椭球方程
-
2023年3月20日发(作者：汉服文化)平面和椭球面相截所得的椭圆的参数方程及其应用
黄亦虹;许庆祥
【摘要】设E:x2/a2+y2/b2+z2/c2=1为一个椭球面,P:px+qy+rz=d为一个
平面.利用Householder变换,证明了E和P相交当且仅当λ≥|d|,其中λ=√(ap)2
+(bq)2+(cr)2.当λ＞|d|时用新的方法证明了椭球面E和平面P的交线e一定
是椭圆,并且给出了该椭圆的参数方程.利用交线e的参数方程,给出了由e所围成的
内部区域的面积公式,进而给出了椭圆的长半轴和短半轴的计算公式.作为应用,又给
出了交线e成为一个圆的充要条件.%LetE:x2/a2+y2/b2+z2/c2=1bean
ellipsoidandP:px+qy+rz=ntheHouseholder
transformation,itisshownthattheintersectionE∩Pisnonemptyifand
onlyifλ≥|d|,whereλ=√(ap)2+(bq)2+(cr)λ＞|d|,thispaper
providesanewproofthattheintersectioncurveeofEandPisalwaysan
ellipse,n
theobtainedparametricequationofeandStokesformula,wederivea
formulafortheareaoftheregionboundedbye,andcomputeitssemi-
plication,wegetnecessaryand
sufficientconditionsforetobeacircle.
【期刊名称】《上海师范大学学报（自然科学版）》
【年(卷),期】2018(047)001
【总页数】7页(P24-30)
【关键词】椭球面;平面;参数方程;Householder变换;Stokes公式
【作者】黄亦虹;许庆祥
【作者单位】上海应用技术大学理学院,上海201418;上海师范大学数理学院,上海
200234
【正文语种】中文
【中图分类】O13;O172
1Introduction
Throughoutthispaper,,+andm×narethesetsoftherealnumbers,the
positivenumbersandthem×nrealmatrices,ationn×1
A∈m×n,e
theidentitymatrixofordern.
Muchattentionispaidtotheverypopulartopicoftheintersectioncurve
ofanellipsoidandaplane[1-3].Yet,littlehasbeendoneintheliteratures
ontheapplicationoftheHouseholdertransformationandtheStokes
formulatothistopic,whichistheconcernofthispaper.
LetEbeanellipsoidandPbeaplanedefinedrespectivelyby
(1)
wherea,b,c∈+andp,q,r,d∈suchthatp2+q2+r2≠0.
ItisknownthattheintersectioncurveofEandPisalwaysanellipse,and
,dueto
thecomplexityofcomputation,itissomehowdifficulttoderiveexplicit
formulasforthesemi-axesof.
ThekeypointofthispaperistheusageoftheHouseholdertransformation
toderiveanewparametricequationof,togetherwiththeapplicationof
theStokesformulatofindthearea|S|oftheregionboundedby;see
rpointofthispaperisthe
characterizationoftheparalleltangentlinesof,whichiscombinedwiththe
obtainedformulafor|S|ult,explicit
formulasforthesemi-majoraxisandthesemi-minoraxisofare
derived;plication,necessaryandsufficient
conditionsarederivedunderwhichisacircle.
2Themainresults
Letv∈seholdermatrixHvassociatedtovisdefined
by
n×n.
(2)
Itisknown[4]thatHvT=HvandHvTHv=In,i.e.,Hvisanorthogonal
hefollowingproperty,theHouseholdermatrixisofspecial
usefulness.
Lemma2.1Letx,y∈nbesuchthatx≠yandxTx=
Hv(x)=y,wherev=x-y.
(3)
Theorem2.2LetEandPbegivenby(1).Then=E∩P≠Øifandonlyif
λ≥|d|,whereλisdefinedby
(4)
ProofLetλbedefinedby(4).Firstly,weconsiderthecasethatp2+q2>
w1=(ap,bq,cr)Tandw2=(0,0,λ)early,w1≠w2andw1Tw1=w2Tw2,so
byLemma2.1wehave
Hvw1=w2,wherev=w1-w2.
(5)
Let
(6)
Thenby(1),(5)and(6),wehave
(7)
d
(8)
Itfollowsfrom(7)and(8)that
(9)
ThismeansE∩Pisnonemptyifandonlyifλ≥|d|,whereλisdefinedby(4).
Secondly,weconsiderthecasethatp=q=case,wehaver≠
followsdirectlyfrom(1)that
thustheconclusionalsoholds.
Thefollowingresultiswell-known,yetitsproofpresentedbelowis
somehownew.
Theorem2.3LetE,Pandλbegivenby(1)and(4)respectivelysuchthat
λ>|d|.Thentheintersectioncurve=E∩Pisalwaysanellipse.
ProofItneedsonlytoconsiderthecasethatp2+q2>
w3=(p,q,r)T,w4=(0,0,)T,v1=w3-w4andHv1betheHouseholdermatrix
definedby(2)whichsatisfiesHv1w3=
(x,y,z)T=Hv1(x1,y1,z1)T.
(10)
Then
d=w3T(x,y,x)T=w3THv1(x1,y1,z1)T=w4T(x1,y1,z1)T=z1.
Therefore,
(11)
Itfollowsfrom(1),(10)and(11)that
(12)
whereA=partitionedasA=,where
A1∈2×om(12)weget
(x1,y1)A1+λ1x1+λ2y1+λ3=0forsomeλi∈,i=1,2,3,
observationtogetherwith(11)yieldsthefactthatinthex1y1z1-
coordinatesystem,theequationoftheintersectioncurverepresentsan
clusionthenfollowsfrom(10)sinceHv1isanorthogonal
matrix.
Theorem2.4LetE,Pandλbegivenby(1)and(4)suchthatλ>|d|.Thena
parametricequationoftheintersectioncurve=E∩Pcanbegivenfor
t∈[0,2π]asfollows:
(13)
ProofWeonlyconsiderthecasethatp2+q2>(2)and(5)weobtain
(14)
Furthermore,by(8)and(9)weget
(15)
Eq.(13)thenfollowsfrom(6),(14)and(15).
AnapplicationofTheorem2.4isasfollows.
Corollary2.5LetbetheintersectioncurveoftheellipsoidEandtheplane
Pgivenby(1).Thenthearea|S|oftheregionSboundedbycanbe
formulatedby
(16)
whereλisdefinedby(4).
ProofLet(cosα,cosβ,cosγ)denotetheunitnormalvectoroftheplane
P,where
(17)
WemayusetheStokesformulatoget
|S|=±○zcosβdx+xcosγdy+ycosαdz,
(18)
where±ischosentoensurethattherightsideof(18)isnon-
at
sintdt=costdt=sintcostdt=0,
(19)
sin2tdt=cos2tdt=π.
(20)
Therefore,by(13),(4),(19)and(20)weobtain
○
(21)
○
(22)
○
(23)
Formula(16)thenfollowsfrom(17)-(18)and(21)-(23).
ConsiderthecalculationofI=○x2ds,whereistheintersectioncurveofthe
spherex2+y2+z2=R2(R>0)andtheplanex+y+z=ofthe
symmetry,asolutioncanbecarriedoutsimplyas
Obviously,themethodemployedaboveonlyworksforthesymmetric
nbyExample2.1below,theparametricequation(13)isa
usefultooltodealwiththenon-symmetriccase.
Example2.1EvaluateI=○x2ds,whereistheintersectioncurveofthe
spherex2+y2+z2=R2(R>0)andtheplanepx+qy+rz=d.
a=b=c=R,Eq.(6)turnsouttobe
(x,y,z)T=Hv(x1,y1,z1)T,
whichiscombinedwith(15)toget
Inviewofthefirstequationof(13)and(19)-(20),wehave
I=○
(24)
andμisgivenby
(25)
Notethat
(26)
sowemaycombine(24)-(26)toconcludethat
I=○
whereλisgivenby(24).
Now,weturntostudythesemi-axesoftheellipsegivenby(13).Let
P(t)=(x(t),y(t),z(t))have
whereλisgivenby(4).SupposethatP(t1)andP(t2)aretwodifferent
pointsinsuchthatthetangentlinesatthesetwopointsareparallel,then
thereexistsaconstantμsuchthatx′(t2)=μx′(t1),y′(t2)=μy′(t1)andz′(t2)=μ
z′(t1);ormoreprecisely,
Itfollowsfrom(27)and(29),(28)and(29)thatsint2=μsint1andcost2=μ
ore,
1=sin2t2+cos2t2=μ2(sin2t1+cos2t1)=μ2,
henceμ=-1sinceP(t1)≠P(t2),andthusP(t2)=P(t1+π).Theobservation
aboveindicatesthat
(30)
wheremax,mindenotethesemi-majoraxisandthesemi-minoraxis
of,respectively,and
(31)
whereg(t)isgivenby
(32)
ascos2t=,sin2t=andsintcost=,where
(33)
By(4)wehave
[(λ(cr-λ)+(ap)2)2+(abpq)2]=λ2(cr-λ)2+2(ap)2λ(cr-λ)+a2p2[(ap)2+(bq)2]
=λ2(cr-λ)2+2(ap)2λ(cr-λ)+a2p2[λ2-(cr)2]=(cr-λ)2[λ2-(ap)2].
(34)
Similarly,wehave
[(abpq)2+(λ(cr-λ)+(bq)2)2]=(cr-λ)2[λ2-(bq)2].
(35)
Wemaycombine(4)with(33)-(35)toconcludethat
A=[(ap)2(b2+c2)+(bq)2(c2+a2)+(cr)2(a2+b2)].
(36)
Theorem2.6LetbetheintersectioncurveoftheellipsoidEandtheplane
Pgivenby(1),andmaxandminbethesemi-majoraxisandthesemi-
whereλandAaredefinedby(4)and(36).
ProofItfollowsfrom(30)-(32)that
(37)
(38)
whichmeansthatthearea|S|oftheregionSboundedbyisequalto
Theaboveequationtogetherwith(16)yields
B2+C2=A2-λ2(abc)2(p2+q2+r2).
Theconclusionthenfollowsbysubstitutingtheaboveexpressionfor
B2+C2into(37)and(38).
Adirectapplicationoftheprecedingtheoremisasfollows.
Corollary2.7Supposethata>b>c>heintersectioncurveofthe
ellipsoidEandtheplanePgivenby(1),and=(cosα,cosβ,cosγ)betheunit
normalvectorofPwithcosα,cosβandcosγgivenby(17).Thenisacircle
ifandonlyifeither‖or‖,where
ProofLetλandAbedefinedby(4)and(36).Bydirectcomputation,we
have
θ4A2-4λ2(abc)2(p2+q2+r2)=[(cr)2(a2-b2)-(ap)2(b2-c2)]2+(bq)4(a2-c2)2
+2(abpq)2(a2-c2)(b2-c2)+2(bcqr)2(a2-c2)(a2-b2).
Sincea>b>c,byTheorem2.6weknowthatisacircleifandonlyif
θ=lently,isacircleifandonlyif
⟺
TheresultstatedbelowfollowsimmediatelyfromtheproofofCorollary
2.7.
Corollary2.8Supposethata,b,c∈+suchthata=b≠he
intersectioncurveoftheellipsoidEandtheplanePgivenby(1).Thenisa
circleifandonlyifp=q=0.
References:
[1]AbramsonN,BomanJ,ntersectionsofrotational
elliposids[J].AmericanMathematicalMonthly,2005,113:336-339.
[2]ectionsofellipsoidsandplanesofarbitrary
orientationandposition[J].MathematicalGeology,1979,11:329-336.
[3]llipsoidandplaneintersectionequation[J].Applied
Mathematics,2012,11:1634-1640.
[4]AlgebrawithApplications(EighthEdition)
[M].Beijing:PearsonEducationAsiaLimitedandChinaMachinePress,2011.