elementos de la geometría de sólidos-en inglés

ofthe pointP arex =a,y =b, z =c, and the point given bythese equationsmaybe found by the following construction : Measure on OX the distance OA =a, and through A draw the plane PNAR and

Irving Strin^han

*th.

f

NOTES ON

INpreparing these Notes I have used the treatisesof Gregory,

Hymers, Salmon, Frost and Wostenholme, Bourdon, Sonnet et

Frontera, Joachimsthal-Hesse, and Fort und Schlomilch

NOTES ON SOLID GEOMETRY.

and O2the plane xOz; Oy and Oz the planeyOz And the position of the point Pwith reference to the origin Ois determined by

itsdistances PM, PN, PR from thezOy, zOx, xOyrespectively, thesedistances being measured on lines parallel to the axesCXv, Oy and

Oz respectively. This system of coordinates in space iscalled The

System of Triplanar Coordinates, and the transition to it from theSystem of RectilinearPlane Coordinates is veryeasy We can best

conceive of these three coordinates of P by conceiving O as the

corner of a parallelopipedon ofwhich OA, OB, OC are the edges,and thepoint P is the oppositecorner, so that OPis one diagonal ofthe parallelopipedon

ofthe pointP arex =a,y =b, z =c, and the point given bythese

equationsmaybe found by the following construction : Measure on

OX the distance OA =a, and through A draw the plane PNAR

and draw the plane PMBR parallel to xOz, and finallylay off OC

cand dnuv the plane PMCN parallel to xOy. The intersection

of these three planes is the point Prequired. (Fig. i.)*

3. The threeaxesOr, Oy, Oz are called the axes ofx} y, andzrespectively; the three planes xQy, xOz, and yOz are called the

*.

*ForFigures seePlates I.andII atendof book.

814023

4 NOTES ON SOLID GEOMETRY.

planes xy, xz and yz respectively The point whose equations are

x a, x =b, x =cis called thepoint (a, b, c).

4 The coordinate planes produced indefinitely form eight solid

angles about the point O. As in plane coordinates the axes Ox

and O>.cUvicieUtte plane considered into four compartments, so in

space coordinates the planes xy xz and yz divide the space

con-slcjttfed ia.;o eightCompartments four above the plane xy, viz :

Q-xyz, Q-xyz, Q-x yz, Q-xyz; and four below it, viz : Q-xyz,

Q-xyz, O-xf

y

f

z, Q-x yz. By an easy extension of the rule of

signs laid down in Plane Coordinate Geometry, we regard all xs

on the right ofthe planeyzas + and on the leftof yzas ; allys

in front ofthe planexz as -f- and thosebehind itas ; allzs above

the planexyas + and those below itas . Wecan then write thepointswhose distances from the coordinate planesare a, bandcinthe eight different angles thus:

In the first Octant, Q-xyz Pl is

(a, b, c)

In the second Octant, P2 is (a, b, c)

In the thirdOctant, P3 is

( a, b, c]

In the fourth Octant, P4 is ( a, b, c)

In the fifth Octant, P5 is

(a, b, c)

In the sixth Octant, P6 is (a, b, c]

In the seventh Octant, I\is

( a, b, c)

In the eighth Octant, P8 is

( a, b, c).

The signs thus tell us in which compartment the point falls,

and thelengths of a, b and c give us itsposition in thesecompart

ments

1. Construct the points I, 2, 3 ; o, i, 2

; 0,0, i

; 4, o, 3.

2. Construct the points i, 3, 4 ; 2, 3, o; 3, o, i ; 2,o, o.

5. The points M, N and R are called the projections of P on thethree coordinate planes, and when the axes are rectangular they are

itsorthogonal projections Wewill treat mainly of orthogonal pro

we are tobe understood to mean orthogonal projections, unlesswe

We will give now some other properties of orthogonalprojections

NOTES ON SOLID GEOMETRY.

5

6 DEFINITIONS

The projection of a line on a plane is the line containing theprojections ofitspoints on the plane

When oneline or several lines connected together enclosea plane

projection ofthe first area

The idea ofprojection may be in the case of the straight linethusextended: iffrom the extremities ofany limited straight linewe draw

perpendicularstoa second line, the portion of the latterinterceptedbetween the feetof the perpendiculars is calledthe projection ofthelimited line on the second line

From this wesee that OA, OB and OC (coordinates rectangular)

arethe projections ofOP on the three axes, ortherectangular coordi

coordinate axes

7. FUNDAMENTAL THEOREMS.

I. Thelength ofthe projection ofa finite right line on anyplaneis

equaltotheline multipliedbythe cosine oftheangle which it makeswith

Let PQ be the given finite straight line,xOy the plane of pro

jection; draw PM, QNperpendicularto it

; then MN is the projection ofPQ on the plane Nowthe angle made by PQwiththe plane

to MN meeting PM in R, then QR = MN, and the angle PQR

PQR. (Fig 2.)

to theplane ofprojection. The area ofABC = - BC x AD, and thearea ofthe projectionA B C = - B C x AD. But B C = BC and

A D = AD ccs ADM Moreover ADM = the angle between the

planes Hence A B C= ABC xcos anglebetweenthe planes (Fig.3.)

2 Nexttake a triangle ABCof which no one of the sides is parallel tothe plane ofprojection. (Fig.4.)

6 NOTES ON SOLID GEOMETRY.

Through the corner C of the triangle draw CD parallel to theplane of projection meeting AB in D Nowif wecall 6 the anglebetween the planes, then from i A C D = ACD cos 6 and B C D

= ABC cosd.

triangles of each ofwhich the proposition is true it is true alsoof

the polygon, i.e., ofthe sum ofthe triangles.

Also bythe theory of limits, curvilinear areasbeing the limitsofpolygonal areas, the proposition is also true ofthese

equal to thefirst line multiplied by the cosine of the angle between the

lines.

LetPQ bethe given line and MNits projection on the line CXv, by

means ofthe perpendiculars PM and QN Through Q draw QR

PQwith Ox, and MN = QR = PQcos PQR. (Fig 5.)

9. Ifthere be threepoints P, P, P" joinedbythe right lines PP,

PP"and PP", the projections of PP" on any line will be equal to

thesum ofthe projections ofPP and PP" on thatline. Let D, D, D" be the projections of the points P, P, P" on the line AB Then D will either lie between D and D" or D" between D and

D. In the one case DD" = DD + DD" and inthe otherDD"=

DD -D"D = in both cases the algebraicsum of DD and DD".

Theprojection is -f or accordingas the cosine of the angleabove

is -f or .

In general ifthere be any number ofpointsP, P, P", etc., theprojection ofPP" on anyline is equal to the sum of the projections of

PP, PP", etc., or, the projection of any one side of a closed po

lygonal line on a straight line is equal to the sum of the projections

oftheother sideson that line

10 USEFUL PARTICULAR CASE

equalthesum oftheprojections on thatline oftlie coordinatesOM, MN,

NP of the point P For OMNP is a closed broken line, andthe projection of the side OP on a straight line mustbe equal to

the sum of the projections of the sides OM, MN, and NP on that

AZOTES SOLID GEOMETRY. 7

11 DISTANCE BETWEEN Two POINTS

Let P and Q, whoserectangular coordinates are (x,y, z] and (x,

y, z

), bethe two points. (Fig. 6.)

Cos #, cos /?, cos y determine the direction of the linein rectan

gularcoordinates, andarehence called thedirection cosinesofthe line

We usually call thesecosines /, m and respectively. Sothe equation (i) is usually written P +m^+n?= i, (i), and when we wishto

speak ofa linewith referencetoits direction, we maycall itthe line

(/, m, ii). Only twoofthe angles a, ft, ycan be assumed atpleas

ft.

13 We can use thesedirection cosines also fordetermining the

position ofany plane areawith reference tothree rectangular coordi

nate planes. For since any two planes makewith each other thesame anglewhich is made by two linesperpendiculartothem respec

tively, the angles made by a plane with the rectangular coordinate

planesare theangles made bya perpendicular tothe plane with the

coordinate axes respectively. Thus ifOPbe the perpendicular to aplane, theangle made by the plane with the plane xy is theangley;

with xz is the angle ft; and withyz isthe angle a Socos a, cos

cos are called also the of a plane That

g iVOTES ON SOLID GEOMETRY.

14 The relation cos2a + cos2

ft+cos2

y = i enables us toprove

an important property ofthe orthogonal projections of plane areas

For let A be anyplane area, and Ax , Ay>

Az its projections on the

coordinate planesyz, zx and ;ry respectively Then Art 7, II., Az

= Acos a; Ay= A cosft; Aa= A cos y.

Squaring and adding we have

AZ

2

+ A,,

2+A.9= A2

15 To FIND THE COSINE OFTHE ANGLES BETWEEN Two LlNES IN

TERMS OF THEIR DIRECTION COSINES (cos <*, cosft, cos y)

AND (cosa, cos ft, cos y).

Draw OP, OQ throughthe origin parallel respectivelytothe given

andtheanglePOQwillbetheanglebetweenthegivenlines (Fig. 7.)

NOTES ON SOLID GEOMETRY g

//+ mm + nri= o (3). (3) is called the condition of perpendicu

Let P (xyz) and Q (xyz) be the two points. (Fig. 8.)

axes respectively a, ft and y. Project the broken line PMNQ on

PQ Thisprojection is equal to TQitself. Hence wewill have

PQ = PM cos a + MNcos ft + NQcos y. (a]

Nowproject the broken line PMNQ on theaxesxyzrespectively

We obtain thus the threeequations

PQcosa = PM + MNcosA.+ NQ cos/*

j

PQcosft= PM cosA + MN + NQcosv V

(b)

PQcos y PMcos/* + MNcos v + NQ)

Nowmultiply the first of equations (3) by PM, thesecond by MN and the thirdby NQ and add them taking (a) into account and we

have

+ MN2

+ NQ2+ 2PM.MN cos A + 2PM.NQcos

/* + 2MN,NQcos v (c)

or PQ2=

(X - X J + (y -/) + (z- zY+ 2(x - x

) (y -y)

Cor Ifoneof the points as Qbe atthe origin then

PO2

x*+y* + z* -f 2xycosA+ 2x2cosjn + 2zycos v (6)

17 Direction Ratios In oblique coordinatesthe position of a line

IO NOTES ON SOLID GEOMETRY.

taking care to note that \ve are using oblique coordinates and call

the line PQ, the line

(/, ;//, n} To find a relation among thesedirection ratios, we divide equation (c) Art 16, by PQ2

. We thushave

i =I 2

+ m* +;/

2+ 2//T2 cosA + 2ln cos /*+ 2mn cos v, (7) thedesired relation

1 8. The coordinates of the point (xyz) dividingin theration:n

the distances between the two points (x yz

1 ) (x"y"z") are

The proof of this is precisely the same asthat for thecorrespond

ing theorem in Plane Coordinate Geometry.

O0 makes with thefixed line Oxin that plane (Fig 9.)

We haveOC =rsin6. Hence the formulaefor transforming fromrectangular to these polarcoordinates are

Conceivea sphere described from the centre O, with a radius = a

and let this represent the earth. Then, if the plane zOx be the

plane ofthe first meridian and the axis of z the axis of the earth,

NOTES ON SOLID GEOMETRY. Ir

20 Distance between two points inspace in polarcoordinates

Let Pbe (rr

6,

q> } and Q(r, 6, <p). ProjectPQ onthe planexy,

MN is this projection, draw OM and ON the projections of OP and

OQ respectivelyon that plane Through P draw PRparallel to MN,then PR = MN. (Fig. 10.)

the other two are given Let these bex and j. So the equation

may bewritten

in which we may attribute arbitrary and independentvaluestox and

y Andtoevery pairofsuch values thereisadeterminate pointinthe

plane xy; and ifthrougheach ofthese pointswe drawaline parallel

to the axis ofz, and takeon it lengths equalto the values ofzgiven

bythe equation, it is clear that in this way we will get a series of

points the locus ofwhich is a surface,.and not a solid since we take

determinate lengths on each of the lines drawn parallel toz. Hence

F (x, y, z) = orepresents a surface in triplanar coordinates.

22 Ifthe equation contains onlytwovariables asF(x,y)= o then

it representsa cylindrical surface.

pendently of0, and x and yare no longer arbitrarybutoneisgiven

in terms ofthe other; to each pairof values correspondsapoint in

the plane xy, and the locus of these points is acurve in that plane

If through each pointin this curve we draw a coordinate parallel to

2, everypointin thatcoordinate has the same coordinatesxandj/asthe point inwhich it meetstheplane xy Hence F(x,y) = o repre

sents a surface which is the locus of straight linesdrawn throughpoints of the curve F(.r, v) = o in the plane xy and parallel to the

NOTES ON SOLID GEOMETRY. 13

axis of z. This locus is either what is called a cylindrical surface

as the equation F(x,y) oin the planexy represents acurve or astraightline.

Forexample, x1

4-y* r

2= o in rectangular coordinates is a

right cylinder wiih circularbase in plane xy (sincejva+y =r*isa

And ax +by c= o isa planeparallel to theaxis ofg, intersecting the planexy in thelineax + by= c.

SimilarlyF(x, z) = o represents either a cylindrical surfacewith

.

tothe axis ofx ora plane parallel to this axis

23 An equation containing a single variable representsa plane orplanes parallel to one ofthe coordinateplanes.

Thus x = a representsa plane parallelto the planejyz

Andas

_/"(.#)= o when solved will give a determinate numberof

values of x, as x = a, x I, x =c, etc., so it represents severalplanes parallel tothe coordinate planers

Thus also F(jy) = orepresents a numberof planes parallel to the

planexz.

And F(z)=o, a numberofplanes parallel to xy.

24 Thus we see that in all caseswhen a single equation is interpreted it represents a surfaceofsome kind or other

Theapparent exceptions to this are those singleequationswhich

z=o This equation can only be

z = o, or x =a,

y b,z =c.

Now these represent three planes, but being simultaneous they

represent thepoint a, b, c.

25 In general two simultaneous equations as

14 NOTES ON SOLID GEOMETRY.

represent a curve or curves, the intersections of the two surfaces

represented by the two equations.

Thus _ 7 r taken simultaneouslywehave seenrepresentastraight

ders F(x,y) = o and F(x, z) =o, ec., etc.

26 Three simultaneous equations

represent points which can be found by

solving the three equationswhich themselves represent different sur

faces

Interpretationof PolarEquations

27 i. r=a represents a sphere having the pole for its centre

Hence the equation (r) = o which gives values for r as r a,

r =b, r c, etc., represents aseriesof concentric spheres aboutthe

as centre

NOTES ON SOLID GEOMETRY.

jcj

2. 6= a represents a cone of revolution about the axis ofzwith

its vertex at the origin ofwhich thevertical angle is equal to 2a

Hence the equation F(6) = o giving values 6= a, 6 = ft, etc.,

represents aseries of cones about the axis ofz havingthe originfor

3. cp = (3 represents a plane containing the axis ofzwhose line

ofintersection with the planexy makes an angle/3 with the axis of

x Hence the equation F(cp) o which gives values q)=(3, cp

=ft, etc., represents several planes containingthe axis ofzinclined

to the plane zOxat anglesft, ft , etc.

4. If the equation involve only rand 6 as F(r, 6) =o, since

F(r, 6) = ogives the same relation between rand 6 forany value

ofcp, ii gives thesame curve inany one ofthe planes determined by

assigning values to (p. Hence it represents a surface of revolutiontraced bythis curve revolvingabout the axis ofz.

Example, r= a cos 6 is the equation of a circle in the plane

xz, or inany plane containing the axis of z. Hence r a cos 6represens a sphere described by revolving this circleabout the axisof*

5. If the equation be F(<p, 0) o for everyvalue of cp thereare one or more values of 6 corresponding to which lines through

the poe may be drawn, andas

cpchanges or the plane fixed by it

containing Oz revolves, these lines take new positions in each

new position of the plane, and thus generate conical surfaces

(a conical surface being any surface generated by a straight line

moving in any manner about a fixed straight line which it inter

6. Ifthe equation be F(r, (p) =o, forevery valueof(pthere areone ormorevalues of r, thus giving several concentriccirclesabout

the polein the plane determined bythe assigned value ofcp. As (p

changes, or the plane through Ozrevolves these values ofr change,

and the concentric circles vary in magnitude The equation thus

represents a surface generated by circleshaving their centresat thepole, whichvary in magnitude as their planesrevolve about theaxis

ofz which theyall contain

7. Ifthe equation be F (r, 6, cp) = o, it represents a surface ingeneral Forif weassign a value to <p

as cp= ft, then F(r, 8, ft)

= owill represent a curvein the plane (p = ft. And as cpchanges

orthe plane revolves about Ozthiscurve changes, and the equation

will represent the surfacecontainingall these curves

!6 NOTES ON SOLID GEOMETRY.

28 Two simultaneous equations in polar coordinates representa

neousequationsrepresent apointor points the intersectionsofthreesurfaces

CHAPTER III.

29 Tofindequation ofa plane in terms oftheperpendicularfromthe

Let OD p be the perpendicular from the origin on the plane,and let itmake with theaxes O,r, Oy and Qzthe angles a, ft and y

respectively Let OP be the radius vector of anypoint P of the

The projection ofOM + MN + NP on OD is equal to the projection ofOP on OD.

The projection ofOP on OD is OD itself, and the projection of

OM + MN + NP on ODisx cos a +ycos ft + z cosy.

Hence we have xcos a +y cosft + z cos y p (12)

30 To find the equation ofa plane in terms ofits

Let the intercepts be OA a, OB =b, OC = c. The equation

(12) may be written

pseca psecft psecy

ButsinceODA, ODB and ODCare right-angled triangles, wehave

psec a = OA a, psec ft= OB = b, psec y = OC =c.

Thereforethe equation becomes

theequation oftheplanein terms ofitsintercepts

18 NOTES ON SOLID GEOMETRY.

31 AnyequationAx + By + Cz = D (14) ofthefirstdegree inx,

y andz istheequationofaplane*

For we may write (14)

Hence (14) is the equation of a plane in oblique or rectangularcoordinates

Hence to find the intercepts of a plane given by its equation onthe coordinate axes, we either put it in the form (13) or simply

raakej> = o and z= o to find intercepton x; z = o and x = o to

find intercept onj/; x= oandjy o to find intercept onz.

"Example. Findthe intercepts ofthe plane 2x + $y 52 = 60

32 Itis useful often to reduce the equation AJC + By + Cz = D

to theform xcos a + ycosft + zcos y =/ in rectangular coordi

nates We derivea rule forthis.

Since both of these equationsare torepresent the same plane, we

""

Hence cos <* =

+ B2+ C2

NOTES ON SOLID GEOMETRY ig

Hence the Rule: If we divide each term ofthe equation Ax + By

=D, bythe square rootofthe sum ofthesquaresofthe

coefficientsofx, y andz, thenew coefficientswill bethe direction cosinesofthe perpendicularto theplanefrom theorigin, andtheabsolute termwill be thelength ofthisperpendicular. Givetheradicalthesign of D.

Example Find the direction cosines of the plane 2x + $y 40

= 6 and the length ofthe perpendicular from theorigin.

Ifthe planesare in the form

xcos a + y cos ft + zcos y = p

then sincethisangle isequal totheangle oftwoperpendicularsfromorigin on the planes the cosine will be(Art.15) cosV= cos acos a

+cos /3 cos/3 -f cos y cos y

+ B2+ C2

2Q NOTES ON SOLID GEOMETRY.

Cor i. Iftheplanesareperpendiculartoeachother, then cosV=o.

ofx,y andzin thetwoequations shallbeproportional

Ex i. Findthe anglebetween theplanes

2. Show that theplanes

dicular to each other

3. Writethe equation representing planes parallelto theplane 3*

34 Tofindthe expressionforthe distancefrom apoint P(xyz)toa

plane (coordinates rectangular)

i. Letthe equation oftheplane be of the form

xcos a + ycosft + z cosy = / when p OD.

Pass a plane through P parallel to the given plane,and produce

OD to meetitin D.

Theequation ofthis plane willbe

Nowlet PM bethe perpendicularfrom P on the given plane

Then PM = OD - OD = 1

-NOTES ON SOLID GEOMETRY. 21

Hence PM = x cos a 4-y cos/3 + z cos y p

And (x1

cosa + y cos /? + cos y p) (20)istheexpression

for the perpendicular fromthe point x yz on the planexcosor +_y

cos ft + cos y =p, the sign being + or accordingas P is orisnoton the side ofthe plane remote from the origin

2. Let the planebe in theform Kx + By + Cz =D

A

V A2+ Bs+C*

Hence theexpression

(ycos -FJF cos /3+ z cos y p] becomes

35 The equation of the plane in the form xcosa + ycos(3

4-cos y = p may be used to demonstrate the following theorem in

projecions

7% volumeofthe tetrahedronwhich has theoriginforitsvertex and

equaltothe threepyramids which have any

point (x, y, z) in theplaneABC fortheircommonvertex and for basesthe

respectively.

ForletAbe thearea of the triangle ABC and

theequation ofits plane

Multiplythis equationby A

Then Acos a.x -f A cos /3.y + A cos y. z = Ap

or -JAcos a. x + ^Acos ft.y + -JA cos y. s

But A cos a, Acos/3, Acosy, are the projections of A on theplanes yz, xz,and xyrespectively, and x,y and z arethe altitudes of

thetetrahedrons which havethese projections as bases and the point

22 NOTES ON SOLID GEOMETRY.

which has the origin forvertex and A forbase Hence the theorem

Calling these projectionsAx , Ay, and Ax , we may writethe equation ofthe planeA^v + Ayy + A^ = 3V. (22)

36 Tofindthepolar equation ofaplane.

Let OP = r, POS = 6, P OM = cpbe the polar coordinates of apoint Pofthe plane (Fig 12.)

=jff, and POD = GO.

Then

yyp

express GJ in polar coordinates conceive a sphere about O ascentre

with OP = ras radius Prolong OD to D" on the sphere. Draw

the arcs ofgreatcirclesSPP, SD"D, MP D and D"P.

The triangleSD"Phas for its sidesSD" = a, SP = 6, D"P= GJ

and angle D"SP= D OP =/3

<p. But

cos D"P =cos SD" cos SP + sin SD"sin SPcos D"SP.

Or

cos GO= cos acos6 + sin asin 6 cos (fi cp).

Therefore = cos a cos 6+ sin a sin 6 cos(f3 cp) (23)is the

polar equation ofthe plane

37 The general equation of the plane Ax + By4- Cz = D may

be reduced to the form

y + C2= i

(24) by dividing bythe absolute term D

And also tothe form

z mx + ny+ c (25) by dividing by C transposingand putting

=m, = n and-^= c. These two forms are very useful in

V^/ V^ v_/

the solution of problems and in finding the equations of the plane

undergiven conditions

Plane tinder Given Conditions

38 i The equation of aplane through the originwill be of thegeneral form Ax + By + Cz = o, forihe equation mustbesatisfied

2 The which contains the axis of z is ofthe

NOTES SOLID GEOMETRY,form Ax + By = o ; a plane containing the axis ofy is Ax + Gar

o; one containing the axis ofx is By + Cz =o

3 The equation of a plane parallel to the axis of z is Ax + By

= D; ofoneparallel to the axis ofjy is AJI* + Cz = D ; oneparallel

to the axis ofxisBy + Cz = D

4 The equation of a plane parallel to the planej/s is A^t: = D;

These equationswehavehadalreadyin theformsx= a,y 3,

39 Tofindtheequation ofa plane containing a given point (a, b, c)

andparallelto agivenplane Ax-fBy + Cz =D (i

)

coordinates (a, b, c) must satisfy (2). Therefore Aa +~B&+ Cc =D.

Hence bysubtraction weeliminate D and obtain

Ax + By + C0 = Aa + B3 + O (26)

therequired equation.

Example Find the equation of the plane passing through thepoint (i, 2, 4) parallel tothe plane 2x + 4y 32= 6.

40 Tofindtheequation ofa plane passing throughthreegiven points(x, y, z

), (x", y", z") and(x

", y",

z

").

Letthe equation ofthe plane be of the form Ax + By + Cz = i,

A, B and C tobe determined by thegivenconditions

Since the plane is tocontain each of the points, we must have

NOTES ON SOLID GEOMETRY.

in theseequationsare the double areas of trianglesin the planesjy0,

xz and xy respectively. Moreoverthese triangles are the projections

of the triangle of the threegiven points, on these planes. Hence

comparingthis equation wih the equation (22)

we see that x ,y ,ss

x">y", 2

,z = 3V

=6V That is =6 timesthe volumeofthe

pyramid which hasthe origin forvertex and the triangleofthe three

givenpoints forbase This equation fullywrittenout is

41 Tofindtheequationoftheplaneswhich contain thelineofintersec

tion ofthetwoplanesAx + By + Cz = D and Ax 4- By + Cz = D.

ThisequationisA# + By + Gs D + K(Ajc+B>+ C 0-D/

)=o(29)

particular value and it is satisfied when A^ + By +C-s: D = o and

isa plane containing their line ofintersection Henceas Kis arbitrary it

(24) represents the planes containing the lineofintersection

ofthe two given planes.

42 When the

identity KU + K^Uj + KaUs = o (30) exists between

the equations U=o, U2 o, U3= o ofthreeplanes, then these planes

easy corollary of Article 41 Also when the equation of the firstdegree in x,y and z contains a single arbitrary constant all the

planeswhich it expressesbyassigning particular values to this constantintersect each other in one and the samestraight line This

line of intersectionmay be at infinityand then the planes are all

parallel

Example I. The planes represented by the equation 6x+T&y +2z

= 3 (M arbitrary) all contain the line of intersection of the two

NOTES ON SOLID GEOMETRY 25

Example 2. The planes represented by 2x + ^y 42= n (n

arbitrary) are parallel

Example The planes $x + \v + 60 = 2 }

KU + K1U1+ K2Ua-hK3U3= o (31) ^/j/j, then thesefour planes

dinates which satisfythe first three U = o, U, = o and U2= o will

44 Example i. Find the equation of the plane passing throughthe origin and containing the line of intersectionof the two planes

Ax + By + C0 = i and A x + B> + Cz = i.

forall the planes containingthe lineof intersection ofthe two givenplanes But as the required plane must contain the origin, the

equation mustbe satisfiedby(o, o, o) Hence we have i K=o.

Ex 2. On the three axes of x,y and z take OA = a, OB b,

OC = cand constructon these a parallelopipedon havingMP as the

edge opposite parallel to OC, and AR in the plane xz the edge

opposite and parallel to BN.

Find the equation to the plane containingthe three points M, N

26 A OTES ON SOLID GEOMETRY.

condition that thisplane shall pass through the pont M (a, b, o).

the points P, B, and C, in the same figure.

Result,

45 Iftwogiven planes be in the normal form as

xcos a+> cosft+z cos y=p and.v cos a +y cos /3 + zcos y = p.

The plane containingtheir line ofintersection is

xcos a+y cos/3+zcos y p-\-K(x cos a +ycos ft +zcos y

-/)=o

xcos a-\-y cos /3+z cos y p (x cos a + ycosft +zcos ;/

-/>)= o which represents ihe two plane bisectorsofthe supplementary angles

made by thegiven planes.

That is tofindtheequationsto theplanebisectorsofthe supplementaryanglesmadebytwo given planes, put their equations in thenormal form andthenadd andsubtract them

Example Find the two planes which bisect the supplementary

angles made by the planes 2.v+3.y Vz= 5 and 3^ + 4^23=

4-Result,

Remark If we place A = xcos (Y+ ycrsft + z cosy p and

A = x cos a + v cos /? + cos y />

.

Then A A =o is the plane bisector of one of the angles be

tween the planes A and A and A -f A = o is that of the supplementaryangle

acommon Let A = A = o and =o be three

NOTES ON SOLID GEOMETRY.

planesinthenormalform,andletthe originbewithin thetriedralangle

formed bythe three ofwhich Ptheirpointof intersection isthe vertex

Then the plane bisectors of the angles made by these planes is

A A o, A" A o, A A"=o. Andas these when addedtogether vanish simultaneously, it follows that these three planes

Wecangivethistheorem another formbyconceivingasphere tobedescribed about the vertex of the triangularpyramidas a centre The

three planes A= o, A = o, A"=o cut the surface of the sphere in

arcs of great circles which form a spherical triangle and the three

planesA A =o, A" A = o and A A"= o cut the sphere in

three arcs of great circles which bisect the angles of thissphericaltriangleand theircommon line ofintersection pierces the sphere in

bisecttheangles ofa spherical triangle cut each other in the samepoint

(the pole ofthe inscribedcircle of thetriangle).

A", B", C" A", B", C" A", B", C"

Hencethe condition thatone of these shall be parallel to the line

of intersection of the other two, or that the planesshall notmeet in

28 NOTES ON SOLID GEOMETRY.

49 We have seen that the equations of two planes Ajr

D = o and A.x+ Bjy+ Cz D = o added together oneorboth

ofthem multiplied by any number give the equation of a plane

which contains the lineof intersection ofthe two given planes. If

we combine these two equationsso as toeliminate x we shall obtain

a planeparallel to the axis of A\ containingthis line of intersection

Ifweeliminatey weobtain a plane parallel to the axis ofycontain

CHAPTER IV.

50 The equations ofany two planes taken simultaneously represent

rePresent a straiSht lmethe c

-ordinatesofevery point ofwhichwill satisfy the two equations.

If we eliminate alternately x and y between these equations we

obtain equations ofthe form

^ planes parallel respectively to the

y ~ nz +

axesOy and Ox which represent thesamestraight line as equations

(35) These non-symmetrical forms (35) are very useful The

planesx= mz +a,y =nz+b are called the projecting planes of the

line on the planes of xz andjar, andthese equations are also the

equations of the projections of the line on those planesrespectively.

tion ofthe projection ofthe lineon the coordinate planexy

The equations (35) of the straight linecontain four arbitrary con

paringthese equations with the equationy mx +b in plane coor

dinate geometry.

Theequations(35) maybe thrown in the form

^_^ =z _^ =,

m n i

which gives us an easychoice of fixing the line bythe equations

ofany twoofitsprojecting planes

51 Tofindtheequationsofastraightline intermsofitsdirectioncosinesandthe coordinates a, b, c ofa pointon the line: (axis rectangular.}

30 A OTES ON SOLID GEOMETRY.

Leta, ft, y betheanglesmade bythe linewiththe coordinate axes

respectively Let / be the portion of the line between any point

(x,y, z) on the lineand the point(a, b,

c). Then /cos a =a- a ;

(37)

Thisform (37) of the equation of a straight line is symmetrical

and is therefore very useful. It contains six constantsbut in reality

only four independent constants, since the relation cos2 o+cos2

ft 4-cos2

y = i holds, and of the three a, b, c one of them may be

assumedat will, leavingonly two independent.

We have seen that the equation (35) may be thrown intothe form(37) So also

(37) may be thrown into the form (35) by finding

from them expressions forjyand x in terms ofz.

Ifthe equations be in theform

direction cosinesofthe line

NOTES ON SOLID GEOMETRY.

given bytheequa

tions

xa yb z c , x a vb zc

and ri =:

We have shown (Art. 15)

cos V = cos a cos a -f cos /3cos /? + cos ^ cos y.

32 NOTES ON SOLID GEOMETRY.

Theseequations may be put in the forms

54 The condition of perpendicularity of two lines given bythe

equations in last article is LL + MM + NN ^ o. (42)

The condition that they shall be parallel (see Art 15)

Thisis derived byeliminating x,y and z from the four equations

Subtracting the third fromthe first we have o (m m }z+pf>

NOTES ON SOLID GEOMETRY 33

the elimination ofx,y and z can be effected more readily by writ

ing (i)= K and (2) = K.

56 The equations ofa straight line parallel to one of the coordinate planesas xyare z =c, v= mx-\-p.

The equations of a straight line parallel to one of thecoordinate

a single point

58 Tofind theequations ofa straight linepassing through twogiven

~ "

34 NOTES ON SOLID GEOMETRY.

The required lineby the first condition will be ofthe form

whereL, M, and Nare tobe determined by the conditions

= o