[PDF] Analytic geometry of space R. CRATHORNE Associate in Mathematics

View - Download Analytic geometry of space

Previous PDF

Next PDF

HmericanflDatbematicalSeries

E.J.TOWNSEND

GENERALEDITOR

MATHEMATICALSERIES

I.Calculus.

$2.50.

II.EssentialsofCalculus.

ByE.J.TOWNSENDandG.A.GOODENOUGH.$2.00.

Illinois.$1.40.

Tables.

theUniversityofOregon.$1.25.

V.PlaneandSphericalTrigonometry.

VI.TrigonometricandLogarithmicTables.

ByA.G.HALLandF.H.FRINK.75cents.

Thefollowingareinpreparation:

PlaneAnalyticalGeometry.

ofWisconsin.

NEWYORKCHICAGO

ANALYTICGEOMETRY

OFSPACE

BY

VIRGILSNYDER,Pn.D.(GOTTINGEN)

PROFESSOROFMATHEMATICSATCORNELL

UNIVERSITY

AND

C.H.SISAM,PH.D.(CORNELL)

ASSISTANTPROFESSOROFMATHEMATICSATTHE

UNIVERSITYOFILLINOIS

NEWYORK

HENRYHOLTANDCOMPANY

1914

COPYRIGHT,1914,

BY

HENRYHOLTANDCOMPANY

J.S.GushingCo.Berwick&SmithCo.

Norwood,Mass.,U.S.A.

PREFACE

fortylessons. furtherstudyofalgebraicgeometry. elementofarc.

CONTENTS

CHAPTERI

COORDINATES

ARTICLEPAGE

1.Coordinates

2.Orthogonalprojection3

3.Directioncosinesofaline........5

4.Distancebetweentwopoints6

5.Anglebetweentwodirectedlines

6.Pointdividingasegmentinagivenratio....

7.Polarcoordinates

8.Cylindricalcoordinates.10

9.Sphericalcoordinatesn

CHAPTERII

PLANESANDLINES

10.Equationofaplane12

11.Planethroughthreepoints.13

12.Interceptformoftheequationofaplane

.....13

15.Anglebetweentwoplanes

16.Distancetoapointfromaplane17

17.Equationsofaline19

19.Formsoftheequationsofaline...

"...20

20.Parametricequationsofaline....21

21.Anglewhichalinemakeswithaplane22

22.Distancefromapointtoaline....23

24.Systemofplanesthroughaline.....25

v

ViCONTENTS

26.Bundlesofplanes29

27.Planecoordinates31

28.Equationofapoint.32

32.Planeatinfinity35

33.Linesatinfinity.35

34.Coordinatetetrahedron.........35

35.Systemoffourplanes36

CHAPTERIII

TRANSFORMATIONOFCOORDINATES

36.Translation38

37.Rotation38

38.Rotationandreflectionofaxes41

39.Eulersformulasforrotationofaxes42

CHAPTERIV

TYPESOFSURFACES

41.Imaginarypoints,lines,andplanes....44

42.Lociofequations........46

43.Cylindricalsurfaces.........47

44.Projectingcylinders47

45.Planesectionsofsurfaces48

46.Cones49

47.Surfacesofrevolution60

VCHAPTERV

THESPHERE

48.Theequationofthesphere........52

49.Theabsolute52

50.Tangentplane56

51.Anglebetweentwospheres65

53.Linearsystemsofspheres.........57

54.Stereographicprojection...69

CONTENTSVii

CHAPTERVI

FORMSOFQUADRICSURFACES

ARTICLE

55.Definitionofaquadric...?.,....

VlllCONTENTS

CHAPTERIX

TETRAHEDRALCOORDINATESARTICLE

88.Definitionoftetrahedralcoordinates109

89.Unitpoint.....no

91.EquationofapointH2

92.Equationsofaline....112

93.Duality113

98.Projectivetransformations120

99.Invariantpoints....121

100.Cross-ratio12i

CHAPTERX

QUADRICSURFACESINTETRAHEDRALCOORDINATES

101.Formofequation......124

102.Tangentlinesandplanes124

104.Theinvarianceofthediscriminant...126

105.Linesonthequadricsurface129

107.Polarplanes132

110.Tangentcone........133

111.Conjugatelinesastoaquadric134

112.Self-polartetrahedron135

114.Lawofinertia13g

115.Rectilineargenerators.Reguli137

117.Projectionofaquadricuponaplane139

118.Equationsoftheprojection......140

120.Transversalsoffourskewlines143

121.Thequadriccone.....143

CONTENTSIX

CHAPTERXI

LINEARSYSTEMSOFQUADRICS

AKTICLEPAGK

123.Pencilofquadrics147

124.The\-discriminant147

125.Invariantfactors148

126.Thecharacteristic15

128.Classificationofpencilsofquadrics151

133.FormsofpencilsofquadricsI63

134.LineconjugatetoapointI65

136.Bundleofquadrics167

138.Singularquadricsofthebundle.....168

139.Intersectionofthebundlebyaplane169

140.ThevertexlocusJ/.170

141.Polartheoryinthebundle171

142.Somespecialbundles173

143.Websofquadrics17&

144.TheJacobiansurfaceofaweb175

146.Webwithsixbasispoints177

147.LinearsystemsofrankrI80

149.Apolarity

.../181

150.Linearsystemsofnpolar/JuadricsI86

CHAPTERXII

TRANSFORMATIONSOFSPACE

151.Projectivemetric

152.PoleandpolarastotheabsoluteI88

153.Equationsofmotion190

156.Birationaltransformations196

157.Quadratictransformations..198

158.Quadraticinversion.201

160.Cyclides........-:..-..203

XCONTENTS

CHAPTERXIII

CURVESANDSURFACESINTETRAHEDRAL

COORDINATES

I.ALGEBRAICSURFACES

ARTICLEPAGE

162.Notation207

163.Intersectionofalineandasurface207

164.Polarsurfaces208

165.Tangentlinesandplanes209

166.Inflexionaltangents210

167.Doublepoints210

170.TheHessian213

171.Theparaboliccurve214

172.TheSteinerian214

II.ALGEBRAICSPACECURVES

174.Orderofanalgebraiccurve........216

175.Projectingcones..........217

176.Monoidalrepresentation........219

181.Singularpoints,lines,andplanes226

182.TheCayley-Salmonformulas226

184.Spacecubiccurves230

188.Rationalquartics242

CHAPTERXIV

DIFFERENTIALGEOMETRY

I.ANALYTICCURVES

189.Lengthofarcofaspacecurve245

190.Themovingtrihedral,246

191.Curvature248

CONTENTSXI

ARTICLE

ANALYTICGEOMETRYOFSPACE

CHAPTERI

COORDINATES

inthefollowingmanner. inYOY,theF-axis;the planesYOZ,ZOXintersect inZOZ,theZ-axis.Dis tancesmeasuredinthe Z Xx

V"n

FIG.1.

consideredpositive;those measuredintheopposite directionswillberegarded asnegative.Thecoordi 1

COORDINATES[CHAP.I.

f* 1 \z JOX

ART.2]ORTHOGONALPROJECTIONS3

EXERCISES

(0,-1,0),(-3,0,0),(0,0,0).

2.Whatisthelocusofapointforwhichx=?

3.Whatisthelocusofapointforwhichx=0,y=?

4.Whatisthelocusofapointforwhichx=a,y=b?

origin.

QofPandQontheplane.

QonI. asIandI,respectively.

COORDINATES[CHAP.I.

Itisrequiredtoprovethat

L

FIG.3a.FIG.36.

1 .Then wehave

PQ=PQ"cosB.

ButPQ"=PQ.

ItfollowsthatPQ=PQcos0.

cosines. jectiononIofthestraightlinePiPn.

ARTS.2,3]DIRECTIONCOSINESOFALINE5

equaltoP^P*;thatis,

EXERCISES

or*xi,1/22/i,zo#!,respectively. theorigintoeachofthefollowingpoints. (1,2,0)(1,1,1)(-7,6,2) (0,2,4)(1,-4,2)(x,y,z)

3.Directioncosinesofaline.

LetIbeanydirectedlinein

space,andletIbealinethrough theoriginwhichhasthesame direction.If,fty(Fig.4) aretheangleswhichVmakes withthecoordinateaxes,these arealso,bydefinition(Art.2), theangleswhichImakeswithFlG4 theaxes.Theyarecalledthe

6COORDINATES[CHAP.I.

\_eo_a_r>_b___c rrr areOA=a,OB=b,OC=c.

Hence,weobtainr=Va2+b2+c2

thepositivesign. =cosa=Va2+bz+c2 b v=COSy=

Va2+62+c2

c

Va2+bz+c2

results,weobtainx,+|l,+,,=lf(1) hencewehavethefollowingtheorem. lineisequaltounity. directedlines,wehave

Ifthelinesareoppositelydirected,wehave

ARTS.4,5]ANGLEBETWEENTWODIRECTEDLINES7

equaltothesumoftheprojections ofPXandOP2,thatis

Bysquaringbothmembersofthese

equations,adding,andextractingthe squareroot,weobtainN

FIG.5.

-i)2+(2/2

EXERCISES

verticesofanisoscelestriangle. verticesofanequilateraltriangle. (1,1,1)and(2,3,4). theoriginandwhoseradiusis2. (a)Pi=(0,0,0),P2=(2,3,5). (6)P,=(1,1,1),P2=(2,2,2). (c)Pi=(l,-2,3),P2=(4,2,-1). (6)cosa-andcos=?(c)cosa-1? line. coordinateaxes.

8COORDINATES[CHAP.I.

linesOPlandOP2havingthesamedi N x2=OM,y2=MN,v

XTheprojectionofOP2onOPtisequal

tothesumoftheprojectionsofthe brokenlineOMNP2onOPl(Art,2).FIG.6.

HenceOP2cos=OM\l+MN^+NP2Vl.

ButOP2=r2)OM=x2=r2\2,MN=y2=rzfji2)NP=z2=

fIBiweobtain r2cos orcos= J (3)

1/1and

Sincesin291cos29,itfollowsthat

sin2B=(Vv, 2)(X2 24-v2
2 (5)

6.Pointdividingasegmentina

givenratio.LetPl= (Fig.7).Itisrequiredtofindthe pointP=(aj,?/,2)onthelineP:P2 suchthatPjP:PP2=m^:m2.Let

X,p.,vbethedirectioncosinesof

thelineP^P2.Then(Art.2,Th.I)wehave

PjPA=x-x1andPP2A=x2-x.

HencePXPA:PP2X=xxl

:x2x=mj:?>i2 FIG7.

ART.6]POINTDIVIDINGASEGMENT9

.-,Onsolvingforxweobtain (6)ml+ tn.ai/1

Similarly,y

4-

PiP2,fn,\=

?L+^2.__ 222

EXERCISES

cosinesareJ t-4=,^=and-2=,^,-A,.V14Vl4Vl4V30V30V30 theirvalues.

2;2,3,C;6,2,3aremutuallyperpendicular.

ofarighttriangle. verticesofaparallelogram. lelogramofEx.7. (3,6,4),(2,-1,5)arecollinear.

10COORDINATES[CHAP.I.

angleitmakeswiththeZ-axis. calledthecenterofgravityofthetriangle. gravity. quadrilateralformaparallelogram. tionanglesa,ft,y.Theposition ofthelineOPisdeterminedby a,ft,yandthepositionofPon thelineisgivenbyp,sothatthe positionofthepointPinspace isfixedwhenp,a,ft,yare known.Thesequantitiesp,a,ft, yarecalledthepolarcoordinates ofP.Asa,ft,yaredirection cos2a+cos2 ft-f-cos2y=1.

FlG-8-

ARTS.8,9]SPHERICALCOORDINATES11

pointPisreferredtotherectangular axesx,y,z,andthefixedplaneistaken asz=andtheic-axisforpolaraxis, wemaywrite(Fig.9) x=pcos0,y=psin=z, inwhichp,0,zarethecylindricalcoordi natesofP.w

FIG.9.

piscalledtheradiusvector,theangle <iscalledtheco-latitude, andiscalledthelongitude.

IfP=(x,y,z),then,fromthefigure

(Fig.10),

OP=pcos(904>)=psin<.

Hencex=psin

<f>cos0, y=Psin <f>sin6, z=pcos<.YFie.10. z p=V#24-2/

24-=arccos=arctan

EXERCISES

1.Whatlocusisdefinedbyp=I?

2.Whatlocusisdefinedbya=60?

quotesdbs_dbs47.pdfusesText_47

[PDF] Mathematiques: FOnction polynome de degres 2

[PDF] Mathématiques: géométrie et "x"

[PDF] Mathématiques: Indicateurs de dispersion et comparaison de séries

[PDF] MATHEMATIQUES: LE COSINUS

[PDF] Mathématiques: nombres en écriture fractionnaire 4ème

[PDF] Mathématiques: Puissances

[PDF] Mathématiques: racines carées

[PDF] Mathematiques: Raisonnment A Partir D'un Algorithme

[PDF] mathématiques: résoudre une équation

[PDF] Mathématiques: Tableau de variation

[PDF] Mathématiques: thales

[PDF] Mathématiques: Thorème de comparaison

[PDF] Mathematiques:calculer a² et b²

[PDF] Mathématiques:devoir maison

[PDF] Mathématiques:devoir maison numéro 5