大學本科數(shù)學專業(yè)行列式在解析幾何中的應用畢業(yè)論文

優(yōu)美,而且便于記憶.本文較系統(tǒng)的總結了行列式在解析幾何中一些重要方程中的應用,并且用行列式的形式給出了一些多邊形面積或多面體體積的計算公式.

1=0-1)

平面上三點P

(x

)(i=1,2,3)共線的充要條件是10

-2)

1=0-3)10

即為(1-2)很自然的我們會問的充要條件是什么呢？我們做一下推廣空間三點P

(x

)(i=1,2,3)共線充要條件是110 1y 1y 10

PP12

PP13

PP12

PP13

1iy 1jy 1k

PP13

=0，由此得證.

平面上不共線的三點P

(x

)(i=1,2,3)所確定的圓的方程可用下面四列式表示x2y2 x2y2 x2y2 x2y2

-4)求的圓的方程為x2y2AxByC0意得x2y2AxByC0 x2y2AxByC0 x2y2AxByC0 把以上四式看成是關于1,A,B,C的齊次方程組,這個方程組顯然有非零解.根110非零解得充要條件得x2y2

x2y2 x2y2 x2y2

平面上四點P

(x

)(i=1,2,3,4)共圓的必要條件是x2y2 x2y2 0x2y2 x2y2

-5)但需要注意的是(1-5)式不是四點共圓的充分條件.例如,當四點共線時(1-5)仍然不共線四點共圓的一個充分必要條件我們已經知道,空間不共面四點可以確定一個球面方程,下面我們將給出與(1-5)相仿形式的行列式表示

空間中不共面的四點P

)(i=1,2,3,4)所確定的球面的方程可以面的五階行列式表示x2y2z2 x2y2z2 x2y2z2 x2y2z2 x2y2z2 其證明的過程同定理1.2.

10

-6)

空間中五點P

(x

)(i=1,2,3,4,5)共球面的條件是

10

-7)其中,x2y2z2,(i=1,2,3,4,5)i i i 11:xx同樣應該注意,條件(1-7)不是五點共球面的充分條件.例如,當五點共線(或者共面)時,條件(1-7)也是成立的

空間中不共線的三點P

(x

)(i=1,2,3)所確定的平面方程為xx xx yy yy zz zz

-8)

-9)(1-8)或者(1-9)稱為平面的三點式方程事實上,將(1-8)或者(1-9)中的行列式按照第一行展開可以知道,(1-8)或者(1-9)

(x

)(i=1,2,3)均滿足(1-8)或者(1-9)式,從

空間中任意四點P

(x

,z)(i=1,2,3,4)共面的充分必要條件是 0

-10)定理1.5空間兩條異面直線的方程如下

zz1zz2所確定的公垂線方程為 XXYxx xx2X

yyyy

zzzz

-11)

1212,Y 212

,Z

1212事實上,(1-11)中第一個式子即為L1與異面直線所確定的平面方程,(1-11)中第二,.我們將L1中的x,y,z代入(1-11)中的第二個行列式;L2中的x,y,z代入第一個平面上不共線的五點P

(x

)(i=1,2,3,4,5)所確定的二次曲線的方程可以階行列式表示 y2

x1yx22x44

y2y2y2y2y2

-12)將(1-12)式中的行列式按照第一行展開,即知(1-12)式表示的二次曲線,又由行列式的性質知道,此二次曲線顯然過P

(x

)(i=1,2,3,4,5)諸點.aab=(a1ia,,a==(a1b==(a2ba3b2)i(a3biiajja[[o;e1,e定理2.1設a,b在右手坐標系中的坐標分別是(a1,a2,a3),(b1,b2,b則ab的坐標是(

aa

bb

a

bb

aa

bb

-1)2 2a2b1)ij(a3ba1b3)ki(a2ba3b2)jk3 1a1b3)j(a2a2b1)k

aa

bb

a

bb

a

bb

從而結論成立因此,為了便于記憶們可以寫作aa定理2.2取定一個仿射標架

],設向量a,b,c的坐標分別是(a1,a2,a3),(b1,b2,b3),(c1,c2,c3),則有(e

aaa

-2)33b2)e==[(a1b(ab)c=[(a

(a

a

(a

a

e3](c1e

a

(a

a

(a

a=aa從而以上結論成立

(

可知a(ab)caa推論設向量a,b,c的坐標分別是(a1,a2,a3),(b1,b2,b3),(c1,c2,c3),則a,b,c的充分必要條件是aaa

-3)事實上,由(ab)c的幾何意義可以知道a,b,c共面的充分必要條件是(ab)c0從而可以得到結論由推論(2-3)我們可以得到四點共面的另外一個證明方法必要條件是

從而,分必要條件為a2aa1aaa==a

從而得到以上結論定理2.3兩向量a,b在空間仿射標架[o;e1,e2,e3]中的坐標分別為(a1,a2,a3)和(b1,b2,b3),則a,b共線的充要條件是aa

bb

a

bb

a

bb

-4)設a,b共線.若a=0,則結論顯然成立.若設a0,于是有實數(shù)k使得bka,所以b

ka=1,2,3),

aa

bb

a

kaka

同理也可以證明其余兩個行列式也等于零知條件知

a a abab12 21

a

a

a因此a,b共線.iizjjx

對任意四個向量a,b,c,d有(ab)(cd)accdbcbd事實上,量混合積的有關性質易證acadbcbd

對不共線的三點P

(x

3為鄰邊的平四的面積是S21

222

y1,

1,事實上,因為PP12

PP12

PP13xx xx yy2 yy3 yy yy zz2 zz3

2 zz3

23

12

23

11

23

11

x11x22x如果已知平面上如果已知平面上n邊形P1P2……Py 1iy 1jy 1k所以,S等于PP12

PP13

的模

S2221 2

以P

(x

)(i=1,2,3)為頂點的三角形的面積為S122

222

-1)

以P

(x

)(i=1,2,3)為頂點的三角形的面積為

-2)需要注意的是,此處P

P

P列式前邊應該加負號,

各頂點的坐標P

(x

)(i=1,2,……n)則此多邊

(假定P

P的回路積是1xS 12x

2+……+

n1n

n1n

n1

n1

-3)三角形的面積公式以寫成2x

用數(shù)學歸納法不難證明(3-3)注2公式(3-3)可以表示成復數(shù)形式1S 2

2

z2z3

……z

其中 zxiy (i21,k1,2,3,……n) ),(),(x2,y2)(,假設ABC三邊所在的直線的方程分別是=1,2,3)則有下面的定理成立定理3.3令三邊所在直線方程的系數(shù)和常數(shù)項所構成的行列式為A31,32,33分別是元素C1,C2,C3在中的代數(shù)余子式ABC2S2

-4)

12,22),(x3,y3)(13,23) 其余可以類推將所得的坐標代入(3-2)S 2 332131

(正負號的選擇使S0)

2 ad 2 33C

(B

22 注上面的推導過程中應用了公式adjAA

：：AixBiyCizD0（i=1,2,3,4）推論三條直線l 0 (i=1,2,3)

-5)請注意,上述條件(3-5)不是三線共點的充分條件.例如,當三條直線中至少有兩條平行時,上述公式也成立

已知空間中不共面的四點P

(x

)(i=1,2,3,4),V2

-6)此結論的證明可以根據三個向量的混合積推論1已知空間四面體的四個頂點的坐標是P

(x

)(i=1,2,3,4),則此空間四的體積為

-7)

空間四點P

(x

)(i=1,2,3,4),共面的充分必要條件是

-8)此結論的證明是顯然的,知空間四面體四個側面的方程為A：：AixBiyCizD0(i=1,2,3,4)四面體的體積為

361 2

4

-9)其中1,2,3,4分別是D1,D2,D3,D4在中的代數(shù)余子式.此定理為定理(3-4)在空間中的推廣,其證明亦與定理(3-4)的證明類似,此處不再推論四個平面i個平面共點的必要條件是

-10)注1(3-11)只是必要條件,不是充分條件.如(i=1,2,3,4)中至少有兩個平面重合或者三個平面平行時,(3.5-2)也成立注2如果1和2的交線是l1,3和4的交線是l2,則(3-11)可以看作l1與l2共面的充分必要條件注3若已知四面體的六棱長分別是a,b,c,x,y,z,則此四面體體積的平方是V2

a2b2z2c2

a2b2c23.6用行列式表示平面上三角形外接圓的面積平面上已知ABC的三個頂點分別為(x

)(i=1,2,3)式知1101(x2y2)2x3yx2y2 x2y2 x2y2 x2y2

x2y2 x2y2 x2y2

1,

x2y2 x2y2 x2y2

1,

x2y2 x2y2 x2y2

三角形的外接圓方程為40x2y2

x3y40

(x2)2(y23)22

242

242

4

22 42于是ABC的外接圓的面積是S

4

22 421

程分別為：

zz1,zz2

們之間的距離為ZXzx

0 0 00

yy

zz

d

通過這一段時間的努力,幾經周折,本文終于完成，基本達到了預期目的,即系統(tǒng)的應用；并且用行列式的形式給出了一些多邊形或多面體的面積或體積的計算公式我想.制,在論文中肯定存在一些不妥的地方敬請各位老師指教[14]許延功.行列式在解析幾何中的表示昌濰師專學報,1992,11(2):55-61.我要感謝我的指導老師.老師繼承了優(yōu)秀的學術傳統(tǒng),對學術極為尊重,治學非常嚴謹.高山仰止,景行行止,在踏入工作崗位之后,我一定會尊重自己的職業(yè),并且也一定會嚴格要求自己在論文撰寫過程中老師提出了諸多建議為我的論文付出了大量的心血,對此我只有用好好工作來報答了老師對我大學畢業(yè)時期所進行的小研究進行了諸多指導,并鼓勵我投身學術.我的任課老師和同學對我的學習和生活都給予了諸多幫助在此一并致謝FromtheworldofradiointheworldtoasinglechipModerncomputertechnology,industrialrevolution,theworldeconomyfromthecapitalintotheeconomytoknowledgeeconomy.Fieldintheelectronicworld,fromthe20thcenturyintotheeraofradiotocomputertechnologyinthe21stcenturyasthecenteroftheintelligentmoderneraofelectronicsystems.Thebasiccoreofmodernelectronicsystemsareembeddedcomputersystems(referredtoasembeddedsystems),whilethemicrocontrolleristhemosttypicalandmostextensiveandmostpopularembeddedFirst,radiohascreatedgenerationsofexcellenceintheworldFiftiesandsixtiesinthe20thcentury,themostrepresentativeoftheadvancedelectronictechnologyiswirelesstechnology,includingradiobroadcasting,radio,wirelesscommunications(telegraph),AmateurRadio,radiopositioning,navigationandothertelemetry,remotecontrol,remotetechnology.Earlythattheseelectronictechnologyledmanyyoungpeopleintothewonderfuldigitalworld,radioshowwasawonderfullife,theprospectsforscienceandtechnology.Electronicsbegantoformanewdiscipline.Radioelectronics,wirelesscommunicationsbegane-worldjourney.Radiotechnologynotonlyasarepresentativeofadvancedscienceandtechnologyatthattime,butalsofrompopulartoprofessionalfieldsofscience,attractingtheyoungpeopleandenablethemtofindalotoffun.Orefromthebedsidetothesuperheterodyneradioradio;reportissuedfromtheradioamateurradiostations;fromthetelephone,electricbelltotheradiocontrolmodel.Becamepopularyouthradiotechnology,scienceandtechnologyeducationisthemostpopularandmostextensivecontent.Sofar,manyoftheoldergenerationofengineers,experts,Professoroftheyearareradioenthusiasts.Funradiotechnology,radiotechnology,comprehensivetraining,frombasicprinciplesofelectronics,electroniccomponentstotheradio-basedremotecontrol,telemetry,remoteelectronicsystems,hastrainedseveralgenerationsoftechnologicalexcellence.Second,fromthepopularityoftheradioeratoeraofelectronictechnologyTheearlyradiotechnologytopromotethedevelopmentofelectronictechnology,mostnotablyelectronicvacuumtubetechnologytosemiconductorelectronictechnology.Semiconductortechnologytorealizetheactivedeviceminiaturizationandlowcost,somorepopularwithradiotechnologyandinnovation,andtogreatlybroadenthenumberofnon-radio-controlareas.Thedevelopmentofsemiconductortechnologyleadtotheproductionofintegratedcircuit,formingthemodernelectronictechnologyleapfromdiscreteelectronicsintotheeraoferaofintegratedcircuits.Electronicdesignengineersnolongerusethediscreteelectroniccomponentsdesignedcircuitmodules,anddirectselectionofintegratedcircuitcomponentsconstituteasinglesystem.Theyfreedthedesignofthecircuitunitdedicatedtosystemdesign,greatlyliberatingtheproductiveforcesofscienceandtechnology,promotethewiderspreadofelectronicsystems.Semiconductorintegratedcircuitsinthebasicdigitallogiccircuitsfirstbreakthrough.Alargenumberofdigitallogiccircuits,suchasgates,counters,timers,shiftregisters,andanalogswitches,comparators,etc.,fortheelectronicdigitalcontrolprovidesexcellentconditionsforthetraditionalmechanicalcontroltoelectroniccontrol.Powerelectronicdevicesandsensortechnologytomaketheoriginaltotheradioasthecenterofelectronictechnologyturnedtomechanicalengineeringinthefieldofdigitalcontrolsystems,testinginthefieldofinformationcollection,movementofelectricalmechanicalservodrivecontrolobject.Semiconductorandintegratedcircuittechnologywillbringusauniversalageofelectronictechnology,wirelesstechnologyasthefieldofelectronictechnologyapartof.70yearsintothe20thcentury,largescaleintegratedcircuitappearedtopromotetheconventionalelectroniccircuitunit-specificelectronicsystemsdevelopment.Manyelectronicsystemsunitintoadedicatedintegrateddevicessuchasradios,electronicclocks,calculators,electronicengineersintheseareasfromthecircuit,thesystemdesignedtodebugintothedeviceselection,peripheraldeviceadapterwork.Electronictechnology,andelectronicproductsenriched,electronicengineerstoreducethedifficulty,butatthesametime,radiotechnology,electronictechnologyhasweakenedthecharm.Thedevelopmentofsemiconductorintegratedcircuitsclassicalelectronicsystemsarematuring,remaininthelargescaleintegratedcircuitotherthantheshrinkingofelectronictechnology,electronictechnologyisnottheolddaysofradiofuntimesandThird,fromtheclassiceraofelectronictechnologytomodernelectronictechnologyofthetimes80yearsintothe20thcentury,thecenturyofeconomicchangeisthemostimportantrevolutioninthecomputer.Thecomputerrevolutioninthemostimportantsignisthebirthofthecomputerembeddedapplications.Moderncomputernumericalrequirementsshouldbeborn.Alongperiodoftime,istodevelopthemassivecomputernumericalduty.Butthecomputershowsthelogicoperation,processing,control,attractingexpertsinthefieldofelectroniccontrol,theywantdevelopmenttomeetthecontrolobjectrequirementsofembeddedapplications,computersystems.Ifyoumeetthemassivedata-processingcomputersystemknownasgeneral-purposecomputersystem,thenthesystemcanbetheembeddedobject(suchasships,aircraft,motorcycles,etc.)inacomputersystemcalledtheembeddedcomputer.Clearly,boththedirectionoftechnologydevelopmentaredifferent.Theformerrequiresmassivedatastorage,handling,processingandanalysisofhigh-speeddatatransmission;whilethelatterrequiresreliableoperationinthetargetenvironment,theexternalphysicalparametersonhigh-speedacquisition,analysisandprocessinglogicandtherapidcontrolofexternalobjects.Itwilladdanearlygeneral-purposecomputerdataacquisitionunit,theoutputdrivercircuitreluctancetoformaheattreatmentfurnacetemperaturecontrolsystem.Thisgeneral-purposecomputersystemisnotpossibleformostoftheelectronicsystemused,andtomakegeneral-purposecomputersystemmeetstherequirementsofembeddedapplications,willinevitablyaffectthedevelopmentofhigh-speednumericprocessing.Inordertosolvethecontradictionbetweenthedevelopmentofcomputertechnology,inthe20thcentury70s,semiconductorexpertsanotherway,infullaccordancewiththeelectronicsystemembeddedcomputerapplicationrequirements,amicro-computer'sbasicsystemonachip,theformationoftheearlySCM(SingleChipMicrocomputer).Aftertheadventofsinglechipinthecomputerindustrybegantoappearinthegeneral-purposecomputersystemsandembeddedsystemsthetwobranches.Sincethen,boththeembeddedsystem,orgeneral-purposecomputersystemshavebeendevelopedrapidly.Althoughtheearlygeneral-purposecomputerconvertedtheembeddedcomputersystems,andrealembeddedsystembeganintheemergenceofSCM.Becausethemicrocontrollerisdesignedspecificallyforembeddedapplications,theMCUcanonlyachieveembeddedapplications.MCUembeddedapplicationsthatbestmeetenvironmentalrequirements,forexample,chip-levelphysicalspace,large-scaleintegratedcircuitslow-cost,goodperipheralinterfacebusandoutstandingcontrolofinstruction.Acomputersystemmicrocontrollercore,embeddedelectronicsystems,intelligentelectronicsystemsforthefoundation.Therefore,thecurrentsinglechipelectronicsysteminwidespreaduseofelectronicsystemstoenablerapidtransitiontotheclassicalmodernintelligentelectronicsystems.4,singlechiptocreatethemoderneraofelectronicsystemsAmicrocontrollerandembeddedsystemsEmbeddedcomputersystemsfromembeddedapplications,embeddedsystemsforearlygeneral-purposecomputeradaptedtotheobjectsystemembeddedinavarietyofelectronicsystems,suchastheship'sautopilot,enginemonitoringsystems.Embeddedsystemisprimarilyacomputersystem,followedbyitbeingembeddedintotheobjectsystem,objectsintheobjectsystemtoachieverequireddatacollection,processing,statusdisplay,theoutputcontrolfunctions,asembeddedintheobjectsystem,embeddedsystemcomputerdoesnothaveanindependentformandfunctionofthecomputer.SCMisentirelyinaccordancewiththerequirementsofembeddedsystemdesign,soSCMisthemosttypicalembeddedsystems.SCMistheearlyapplicationoftechnicalrequirementsinaccordancewiththedesignofembeddedcomputerchipintegration,hencethenamesinglechip.Subsequently,theMCUembeddedapplicationstomeetthegrowingdemandsofitscontrolfunctionsandperipheralinterfacefunctions,inparticular,highlightthecontrolfunction,sohasinternationalnamethesinglechipmicrocontroller(MCU,MicrocontrollerUnit).2MCUmodernelectronicsystemsconsistingofelectronicsystemswillbecomemainstreamMCUisadevice-levelcomputersystems,itcanbeembeddedintoanyobjectsystemtoachieveintelligentcontrol.Smalltomicro-machinery,suchaswatches,hearingaids.Low-costintegrateddevice-level,low-toafewdollars,tendollars,enoughtospreadtomanycivilianSCMappliances,electronictoystogo.SCMconstitutesamodernelectronicsystemshasin-depthtothehouseholds,arechangingourlives,suchashomeaudio,televisions,washingmachines,microwaveovens,telephones,securitysystems,andairconditioners.SCMinnovationtheoriginalelectronicsystems,suchasmicrowaveovensuseSCM,itcaneasilysettheclock,theprogrammemory,powercontrol;airconditionerafteruseofSCMisnotonlyconvenientforremoteparametersetting,runningautomaticallytransform,frequencycontrolcanbeachieved.Atpresent,manyhouseholdappliancessuchasVCD,DVDonlysinglechiptoachieveitsfunctionmayoccurbefore.3EmbeddedSystemsledtheentireelectronicsindustryCurrentelectroniccomponentsindustry,inadditiontomicroprocessors,embeddedsystemdevices,themostmodernelectronicsystemsaroundthesupportingcomponentsindustries,suchaskeysusedtomeetthehuman-computerinteraction,LED/LCDdisplaydrivers,LED/LCDdisplayunits,voiceintegrateddevice,etc.,tomeettherequirementsofdataacquisitionchanneldigitalsensor,ADC,dataacquisitionmodule,signalconditioningmodulestomeettheservodrivecontrolintheDAC,solidstaterelays,steppermotorcontroller,frequencycontrolunit,etc.,tomeetthecommunicationrequirementsvariousbusdriver,levelconverters.Electroniccomponentsintheembeddedsystemsworld,drivenbyembeddedapplicationsalongfullymeetrequirementsofmodernelectronicsystemsdevelopment.Thismakestheoriginalclassicworldofincreasinglysmallelectronicsystems.Practitionersinthevariouselectronicsystemstomodernelectronicsystemsasearlyaspossibletostay.5,SCMwillcreateanewgenerationofelectroniceliteIfthe50's,radiohascreatedseveralgenerationsoftheworldelite,thentoday'sSCMwillcreateanewgenerationofe-worldelite.AsinglechipwithyoutotheintelligentelectronicsIfweasadeadclassicelectronicsystemelectronicsystem,thentheintelligentmodernelectronicsystemsisa"life"oftheelectronicsystem.ApplicationSystemofhardware,electronicsystems,"body",microcontrollerapplications,theapplicationgivesit"life."Forexample,inthedesignofintelligentmachinesmonitordisplay,itcanbootthesystemself-testresultsshow,nottoentertheworkshowsavarietyofstand-bystate,equipmentrun-timedisplayrunningprocesses,workcanbedisplayedaftertheendofthecurrentresults,selfresults,rawdata,reportsandothervariousprocessing.Unattended,itcanrunautomaticallygivenavarietyoffunctions.Intelligentelectronicsystemsfortheendlessrealm,oftenwithoutadditionalhardwareresourcescanachieveallkindsofrenovatedfunction.Itisalsopresentinmanyhouseholdappliancesfeaturealargenumberofadditionalfactors.2singlechipcomputerwithyoutotheindustrialareaThe21stcenturyisthecenturyofhumanityintothecomputerage,manypeoplearenotusedinthemanufactureofcomputeristhecomputer.Peopleusingthecomputer,onlythepeopleengagedinembeddedsystemapplicationsreallyintotheinternalcomputersystemhardwareandsoftwaresystems,canwetrulyunderstandthenatureofthecomputer'sintelligenceandgrasptheknowledgeofintelligentdesign.MCUapplicationsstartingfromthelearningtechnologyapplicationsintoday'scomputersoftwaretraining,hardwareandtechnicalpersonnelofoneofthebestroads.3SCMbringyouintothemostattractiveinthedigitalworldCharmingsinglechiptoenableyoutoexperiencethetruemeaningofthecomputer,youcandesignintelligentmicrocontrollerhands-ontoys,differentapplicationscanbedesignedtoachievedifferentfunctions.Bothsoftwaredesignandhardwaremakingthere,bothmentalandphysical,butalsohands.Primarylevelcandevelopintelligenttoys,withmacroprogramming.Intermediatelevelscandevelopsomeintelligentcontroller,suchascomputermouse,smartcars,allkindsofremotecontrolmodel.Highlevelscanbedevelopedrobots,suchasrobotsoccer,thedevelopmentofindustrialcontrolunits,networkcommunications,andhigh-levellanguagewithassemblylanguageordesignapplication.Microcontrollerandembeddedsystemsaroundtheformationofthefutureoftheelectronicsindustry,willprovideavastworldofelectronicfans,anevenbroaderthanthecurrentwirelessworld,richer,moredurable,moreattractiveinthedigitalworld.Plungeintothemicrocontrollerintheworldto,willbenefityourlife.Currently,therearefoursinglechipattacktechnique,namely:(1)softwareattackThetechnologyiscommonlyusedprocessorcommunicationinterfaceanduseprotocol,encryptionalgorithmorthealgorithmofsecurityvulnerabilitiestoattack.ThesuccessofsoftwareattackisatypicalexampleoftheearlyATMELAT89CMCUattacks.Attackersinglechiperaseoperationofthetimingdesignflaw,erasetheencryptionusedbyourselveslockedinplace,thenextstopon-chipprogrammemorydataeraseoperation,thusbringingintotoocloseasinglechipSCMnotencrypted,andthenusetheprogrammer toreadoutchipprogram.(2)electronicdetectionofattacksThetechnologyisusuallyahightimeresolutiontomonitortheprocessorduringnormaloperationofallpowerandinterfacesimulationfeatures,andby electromagneticradiationcharacteristicsofittoattack.BecauseSCMisanactiveelectronicdevice,whenitexecutesadifferentcommand,thecorrespondingchangesinthepowerconsumptionaccordingly.Thisthroughtheuseofspecialelectronicmeasuringinstrumentsandmathematicalstatisticalanalysisanddetectionofthesechanges,youcanaccesskeyinformationspecificmicrocontroller.(3)faultgenerationtechnologyAbnormalworkingconditionsofthetechnologyusedtomaketheprocessorerrors,andprovideadditionalaccesstoattack.Producethemostwidelyusedmeansofattack,includingthefaultoftheimpactandtheclockvoltageshock.Lowvoltageandhighvoltageprotectioncircuitattackcanbeusedtoprohibittheworkofprocessorexecutionerrorsorenforcementaction.Clocktransitionmayresetthetransientprotectioncircuitwillnotdamagetheprotectedinformation.Powerandclocktransientstransitioneffectsincertainsingle-processorinstructiondecodingandexecution.(4)probeThistechnologyisdirectlyexposedtochipconnection,andthenobserve,manipulate,interferewithsinglechiptoachievetheattackpurpose.Forconvenience,thesefourpeoplewillattacktechniquesaredividedintotwocategoriesistheintrusiontypeattack(physicalattack),suchattackrequiresdestructionofpackage,thenusesemiconductortestequipment,microscopesandmicro-positioningdevice,inaspeciallaboratoryspendhoursorevenweekstocomplete.Allofthemicro-probetechniquesareinvasivetypeattack.Theotherthreemethodsarenon-invasivetypeattack,attacktheMCUwillnotbephysicaldamage.Insomecases,non-invasive-typeattacksareparticularlydangerous,butbecauseofnon-invasivetypeattackscanusuallybemadeandthenecessaryequipmenttoupgrade,soitischeap.Mostnon-invasivetypeattackrequirestheattackerhaveagoodknowledgeofprocessorsandsoftwareknowledge.Incontrast,theinvasivetypeofprobedonotneedtoomuchoftheinitialattackofknowledge,andusuallyasetofsimilartechnologyavailabletodealwithawiderangeofproductsMCUgeneralprocessofinvasion-typeattackInvasivetypeofattackisthrownoffitsfirstchippackage.Therearetwowaystoachievethisgoal:thefirstoneiscompletelydissolvedoutchippackage,exposedmetalconnections.Thesecondisonlyremovedtothetopoftheplasticpackagesiliconcore.Thefirstmethodistheneedtobindtothetestfixtureonthechip,usingbindTaiwantooperate.Thesecondmethodrequirestheattackerinadditiontoacertaindegreeofknowledgeandnecessaryskills,butalsothewisdomandpatience,butoperaterelativelyeasy.Abovetheplasticchipscanbeopenedwithaknife,epoxyaroundthechipcanbeerodedbyconcentratednitricacid.Hotconcentratednitricacidwilldissolveoutwithoutaffectingthechip,chippackagingandconnection.Thisprocessusuallyverydryconditions,becausethepresenceofwatermayerodethealuminumwireconnectionshavebeenexposed.Then,inultrasoniccleaningofthepoolfirstchipwithacetonetoremoveresidualnitricacid,thenwashedwithwatertoremovesaltanddried.Noultrasoundpool,aregenerallyskipthisstep.Thiscase,thechipsurface,abitdirty,butdonotaffecttheoperationofUVeffectsonthechip.ThefinalstepistofindthelocationoftheprotectionfuseandfuseprotectionunderexposuretoUVlight.Generaluseatleasta100timesmagnificationmicroscope,fromtheprogrammingvoltageinputpinoftheconnectiontrackingin,tofindprotectionfuse.Ifthereisnomicroscope,theuseofdifferentpartsofthechipisexposedtoultravioletlightandobservetheresultsunderthesimplesearchmode.Operationappliedopaquepapercovertoprotecttheprogrammemorychipsarenoterasedbyultravioletlight.WillprotectthefuseexposedunderUVlight5to10minutestodestroytheprotectionbitoftheprotectiveeffect,useasimpleprogrammercandirectlyreadthecontentsofprogrammemory.TheuseoftheprotectivelayertoprotecttheMCUEEPROMcell,usingultravioletlightresetprotectioncircuitisnotfeasible.ForthistypeofMCU,thegeneraluseofmicro-probetechnologytoreadthememorycontents.Inthechippackageisopened,thechipplacedunderthemicroscopecaneasilyfindfromthememorycircuitconnectedtootherpartsofthedatabus.Forsomereason,thechiplock-bitprogrammingmodeisnotlockedinthememoryofthevisit.Advantageofthisflawonthedatalinestoprobetheabovedatacanbereadallyouwant.Inprogrammingmode,restarttheprocessofreadingandconnectprobetotheotherdatacanbereadonlineprogramanddatamemory,alloftheinformation.Thereisalsoapossiblemeansofattackistheuseofmicroscopyandlasercuttingmachinesandotherequipmenttofindthefuseprotectiontothispartofthecircuittracingandlinkingallthesignallines.Becauseofthedesigndefects,solongascutofffromothercircuitprotectionfusetoaonesignalline,youcanbantheentireprotection.Forsomereason,thisthreadisveryfarfromtheotherline,sotheuseoflasercuttingmachinecancutthewirewithoutaffectingtheadjacentline.Inthisway,usingasimpleprogrammercandirectlyreadthecontentsofprogrammemory.Althoughthemostcommonsinglechipmicrocontrollerhasfuseblowninsidethecodeprotectionfeatures,butbecauseofgenerallow-endMCUisnotpositioningtheproduct