrthdthankhxngsupepxrkhwxdrikthimicudsunyklangthi 0 0 0 khux x 2 a 2 1 ϵ 1 y 2 b 2 1 ϵ 2 z 2 c 2 1 ϵ 3 1 displaystyle left x 2 over a 2 right 1 over epsilon 1 left y 2 over b 2 right 1 over epsilon 2 left z 2 over c 2 right 1 over epsilon 3 1 hrux inrup x 8 ϕ displaystyle x theta phi a sign cos 8 cos ϕ cos 8 cos ϕ ϵ 1 displaystyle a operatorname sign cos theta cos phi cos theta cos phi epsilon 1 y 8 ϕ displaystyle y theta phi b sign sin 8 cos ϕ sin 8 cos ϕ ϵ 2 displaystyle b operatorname sign sin theta cos phi sin theta cos phi epsilon 2 z 8 ϕ displaystyle z theta phi c sign sin ϕ sin ϕ ϵ 3 displaystyle c operatorname sign sin phi sin phi epsilon 3 ody p 2 ϕ p 2 displaystyle pi over 2 leq phi leq pi over 2 aela p 8 lt p displaystyle pi leq theta lt pi singthisupepxrkhwxdrikaetktangipcakphiwkalngsxngkhux elkhykkalng ϵ 1 ϵ 2 ϵ 3 displaystyle epsilon 1 epsilon 2 epsilon 3 odythikha ϵ 1 displaystyle epsilon 1 aela ϵ 2 displaystyle epsilon 2 nnmiphltxrupranginaenwnxn swn ϵ 3 displaystyle epsilon 3 nnphltxrupranginaenwtng dngaesdnginrupdanlang ϵ 3 4 displaystyle epsilon 3 4 ϵ 3 2 displaystyle epsilon 3 2 ϵ 3 1 displaystyle epsilon 3 1 ϵ 3 0 5 displaystyle epsilon 3 0 5 ϵ 3 0 1 displaystyle epsilon 3 0 1 ϵ 3 0 displaystyle epsilon 3 0 ϵ 1 ϵ 2 0 displaystyle epsilon 1 epsilon 2 0 ϵ 1 ϵ 2 0 5 displaystyle epsilon 1 epsilon 2 0 5 ϵ 1 ϵ 2 1 displaystyle epsilon 1 epsilon 2 1 ϵ 1 ϵ 2 2 displaystyle epsilon 1 epsilon 2 2 ϵ 1 ϵ 2 4 displaystyle epsilon 1 epsilon 2 4 ihepxrkhwxdrik aekikh ihepxrkhwxdrikepnswnthikhyaytxcaksupepxrkhwxdrikihmikhwamsamarthinkarcalxngphiwthisbsxnyingkhun odysupepxrkhwxdriknnepnephiyngkrniphiesskhxngihepxrkhwxdrik ihepxrkhwxdriknnsamarthekhiyninrupsmkarthangkhnitsastrdngtxipni i 1 N l i x y z 1 ϵ i 1 displaystyle sum i 1 N l i x y z 1 over epsilon i 1 ody l i x y z a i x b i y c i z d i displaystyle l i x y z a i x b i y c i z d i aela N 3 displaystyle N geq 3 ϵ 1 ϵ 2 ϵ 3 1 displaystyle epsilon 1 epsilon 2 epsilon 3 1 ϵ 1 ϵ 2 ϵ 3 2 displaystyle epsilon 1 epsilon 2 epsilon 3 2 ϵ 1 ϵ 3 2 ϵ 2 0 2 displaystyle epsilon 1 epsilon 3 2 epsilon 2 0 2 nxkehnuxcakrupaebbkhxngihepxrkhwxdrikkhangtn aelwkyngmikarphthnaephimetimkhwamsbsxnkhxngruprangihepxrkhwxdrik eriykwa khxmophsithihepxrkhwxdrik hrux ihbridihepxrkhwxdrik odyswnthiephimxacxyuinrupphhunamkhxngelkhchikalng i 1 N p l i p o l x y z 1 ϵ i m 1 M w m e j 1 N e l m j e x p x y z 1 ϵ m j 1 displaystyle sum i 1 N p l i pol x y z 1 over epsilon i sum m 1 M w m cdot e sum j 1 N e l mj exp x y z 1 over epsilon mj 1 phcnthiephimekhama miphlinkarprbaetngrupthrngkhxngphiwechphaathi echnichinkarephimhlumhruxrxybum dngaesdnginphaphdanlang displaystyle Rightarrow ihepxrkhwxdrik phaphkhxmophsithihepxrkhwxdrik odykarephimphcnkhxngelkhykkalng 1 phcn 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