一 : T P H

那是去年的5月15号，就那样出现在你面前了 当我意识到你真的是女孩，我已经一发不可收拾了，我怎么就这样给弯了呢 煎熬 绝对是煎熬整整三天晚上的睡不着 脑子里面全是你的画面我才知道，我原来真的是LES
T
女同志族群中，“T”指特质倾向于阳刚，或外貌喜欢作男性化/中性化装扮者（“T”可能来自英文的Tomboy）； “婆”指装扮、行为、气质阴柔的女同志（又称为P）。(www.61k.com］台湾女同志文化中的T／婆之分，相当于西方的Butch／Femme。“婆”这个词在最早是相对于T 而来，指“T的老婆”。但近年来，婆的主体性已经逐渐浮现，用以泛指气质较阴柔者；Uncle指年长的T。这个词起源于60年代，是女同志用来对长辈的尊称；“拉子”、“拉拉”或“蕾丝边”指女同性恋者（由Lesbian谐音而来），最早出现在台湾作家邱妙津的女同志小说《鳄鱼手记》里，于1990年代开始流行于台湾。
代表杨杨

P
在同性恋中，P也就是婆
女同性恋中扮演老婆的一方，也就是女性化的一方，与T相对应。
女同志族群中，“T”指特质倾向于阳刚，或外貌喜欢作男性化/中性化装扮者（“T”可能来自英文的Tomboy）；“P”指装扮、行为、气质阴柔的女同志。台湾女同志文化中的T／P之分，相当于西方的Butch／Femme。“婆”这个词在最早是相对于T 而来，指“T的老婆”。但近年来，婆的主体性已经逐渐浮现，用以泛指气质较阴柔者。

H
H是指“不分”，即不分“T、P”，严格的说H才是真正的女同性恋者，因为从外表和打扮是看不出谁是T谁是P的，这些只有在双方交往互动中才能有大致的了解。
P是指女同性恋者中性格或能力略微弱势，倾向于被照顾的一方。
T是指女同性恋者中性格或能力略微强势，倾向于照顾P的一方。

亲们是T?P?H?

怪不得爷老说我有当T的潜质，原来还是H```⊙﹏⊙b汗

扩展：tpg / tph是什么意思 / tph总石油烃

二 : = [p] ′ (t) [p] ′.

p.t = [p] ′ (t) [p] ′.

p.t = [p] ′ (t) [p] ′.

p.t = [p] ′ (t) [p] ′.

p.t = [p] ′ (t) [p] ′.

p.t = [p] ′ (t) [p] ′.

p.t = [p] ′ (t) [p] ′.

p.t = [p] ′ (t) [p] ′.

p.t = [p] ′ (t) [p] ′.

p.t = [p] ′ (t) [p] ′.

三 : = [p] ′ (t) [p] ′.

RegularCurvesinThreeDimensionalSpace

MarcH.MehlmanDepartmentofMathematicsUniversityofPittsburgh,Johnstown

27February2000

1TangentialandNormalDirectionstoaCurve

LetCbeadi?erentiablecurveinR3,i.e.a“route”andanorientation.Letp??(t)=x(t)???+y(t)???+z(t)??kbeaparameterizationofC.Thatisp??(·)travelsthesamepathasCandinthe??defcorrectdirectionandthuscanbethoughtofasthepositionvector.Similarly,??v(t)=[??p](t)def????isthevelocityvectorand??a(t)=[v](t)istheaccelerationvectorattimet.

????????????????De?nition1.1Thetangentvectortop??(·)attis[p](t)=x(t)???+y(t)???+z(t)k.If[p](t)=??0,????def[p](t)??theunittangenttoCatp??(t)isT(t)=.??[p](t)??

Thinkofp??(t)asajourney.Thejourneyisregularifandonlyifyouneverstopandyouarealwaysturning.

De?nition1.2Aparameterization,p??(·)ofacurveCisregulari?theparameterizationhas????????(t)=??twocontinuousderivativessothat[p](t)=??0andT0.Acurveiscalledregulari?ithasatleastoneregularparameterization.

Example1.3Givenp??(t)=etcos(t)???+etsin(t)???+et??k,itstangentvectoris

??[??p](t)=(etcos(t)?etsin(t))???+(etsin(t)+etcos(t))???+et??k.

1

21TANGENTIALANDNORMALDIRECTIONSTOACURVE

??Thenormof[??p](t)is

????[??p](t)??=??(etcos(t)?etsin(t))2+(etsin(t)+etcos(t))2+(et)2=√t(1)

anditsunittangentis

tt??tttsin(t))???+(esin(t)+ecos(t))???+ekcos(t)?e(e??(t)=Tet√??????(cos(t)?sin(t))???+(sin(t)+cos(t))???+k.=Noticethat:

??(t)·T??(t)=1T??(t)·T????(t)=0??T??(t)⊥T????(t).??T

??(t)isde?nedbyDe?nition1.4Foraregularcurve,theprincipalnormalvector,N

????defT(t)??N(t)=.??(t)????T

??(t)fortheaboveexampleonegets:Example1.5IncomputingN√????(t)=T[(?cos(t)?sin(t))???+(cos(t)?sin(t))???]??√????(t)??=[(?cos(t)?sin(t))]2+[(cos(t)?sin(t))]2)=??T√[(?cos(t)?sin(t))???+(cos(t)?sin(t))???]??N(t)=2[(?cos(t)?sin(t))]+[(cos(t)?sin(t))]2)√[(?cos(t)?sin(t))???+(cos(t)?sin(t))???].=(2)(3)(4)

??(t)isaunitvectorperpendiculartoT??(t).TheprincipalWeobservedfrom(2)abovethatN??(·)is????(·)pointsinthedirectionthatTnormalpointsintotheinteriorofthecurvesinceTmovingtoward.

??(t)andTheorem1.6Theacceleration,??a(t),canbewrittenasalinearcombinationofT??(t).Inparticular,N

??(t)T??(t)N??(t)+????(t).??a(t)=??a(t)·Ta(t)·N????????(5)

3

Proof:??(t).ThusNoticethat??v(t)=????v(t)??T

????(t)v(t)??T??a(t)=[??v](t)=??????????

??(t)+????????(t)=????v(t)????Tv(t)??T??(t).??(t)+????????(t)??N=????v(t)????Tv(t)????T

??(t)⊥N??(t),Equation(5)nowfollowsfromthefactthatsinceT

a(t)+ProjNa(t)

.??a(t)=ProjT??(t)????(t)??

??(t)T??(t)a(t)·TDe?nition1.7Thetangentialcomponentoftheaccelerationis??aT(t)=??????def????(t).a(t)·N(t)Nandthenormalcomponentoftheaccelerationis??aN(t)=??def????

WhilePaulNewmanmayormaynotbeabletodovectorcalculus,asaracecardriverhecertainlyknowswhatthetangentialandnormalcomponentsofaccelerationare.Thetangentialpartofaccelerationisthatcomponentthatliesinthedirectionthatthecarismoving.Discountingthee?ectofhillsandwindresistance,itoccurswhenPaulNewmanstepsonthegaspedalorthebrakes.Toomuchofitcanmakehiswheelsspinorskid.Thenormalcomponentofaccelerationcausescentrifugalforce,theforcethatPaulNewmanfeelspushinghimtotheoutsideofacurve.Whenthisforceistoogreat,itcancauseacartospinoutonacurveandcrashintothestandscausingagreatdisaster.Evenwhenacaristravelingatconstantspeedthiscomponentofaccelerationispresent.Itcomesfromthechangeinthedirectionofvelocity,notitschangeinmagnitude.

??(t)def??(t)×N??(t).TheorderedvectorDe?nition1.8Thebinormalvectortop??(·)attisB=T??(t),N??(t),B??(t))iscalledtheFrenetframeofCatptriple,(T??(t).

??(t)isaunitvectorperpendiculartoT??(t),N??(t)andtheacceleration,??NotethatBa(t).Furthermore,theFrenetframesatis?estherighthandrule.

Example1.9ToobtaintheFrenetframeoftheaboveexample(s)oneneedstocompute??(t).B

??(t)=T??(t)×N??(t)=B

√=?????????√??????√(cos(t)?sin(t))????(?cos(t)?sin(t))??????(sin(t)?cos(t))????(cos(t)+sin(t))???+2k.???(sin(t)+cos(t))√(cos(t)?sin(t))√??????k??√0??????????(6)

41TANGENTIALANDNORMALDIRECTIONSTOACURVEOnecanviewaFrenetframeasthenaturaldirectionsthatsomeonetravelingCinacarwouldbeawareof.Ifthecarisonatrackandhasnowindows,thepersoninthedriver’sseatwouldonlybeawareofsixthings:

??(t),1.theforwarddirection,T

2.thespeedgivenbythespeedometer,????v(t)??,

??(t),pointingintoacurve;thesideoppositethesideoneispulledtoby3.thedirection,Ncentrifugalforce(thismaynotbethedirectionstraightthroughtheleftorrightdoorsiftheroadbedisnoteven,i.e.itcouldbebankedforexample),

4.howcurvedtheroadisbycomparingthenormalcomponentofacceleration(propor-tionaltothecentrifugalforce)withthespeed,

??(t),(this5.thedirectionperpendiculartotheforwarddirectionandtheacceleration,Bneednotbeexactlythedirectionofthroughthetoporbottomofthecar–seeremarkinitem3)and

6.thefeelingofqueasinessthegyroscopesinone’sinnereargiveswhenoneisnottravelinginjustoneplane1.Inparticular,onecansensehowmuchoneistwistingoutofthe??(t)andN??(t).planedeterminedbythevectorsT

Alloftheitemsaboveareindependentofthee?ectsofgravity!Allbutitems4and6havebeende?nedsofar.Thesetwoitemswillbecalledcurvatureandtorsion,respectively.

Onecanrelatetheperspectiveofthedriverofcarwithnowindowstotheperspectivemodernphysicsoftentakes.Whileinrealityonehastheluxuryofviewingtheroadfromanearbymountainormapandcanseehowitisshaped,thedriver’s?rstimmediateimpressionisthatthereisonlyaforwarddirectionandabackwardsdirection,i.e.,lifeconsistsofastraightline.Similarly,ifwewhereable(wearenot!)toviewouruniversefromsomeothervantagepoint(wherewouldthatbe?),wecouldseeourfourdimensionaluniverse(threedimensionsplustime)assometypeoffourdimensional“curve”insomehigherdimensionalspace.Fromthisviewpointwemightbeabletoseethatrealityiscurved,eveniftheperspectiveofsomeonelivinginrealityisthatrealityisniceandstraight.

Itturnsoutthatthoseofuswholiveinreality(thatisallofus)canconcludethatouruniverseisindeedcurved,justasourdriverofthecarwithnowindowscaneventuallyconcludethattheroadheistravelingiscurved.Thedrivermakeshisconclusionbynoticingthecentrifugalforceheisexposedto.Amoderndayphysicistcannoticethewaylightisbendwhenisthepresenceofamassiveobject.(Actually,thelightisnot“bending”–it

Thisisnauseatingfeelingsomepeoplepaygoodmoneyforwhenpurchasingatickettorideonarollercoaster.Itisthefeelingonegetswhenroundingalevelcurve,therollercoastersuddenlydropsorclimbs.1

5

travels“straight”.Itisthespace-timecontinuumthatis“bent”bythegravityofamassiveobject.)

2CurvatureandTorsion

Ourmaninthecarwithnowindowshasawayofmeasuringhowsharpacurveacurvemaybe.Ifthenormalcomponentoftheacceleration(thecentrifugalforcehefeels)isgreatthenthecurvemustbesharp.Ourmaninthecarmightwanttomeasurethecurvatureofthecurvebyassigningtoeachpointonhisroadthemagnitudeofthenormalcomponentoftheacceleration.Theonlyproblemwiththisschemeisthatthefasterhegoesthegreaterthenormalcomponentoftheacceleration.Thesamepointinthesamecurvewillgenerateasmallernormalcomponentofacceleration(lesscentrifugalforce)at20milesperhourthanat60milesperhour.Anyonewhohasbeeninacarknowsthis.Thewayaroundthisdilemmaisthatthemaninthecardecidestoonlymeasurethenormalcomponentoftheaccelerationwhenthecarisgoingonemileperhour(speedequaltoone)andthende?nethemagnitudeofthenormalcomponenttobethecurvature.

De?nition2.1Letw??(s)=x(s)???+y(s)???+z(t)??kbeaparameterizationofCsothatthe??wcurveistraveledatspeedone,i.e.??[w??]??(s)??=1.ThusT??]??(t).Thecurvatureat??(t)=[w1def????(s)=??T(s)??.Theradiusofcurvatureatw??(s)isde?nedtobew??(s)isde?nedasκw.??w??w??Onewishestocomparethecurvatureatsomepoint,p??(t),onacurvetothecurvaturefoundonsomeparticularcircle.Theproblemiswhatcircle,i.e.,whatradiusshouldoneuseforthisreferencecirclewhosecurvaturematchesthatofthecurveatp??(t).Itisaneasyexercise(whichwedoinExample2.4)toseethatacircleofradiusrhasthesameradiusofcurvatureeverywhereandthatradiusofcurvatureisjusttheradius,r.Thusoneshouldnotbeabletodistinguishbetweenthecentrifugalforceonefeelsatp??(t)onourcurve,orthecentrifugal1forceonefeelstravelingaroundacirclewhoseradiusisequalto2.Ifoneistravelingatunitspeed,oneexpectstotravelsdistanceinsunitsoftime.Ifoneistravelingatunitspeed,onecoulduseone’swatchasanodometer!Thisleadsustothestudyofdistancealongacurve,thatisarclength.

Theorem2.2Givenaregularparameterizationofacurve,p??(t),thearclengthofthecurve??t????betweenp??(a)andp??(t)wherea≤tisdenotedbys(t)andisgivenbys(t)=a??[p](u)??du.

Hereweusethenotationκ(t)=κw??(t)andκw??(s(t)),i.e.κ(t)isthecurvatureofCatp??(s)isthecurvatureatw??(s).2def

62CURVATUREANDTORSIONProof:Letp??(t)=x(t)???+y(t)???+z(t)??k.Then

s(t)=??t??

a[x??(u)]2+[y??(u)]2+[z??(u)]2du=??ta??[??p](u)??

du.??

TheaboveTheoremsimplysaysthatintegratingthespeedhvesthedistancetraveled.AnimmediateconsequenceoftheabovetheoremisthefollowingcorollaryandtheSecondFun-damentalTheoremofCalculusisthat:

??ds??Corollary2.3(t)=??[p](t)??.TheabovecorollarypointsouttheneedtoparameterizethecurveCbyarclengthinordertotravelatspeedone(asinthede?nitionofcurvature).Inthefollowing,picturetwodi?erenthikesalongC

.

The?rstisahikedescribedbyp??(t).Itisahikeahumanlikeyoumighttake:whereyougoatdi?erentspeedsatdi?erenttimes;fastwhenyouarebeingchasedbywildanimalsorslowwhenyoupassanudistcolony.Thesecondtrip,w??(s)couldbeonetraveledbyrobocopwithhiscruisecontrolsettospeedone.Theabovetheoremtellsusthathetravelssdistanceinsunitsoftime.Robocopcoulduseaclockforanodometer.Onede?nesthefollowingtwo

7

functions(theyareinversesofeachother):

t(s)=thetimeyoupassthepointwhererobocopwasattimes

s(t)=thetimerobocoppassesthepointwhereyouwereattimet.

Oneseesthatw??(s)=p??(t(s)).

Inthefollowingexamplethecurvatureandradiusofcurvatureofacircleofradiusriscomputed.

Example2.4Letp??(t)=rcos(t)???+rsin(t)???for0≤t≤2π.Wewillcalculatethecurvatureandradiusofcurvature.Noticethatp??(·)describesahikeatconstantspeedaroundacircleofradiusr.Sinceittakes2πunitsoftimetotravelthe2rπdistance,onemustbetravelingataspeedofr.

Ourmethodofcalculatingthecurvaturewillbestraightforward.Theorem2.5willdescribeaneasierwayof?ndingcurvaturethandescribedhere.Ourcurrentmethodwillbeto?ndw??(s)=p??(t(s))andthentousethede?nitionforw??(s),atriptakenatunitspeed.Inkeepingwiththisstrategy,noticethat

s=??t??

0defdefr2sin2(u)+r2cos2(u)du=rt.

sThinkingoftasafunctionofsonehast(s)=.Thus????????ssw??(s)=p??(t(s))=rcos???+rsin???.Furthermore,

ss??wT??]??s=?sin???+cos?????(s)=[w????????ss11????(s)=?cos???????.Tsinw??????????and

1????Thusthecurvaturefunctionistheconstantfunctionκw(s)=??T(s)??=andtheradiusof??w??convergenceisr,theradiusoftheoriginalcircle!

Aneasiermethodofcalculatingthecurvatureisgiveninthefollowingtheorem.

Theorem2.5Givenaregularvector-valuedfunction,p??(·),thecurvatureatp??(t)isgivenby

????????????????[p](t)×[p](t)????T(t)??=.κ(t)=3??[p](t)????[p](t)??

82CURVATUREANDTORSIONProof:UsingCorollary2.3oneseesthat

??(t+?t)?T??(t)T????(t)??????(t)????T??T=≈??[p](t)??????(t+?t)?T??(t)?TT????(s(t))??=κ(s(t))=κ(t)==??T=Letting?t→0onehasthe?rstequality.

??????(t).ThusTogetthesecondequalitynoticethat[??p](t)=??[??p](t)??T

????????????????(t)??N??(t),????(t)=??[????(t)+??[????(t)+??[??p](t)????T[??p](t)=??[??p](t)????Tp](t)????Tp](t)??T

fromwhichonegets

????????????[p](t)??????????????[p](t)×[p](t)=??[p](t)??×[p](t)??[p](t)????

??(t)×[??=??[??p](t)??Tp](t)

????????????????(t)×??[????(t)??(t)×??[??=??[??p](t)??Tp](t)??T(t)+Tp](t)????T(t)??N??????????????????????

????????(t)??B??(t)??=0+??[p](t)??2??T????????(t)??B??(t).=??[p](t)??2??T(7)

??Thesecondequalitynowfollowsfrom2??????????????(t)????????????[p](t)????T(t)??B????????????T(t)??B(t)????T(t)????[p](t)×[p](t)??===.33

??[p](t)????[p](t)????[p](t)????[p](t)??????????

??(·),isgivenbyLemma2.6GivenacurveCparameterizedbyp??(·),thebinormal,B

??????[??p](t)×[??p](t).??[p](t)×[p](t)??

Proof:Thisfollowsfrom

(7).

Example2.7Wewillcalculatethecurvatureforouroriginalexample.Using(1)and(3)thecurvatureis√√??????T(t)??==.κ(t)=te??[p](t)??

9

De?nition2.8Theosculatingplaneattimet0ofacurveCparameterizedbyp??(·)isthe??(t0)andN??(t0).planethroughthepointp??(t0)andparalleltothevectorsT

Onemaynowtrytomeasurethetendencyofacurveattimet0tostayintheosculatingplane.Ononehandtheaccelerationiscompletelycontainedinthisplane(seeTheorem1.6).Ontheotherhand,manycurvesdonot?tintojustoneplane.Thebestwayofkeepingtrackofanyplaneislookataunitvectorperpendiculartothatplane.Afterallthereislotsofdirectionsparalleltoaplane,yetonlyone(modulo180?)thatisperpendiculartothe??(t0)isperpendiculartotheplane.Thusagoodplane.Fortheosculatingplane,thevectorBmeasureofhowmuchtheosculatingplanechangesindirection(ignoringtranslations)with??(·)withtime,i.e.??B????(·)??measureschangeinthetimeisthemagnitudeofthechangeinB????(·)??istheabsolutevalueofthetorsion.osculatingplane.InDe?nition2.10oneseesthat??B

BeforegivingDe?nition2.10weneedtoobserve:

??(t).????(t),isascalarmultipleofNLemma2.9Thederivativeofthebinormal,B

Proof:????(t)⊥T??(t)andthatB????(t)⊥B??(t).Itsu?cestoshowthatB

??(t)followfromthefactthatB??(t)·T??(t)=0sinceafter????(t)⊥TThestatementthatBdi?erentiatingbothsidesonehas

????(t)·T??(t)+B??(t)·T????(t)=B????(t)·T??(t)+B??(t)·[??T??????N??(t)]=B????(t)·T??(t),0=B

??(t).????(t)⊥Ti.e.B

??(t)followsfromacalculationidenticaltothatperformedin????(t)⊥BThestatementthatB(2).

Thefollowingde?nitionnowmakessense.

De?nition2.10Thetorsionisde?nedtobebethereal-valuedfunctionτ(·)suchthat??(t).????(t)=?τ(t)NB

Example2.11Againusingour?rstexample,(6)and(4)onecalculatesthetorsionas,√√????(t)=(cos(t)+sin(t))???+(sin(t)?cos(t))???B??(t)=?τN√??√??=?τ(?cos(t)?sin(t))???+(?sin(t)+cos(t))???.Thusτ(t)=√.

10REFERENCES3SummaryUnitTangent

PrincipalNormal

Binormal

Curvature(t)==T??(t)=N??[p](t)??????=??T??(t)????(t)×T??(t)=B??????p](t)????[??p](t)×[??

??[p](t)??3??????p](t)[??p](t)×[????[p](t)×[p](t)????(t)=T??(t)×N??(t)==Bκ(t)=????(t)????T??[p](t)??=RadiusofCurvature=????(t)=?τ(t)N??(t)=τ(t)suchthatBTorsion

4Exercises

√t??kInthefollowingtwoproblemsletp??(t)=t????ln(cos(2t))???+

1.WhatistheUnitTangentvector,T(0),top??(·)att=0?

2.WhatisthePrincipleNormalvector,N(0),top??(·)att=0?

Inthefollowingtwoproblemslet

p??(t)=(t3?t)???+(t2?8)????

3.WhatistheBinormalvector,B(2),top??(·)att=2?

4.Whatisthecurvature,κ(2),ofp??(·)att=2?

5.Whatisthenormalcomponentofaccelerationattimet=2?8??kReferences

[1]BarrettO’Neill.ElementaryDi?erentialGeometry.AcademicPress,NewYork,1966.

[2]GeorgeB.ThomasJr.CalculusandAnalyticGeometry.Addison-WesleyPublishingCompany,Reading,Massachusetts,fourthedition,1968.

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