-/ VI DESCRTPTTVE GEOMETRy/,^,

] )ecafr.#iryo*, n'Nl_2\ \'/(wIOI.. ,/\,I ^-- rr

I E rgeering Stqdents' " , / v

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SY 1999 - 2000 Edition

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CITED SOIIRCES / REFERENCES

l. Descriptive Geomehy (krsbuctional Pamphlet). pt:PI"d- by. the Department ofMechanics, United States Air Force Academy, Coiorado, dsat June 1958' (Unpublished)

2. Lecture Notes Taken During Engineering Drawing 228 (Dept of Mech) classes at the US' Air Force Academy, Fall Term of 1964' (Unpublished)

3. Solid Geomeby. Revised ed- F. Eugene seymour and PaDl smith, MacMillan company'Nerv York: 1959. (ISBN 971-103-168-x)

. 4. Student,s Classroom Exercise Plates in Engineering Drawing at U.P' Diliman, School-Year1995-1996.

5. Technical Drmving. 9h ed. Frederick E. Giesecke et al, MacMillan Publishing Company'New York: 1991. (ISBN 971-1055-58-9)

DOCUME,NT ATION TERMIN OLOGY

1. Source - Minor or no change from the cited material

2. ReGrence - Major or very significant alteration of the cited material

3. New (or no doctrmentation cited) - Material is original, or is based on all citedreferences

DESCRTPTTVE GEOMETRYond

TECHNTCAL DRAWINGfor

Enginee?ing StudentsIIt!

&_r'

RI Compiled, Arranged and Annotatedi ,by

ROBERTO S. BARANGANB.S. Engineering Science (1966), USAI Academy, USA

Lecfu rer, PAF Colle ge of Arronautic s, I97 7 - 197 8Senior Lecfurer, University of the Philippines Dilim'. College of Engineerin g 1995-1997

and

ROBIN M. BARANGANB.S. Computer Engineering (1993X University of San Carlos, Cebu City

Insbuctor, Asiaa College of Technolory Masters ia Coryutcr Science Prograrn, 1996ceteacher, university of san carlos Masters in Eosineering program, lgg6-lggT

Faculty Chairman & Computer Insbuctor, Divine Mercy Comprter Collegg 1997I.T. Coordinator & Computer Instructor, Systeos Technology Instibrte, l99g

1,*

lu

ACKNOWLEDGEMENTSThis booklet cvolvcd initially from a compilation of lccture manuscripts

used by onc of the authors

in conducting crasscs in Enginecring D**id;iin oilirrl an"t u lirftce number of copies of the first

edirion (199G19971** *in" ,uuiiubl" to G? rip-fngineering Pi*int m9""T' very helpful zuggestionswere received by the aulhors from the up e"gir,*ri"i.q"*q pcffient facutty

which led to significant

*pr"*,o*O fi'i"g i"-rp"*t l i" tbe zubfruent editions of lhis booklet'

The authors also wish to acknowledge the temcndous value of the lectures and "poop sheets''

given to the first+ited author by rhe f*"b';ithi rtaon*i-.t' genarrnelt at Jhe Yt oit Force Academy

during his undergradualc (ca.de0 y*r, Jf,if" *ing up tngin-eering Drawing thereat in 1964' The

methods and techniques of instruction u*n W tho;faculty members have a very strong bearing in the

selection and arrangement of the zubject rr*r rn., in this uoott"t, despite those rong intervening years'

chapters 4 lo 7 are dedicated to these people. For chapte r 'z

ia 3' as well as lhe Appendices'

TECHNTCAL DRAWTNG by Frederick ol#i".rrr (r;';ittioni*o soiio ceometry by seymour and

i*irft @evised edition) were used as the primary reference '

DuringtheschoolYear|997_1998,&coPyofthg.draftofthisbookletwassentiJotheDearrofEngineering of each of the eight major universities and colleges i" lut"tto cebu' with a request

for their

cvartration and rccommcndations so es to makc the u."n.t *it"ule for thcir respcctivc engincering

drawing subjects. Almost without exception, ea.ch fean endorsed his copy to their Architecture

Deparhnent which handles engineering a**ing zubjttts in toi itUu engineering schools' Except for the

University of San cJo, and tie Uoiu"rrity oiifr. Vi*yri.*t o. p.uiO.A r"elevant L-mments' the rest of the

schoors pcrceive the book as having ritu" "ir""*..

or appricability to ttr"ir current engineering curriculum'

I-astly, we 8I proud of thc invaluable contribution of our youngst daughter (of the first{ited

author) and younger sistcr (of ffr" ,..o.ri*if"O author)' Tarah Atd in fhe computcr-processing of the

original mantrscriptand its subsequentcditions' f1l\'..'- t)^w{d*-- RS Barangadb^fu P""rc^r\pJRMBarangan 0

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FOREWORI)This booklct was initialty madc in orqgrlo providc cnginccring &rwing shrdsnb at Ilp Dilimm

witi a singlc rcfc-rcncc matcrial drat contain,s all_&g lasic conccpts and[rr.inciptciUat tc sflrdcn6 nccd in-ordcr- to accomplish_ th9 rcgrhcmcutl of trc frcshan cnginiaing O"r*ltg cunicul'nr- lo fr.f U*booklct is a surnmn'izcd collcction of trc rescrchcd lccturJnotes ricd in cliss presentatiors e,rrioi ttiSchool-Year 1995'1996.

^During fte dweloprned{ arohrtion of tris boo\ sevcral materiels tryonj tenccds-of frc cnginccring frcshmcn wcrc latsr addcd in ordcr to hc$ offrcr cnginccring &awing io'rtuCo*in their own presentations, per zuggcstion of thc insructors themsjbcs.

This cdition 1o^ntainl proccdurcs in thc conskuction of two-dimsnsional gcomctic figurcs(Chsptef 2); review 9f the principles and p,roce&nes in drawing Orrhographic projeciion, tCtmpi&lj;chracteristics md relationships of lines in tlree dimensions (chapter q); ch-aracterisiics md ritationsnipiof plancs/surfaccs in thrcc dimsnsions (Chaptcr 5); gcomcbic consFuctions in tbrcc dimsnsions (Chrpi:;6)' iltusrative problem-solving tccbniques (Cfaqter ?); tcchniguel mg procedses in making tni sin'gf.-shokc, vsrtical lcttcring (Appcndix A); thc basic rcguhemcnts in dimcnsioning cnginccriig e"*fig'(4pp*dit B); and sclcctcd two-dimcnsional tangcncics and tangenry-bascd Jonsfrrction iroccdure"s.Claptcr 1, on the other hand,

-gv:t thc scope of DESCRIPTfvE boMEIRy ar a ,ubi.cf anddiffcrcntiatcs it from othsr rclatcd subjccts. For cxarnplc, whcrcas Analytic Gcomcty malpcs fcomericproblcms tt[ough algcbraic calculations, Dcscriptivc Gcomctry dctErmincs acfiral mcnstna]tion andprojective properties of objects through orthographic projections, or precise drawinp of thesc pro;..iio*.Conccpts and proccdurcs c4plaincd in ftis book arc vcry basic and cncapzutatcd but arc csscntialh 1.. graphical sohrtion t9 Tarry levels of engineering probftms. Since enfineering aawint ls n*graphical mcthod of prccisety stating cngincering facts, through thc application oiOfSCnfoffVfGEOMETRY, the concepts and procedures presented in &is book are, ttr-"rrfore, basicalty useful anJ

lelevant io t* production of cngineering. drawings through Computer-Aided Drafting and Desig'(CADD), and in othcr computcr graphic applications.

TABLE OF CONTENTSTTTLE

INTRODUCTION......Abbreviations and SYnbols

PLA}.IE GEOME1RIC C ONS]}.U CTION SDividing Lines/Anglee Into Equal Ptrts """"'Drawing Parallel and Perpendiculr Li99s" " " "Constru"c ti on o f Tri angltt, S quurt u md Re ctangl"s" " " " "'Drawing Tangent Lines to CirclesDrzwing Circles krside/Outside Potygons " " " " " "Conskuction of Ellipses ---------"""

PROJECTIONS USED IN ENGINEERING DRAWINGIso mebi c Proj ri cti ons/DrawingsMulti-View Proj ecti onslDrawinPMissing Viewsll ines ....-.-- '.---. 'Auxilia'y Views """"":""""'

PAGECHAPTER

t

IL

I4

59t2161923

u.

IV.

v.

7633394347

Sectional Views

LINES IN SPACE s,,Classification ofl.ines

Je

Prallel,Intersectinf p"tp""aif"lr and Skew Lineg 54

PLANES IN SPACE 6A.Definition ofPlanesRelationships of Lines andPlanes

65

FUNDAMENTAL C ONSTRUCTIONS IN SPACELine Parallel to a Given Line and ffougl a Given Point

'.U,

True-Length View andPoint View of aline t t

Line From , cirr"n poiot rerpendicd-;;; ci*,.n Line .....---. .. 781q

Estsblishment ofPlmes in SPace '',

Plane Parallel to One Line I to Tlvo Skew Lines 8?'o?

EdgB View andTruo Shrye Visw ofPluree oJ

IncatingPiercingpoint ofatitre on aPlane 86

Line Perpendiculr to aPlure 88

PlaaeThougbaPointandPerpendicularts aline 89

Constuction of "

Sofia (Cir*io Cone) 90

PROBLEM-SOLVINCIntersection of Two Planes (Touching aad NON-Touching)True Angle Between a Line and aPtire / Between T\ro Plues """"""""'Snortegtbishnce Between Point-md-LinelPoist-and-Plane

929597

vlt

!f*

III

t

l

Shortest Distance / Eorizontal Line Between Tlvo Skew LinesPlaas Prallel to a Given Plae and Tkough aPoint ................. 102Plane Perpendicular to a Given Plme .... 103Line of IntersectionBetweenPlme aad Solid (Pj'ranid) .......... 104Plane Perpendiailarto aGiveo Line .,....... ............... 106

APPENDIX A (Single-Sboke, Gothic Vertical Letters mdNumerals) ......APPENDIX B @imensioning)APPENDI X C (Two-Di mens ional Tange nci es and Tange ncy-B as e dConshuctions) ...........

AIB1

cl

III.I

I,fr

Yll

A. BASIC STUDENT REQINREMENTS ( CLASSROOM )

1. Drawing Board, Desh or Table2. Drawing orMechanical, Pencils:

a. ZWmlead for initial construction linesb. HB/F lead for final drawing lines

5W6H lead for work requiring extreme accuracyB/2B lead for free-handiketching and lettering

c.d.

aJ .

4.5 .6.7 .8 .

Pencil Erasers: Rubber, or rubberized plasticStraight-edge with millimeter scaleCompassDividerProtractorIrregular, or Frenctr, Curve

COMPUTER-AIDED DRAWING ANDREQUIREMENT

DESIGN(CADD) Nt rNnduM

B.

Irt

Erti -

NDESIRABLE ADDITIONAL EQUIPMENT ( HOME )

l. T-Square2. Triangles: 30" - 60" and 45o ( 4" to 6" sides )3. Metric Triangular Scale4. Erasing Shield

C.

l. Computer ( Minimum Spec: 486 or better ) .2,. Digiiizer/Graphics Tablet, Lightperg Trackball' or loystick3. Monitor: with Raster Scan, 6r Vector Refrestr, Display device4. Dot-Matrix/Laser Printer. OR Plotter5. Alphanumeric KeYboard

vr1l

l-

CHAPTER IINTRODUCTION

Mmy engigeering problemr cm be solved moro easily by grryhical tran by mathematicalsolutions. For example, a sheet mehl prt can bs laid out graphically on a flat surface fairly easily,whereas it would be more difficult (and less &scriptive) to describe the outline of that partmathematically. The clearance between conbol c$les of a machine can be determined anddescribed graphically, and again it might be more difficult (and less easy to visualize) to describetre clearance between tre cables mdhematically. The grryhical solutions to geomehic problems iscall ed DESCRIPTIVE GEOMETRY.

To more ftlly appreciate the relationship behveen Descriptive Geomeb;r and EngineeringDrawing the following Webster's Dctionary definitions are necessry:

Geomeh-v -- That branch of mathematics which investigates relationships, properties andmeasurements of solids, surfaces, lines and angles; be science ftat heats on the properties andrelations of spatial magnitudes; the theory of space and figures in space.

Plane Geomebv -- That branch of geomehy dealing with plane figures.

Solid Geomebv -- The geomeby of solid figrrres.

Analytic Geomebv -- The branch of geomeb-y in which position is indicated by algebraicsymbols, and solutions are obtained by algebraic analysis.

DESCRIPTIVE GEOMETRY -- The theory ofgeomeh-y heated by means of projections,specifically, the theory of projecting an exactly defined body so as to deduce both projectirre andmebical properties from irc projection

Graphics -- The art of making drawings in accordance wi0r mathematical rules anddreories.

Perspective Dra,vinq -- The r-t of pichring objects, or a scene, in such awwy a.s to show.them as they appear to the eye with reference to relative distance or dep$-

Oblique Draving -- The art of picturing objects in such a way as to show drem in bueshape in only one (frontal) view, uftile slanted by 45 degrees on the other two dimensional a,\es.

Isometric Drawing -- A method of drawing figures md maps so that thee dimensions areshown not in perspective, but in their achral meafirements.

Orthonraphic Projection -- A projection in which 6e projection lines are perpendicular toSre plane ofprojection.

Sketches -- Free-hand draring using only pryer and pencil, and uzually re not made toany scale.

Drawin$ -- A term applied generally to all drafting activities, but more specifically tot op u.ilgsrdiory scale, .r difrtroiiated from trose using special scales (or actual projections),or to those done fee-hmd (sketches).

projections -- The representation of fre surfaces of m object on apichre plane- Behindevery dr"*i"g

"f an object iJa space relationship involving fotn imaginary things:

1. The observer's eyes, or tre station point2. The object3. The plane orplanes ofprojection4. The projectors, also called visual rays or lines ofsight

Wtrere the observer is relatively close to the objec! and ttre projectors form a "cone" ofprojectors, Sre resultingprojection is known r".p."rp".tit". If the observeC.s.eye is imagined asinfinitely distant fromlhe o-bject and the plane o-p-iotion, the projectors will be parallel, so theq,'pe ofirojection is lqrown as parallel roiection If the projectors, in addition to being parallel toeach other, are perpendicula'(*"r.D i" th" pt-e of projettion, dre result is an qrhe^sgph!!, orri$t-angle. proj""iion. If fte projectors a-e parallel to each other but oblique to the plane ofprojection, tre result is an oblique projection

It can be said therefme ttrat orthosaphic proiection (multi-view or a:(onomebic) is amethod of representing spatial urr*gr@ descriptive gqomet--v is the methodof interpretrrg (or A"#iUiogl the shapes, sizes and positions of solids, surfaces, lines and angleswhich are so repesented- In-making lhese interpretations, the theorems and concepts of geomebyre applied.

Since engineering drawing is Sre graphical method of precisely stating engineering facts^concerning three dimensional objece o"

"puti"t arangements, it involves the application of

descriptive geomehy.

Just as there 6.e certain findanental theorems of geomeh-y, there are also certainfundamental conshuctions, or processes, of descriptive geomety upon vrhich the subject isdeveloped These re:

1. Consbruction of a line 0nu a point prallel to a given line2. Conskuction of a line tbru a point perpendiculr to a given line3. Constiuction of a line tfrru a point intersectiqg a given line at a specified point4. Establistrnent ofaplane in space5. Consb:uction of a line parallel to agiven plare; aplane parallel to agiven line6. Conskuction of a line perpendicular to a given pi-r; a plane perpendiculr to a given

line7. Detennining the point at uihich a line pierces a plane8. Determining the line of intersection ofhro planes9. Determining the shortest distance beturcen two SIG\M lines10. Detennininl tle shortest HORIZONTAL distance between two SI{EW lines

11. Construction of aplane parallel to agiven plae12. Construttioo of aplme perpendioltar to agiven plaoe13. Determiningthe bue angle betweentwo intenecting linesi+. pttt*inini tne gp rqgle between a line ad aplmeiS. pututtioiol thu angle (dihedral) betweeo two planes16. Establishment of a solid in sPrce

The above processes pimrily require he execrfion of the following graphical operations

uti lizing corrosponding goomohi-c concepts:l. Co*t*tEitiof theNormal (orTirue I-ength) Vi* of aline;7. Consruction of the Point View' orEnd-View' (PU oftre line;3. Cons[uction of 6e Blge View (EV) of aplanel-4. Conshction of the Noimal View (finre Slrye View) of a_plane; md5. Construction of First, second, rrriio, and otirer Auxiliry views, as needed,

in order to

aftain the requirements oftbe problem at hatrd

Thispanrphletcoverssomeofthefundamentalsofdescriptirregeomehyandsomeofthepracticalapplicatioril.;;;;fi ;-q:,gl""::t*:',,i:"'*3"*i;f T.ltr*t*:f *ilHilt ffi'o#iti; ##Jr'ffirobr",o, .* b, readily solved tbrougb manual drar'rdns orthrough corrputor-AidJ;*fting anaDesign (CADD) using softwa-e zuchrdAiltocaD:-----=--\-r{K(JC'- RS. Barangan

l-!"^,R.M. Barang#

July 1996

CPDS/TSEVLOILOSPPPVRP

FRPPRPHRP

T (or PT)TATLTS

ABBREVIATIONS AND SYMBO LSCuttingPlaneDidortd/ForEshorterrd Dirnrrsiorr of obj edEdge View of a planeLine of IntersedionLineof-Sight" cr direction of viewPiercingPoint of a line tlnougfr a plmePoirt View, cr end view, of a lineReference Plarn

Frortal ReferencePlurProfile Reference PlaneHcrizontal Referere e Pl an e

Poirt of TmgencyTn-re Angle cr- Ditre&alTrue Lengtlr of a lineTrue Shape (and Size) of a plane surface

Poir:t in space (also A B. P. Q. X dc )

Inlersection of trpo rcs

Finite end-poirt of a lirIJ Perpendiculr lines

Center line; Axis of rotdion

R P Referenct Plarrc, dirrrrniorc rneasured r:pward

RP Reference Plare, dirrersiorn rrreasured downward

Hidden lircfidersion linetine of Intersectior(t id&n)

Irnagrrary lire;Corsh'udion lirre; Projection line. . Probable line

_____ f,tffJt Point of a line cfierin left-side pcrtion is visible, while right-side is

' - ' Visible lirre of inlersection

L- --J Cut ingptanefasect ionalViess

A.

CHAPTER IIBASIC GEOMETRIC CONSTRUCTIONS

BISECTING AGI\TEN LINE AB:

L Using poirt A as center md any radius R grcdcr thm onehalf of the lengh of Une AB, &aw an rce$cnding to both sides of Une AB'

Z. Using poirt B as center and the srrp raditrs & draw a second rc irtersecting the first ec at Poin! C andat Point D.

3. Draw a line corrrectjngPoints c and D irtersecting Line AB aL Point E (Lirr cD is perpendicular toLine AB).

(Source: Reference 5, Page 122)

B

\q /'\, , , , , \/ '

i*n IiF--t ' \

i_-I---l- B\ /\ r\ , ,X./ b'

1 . UsingPoiri E as crigin, draw another lirre EG making an angle of 30 to 45 de grees with Line EF'

?.. Starti"ng from Poirt L and using any conveniert length lay or:l points l, 2, and 3 on Line EG af equal

dis:nces.3. Draw a line connectingPoint F a-rdPoinl 34 . Draw a fouth line pa.utt.l to lirc 3F, passing tlroudt p9h ?, a-rd intersecting Line EF d. Point 11

5. Draw a fifth line also parallel bo Line if, p.*tit g tt-ugh Poinl l, and inta-secting Lirre EF' alPointIC

(Sorrrce: Relerence 5, Page 125)

\\\ F

II

l+

\ _t \\

c.I . Locate certc (Poirt O) of Arc AB, aod draw e tmgert (LitE lp d' Poirt' B'z. Draw ctrcd li"" AB

"Ja.*-,a ir L poirt c, m"tia ab t-+rl t" ure'half of AB'

3. UsingPoirtCasccr&rurdndi.5eq'.raltotLdistieeC,&as-a."offingturgertLineBFd'PoirtD.

(Source: Rcfercncc 5, Fe$ l4l)F

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k

-v-t ,Y )ecO / /

D

divieion pointr asPoints l, 2. md 3.7. UsingPoirt 1 as ctnter, andradus equal to 1'A draw the Arc AC'3. Arc BC is PracticallY eqal !o AB'(Sor:rcc: Refercnce 5, Page 14l)

x

O - -r-

O - -

\

1, Draw line AB tangert to Arc BD atPoirt B. Divide Urr AB irrto four eqtral segmerls' makjn3 tlre

EAB. DIVIDING AN ANGI,E INTO EQUALPARTS:

I . Usingpoirt A as center end any radius & drsw an rc o.dling Si& AB at Poirt' D and Si& AC at Poirt F.2. using poirt o as ceffcl. and any radius Rl , &as a e econd 8rc asay horn Point A 8nd beturecfi sides AB

sndAC.3 . ;;G tt

"

,'.* rediw Rl md poin! F ar ccr{cr, &aw a third rc that irkr:seds the sccond arc al Poirt G.4. Line AC UiseCs Angle BAC and Arc DF'(Sourcc: Rcfccncc d Page 123)

L Usingpoiri O as center md any radius R, draw an rc cuBing Si& OM *PointN and Side OP alPoinlL.

Z. Cornectingpoirt N Lo Poirt L, divide the resulting chordlirr (Chcd tN) irfo tlrce eqr:al segrrnnts.3. Designstc tlr [wo new points e*ablishcd on Chord LN as Poinl 3 and Poird' T. Corrst Poirts S and T

with Poirt O.4. Line OS and Line OT divide Angle MOP, Ctrcrd I.,}{, and Arc LN into tlree approximatety equal parts.(rd

(z-a)II t J

0)

G.

AC. DEF1INING A CIRCI.E OR AN ARC

ffi poir,tr (poirrs I P,:1 I t*:T n:trK.from each other as possible'

Itsfaousu u[lt el rvq lrys'prit*o chordlirrscorrrctingPoirtAbryk

. . ..

X. Or* p.tp.ndiorlar biuotttt, * and GItr to the chrd line'

3. The irnrrsecri."ortr,.illffi,a;i;;J*,PoirtD, is the ccrrtcrof th rc'

(Sourcc: Rcfcreocc I Pagc l3a)

DRAWDIG A CIRCLE ltlKuu\rl1 rrrrrle r \

l. coru.recl poin! A to point B, 'd poirt B to c. Draw perpendiculr b'ise

AP r c

\ r . /

A J t . / c B

- r l -} t '

I

P*' l

I/ 11 l

R I

B.

'_-_')

Ia

II

II

ft

lb

Il - -

I

IIII

II

IIh

II

---I--Ftt

t l \ .r t' St1 /t ll t

_GAB

hB

BB. DRAWNG PARALLEL LINES:

LINE PASSINC TIIROUGH POINT P A}qD PAIT]\LLEL TO LINE AB:l. UsingPoirtP as center md anyradius R gfealer than ttte digtarce FcrnPoinlP !o LirE AB, draw m rc

crd,irry Lirr AB at Point C.Z. UsingPoirtCascenterandthesarrrradirs&drammothrrcpassingtlroughPoirtPmdo.rtingLine

AB atPoinlD.3. UringPoirtCaeainascerterandthedistarebehreenPointPandPoirt,DrradiusRl,drawathirdrc

cr.dting tlx frst erc d Point F,4. The line cornedingPoirt P to Poirn F is parallel to Une AB.(Source: Refcreoc 5, Page l?4)

LINE pAR.0U..EL TO. AIID AT A GIVEN DISTAIICE FROM. LINE AB:

l. UsingPoirt A (of Line AB) as center and the given did.ance ! as raditx, &aw an arc on orr si& of LineAB.

Z. UsingPoirt. B (of Lirr AB) as center and the srne disterre S r radiu, draw a second sc on ttr saf,nesi& of Une AB.

3 . Draw a line tangent bo both ttE fn$ and the second acs.4. the new line is parallel to Une AB'(Source: Refecoce 5, Page 125)

a-s\\

c.BC. DRAWING PERPENDIqULAR LINES

/ b/ ' D

P+

U .

1

3.

II

aA JI = {OSire 10-] l = 2 O m

l 1

0CC. CONSTRUCTION OF TRIANGLES :

Iey out Si& A in the desired pcitionUsingoneen@ia@ndPoirt l )ofsideAasccrterrrdthclcn$hofSi&Basradius,drawsf iarcon

iffi#d:XT*r,* A and the rengh of side c a radius, draw a second rc intcrsectins the 6rd rc

at Poinl3.4, ConnectingPoirt 3 with Points I dld 2 e*ablishes Triangle ABC'

(Sorrrce: Refercncc 5, Pagc l??)

S i le A

\

(t- z)

1. usingthe lengh of the given hypotenuse (side aB) * djlTg' draw a sernicircle'Z. Using one endpoirt (E ,Soirt A) of tlrc f"lp".,G1side AB) as cenbef and the length of

Side AC

radir-t!, &aw an a'c inbsectirylt}r scrni'circle atPoirt C'3. corurecrirypoirt c with En$oi*s A a-rd B .Juti"tes tlp &sired Right TrienS,le ABC'(Souce: Refersnce 5, Page 128)

- - - \ 1 S i d e F o(HYPotenuse)A _ S i d e C

(r)

1 .2.

3 .

?.3side CSide F

72( i )

(B.

(+)

( ' )

l 2

pRA\rlntc ,Jq EQUTIAI{GULIiIr TRIAT|IGLE VJTIH ONE SIDE ON GMI{ Ln{E AB:L Drarr a straidrt Lim AB, md designate rPoint, O ner cne end Using Poirt, O a certer, md any

conveniertradir.rs R, draw an arc cuting LirE AB etPoirf, C; end efend thc rc W b 90 degrccs (fr,omPoirt O) in ttr d$ired si& of Une AB'

2,. UsingPoirt C as crnter, rrd the sarrr radiun & &aw a second arc irferxdingthe Ent rc atPointD.3. CorrrectPoint O !o PointD md Poirt C !oPoirtD to compl& the egiangular tsimglc.(Sourcc: Rcfcreoca 5, Pagc 129) 1

INSCRIBING A}.I EOTILATERAL TRIANGLE IN A CIRCI.E OF RADIUS R:l. Desigrrate anyPointAinthe circurnfererre of the circle;PointD is located althe opposibe md of the

dia'neier lirn.Z. Using Poirt A as cenLer, and radius R equal to the radiu of the circle, dr;ur 8n arc cr-tting tlre

circumference of tle circle atPoinlB and at Point C'3. ConnectPoint D to Point B, Point B to Point C, and Point C bo Poirt D to complde the biangle.(Sor:rce: Reference 4)

-

ge--.

A- - - -

1 3

F \

cD. CoNSTRUCTION OF PARALI.g,oGRAI\{S ffouR-sIDD POLYGON$:DRAWING A SOUAREWITII SIDE AB GIVEN:

1. Draw Side AB in the desirt pcition CorEhuct Une BE perpendiculr to Side AB ard originating fr'om

Poirt B,Z. UsingPoirn B as center rrd AB as radius & draw m rc cr:Eing Une BE at Poid C' Using Points

A and

C as centtrs, and the sarrr radirs R in both oper*iorn, draw two arcs irrtcrsecting each other d'Point D'

3. Cornect Point C to Point D and Poirt A to Poirt D'(Sotnce: Refercnce 5, Page 129)

NSCRIBING A SOUARE IN A CIRCLEWTH RADIIJS R;

l. Draw the circle withPoirt E as centJer. Draw Line AB tlnough Point E cutring the circle d Points G and

It?. Draw Line c.D perpendicutr to Lirr AB, and passin3 tlrough Poinl E, cr-ilting ttrc circle

ar Pohts M

a-rdN.3. ConnectPoints GtoI4 Mto }1 H!o N, mdNto G'(Source: Reference 4)

I

),1(

B

z

\a.t

7

3 .

Draw the diagonal BD md bisect it af Poirt O. UsingPoinr O as centef,, &av a circle passing tlroudrPoirt B urd Poiri D. Lirn BD ig a dirndcr'urirg p"irr" B urd D as cntert, md lcnglh of side Bc as radiu, draw two rcs cttting the circle * PointC rrdPointACgnn{fgnrBb-Point C, C-!o D,D,!o A 1d AtoB tolltaeUerectaull

B-C

- - - - \

(s)

r trL

1, Draw Side AB in the dcired crierialion Corsbruct Angle 0 d. Point B, a-rd desigrate Lire BF as the si&

makingthe Angle 0 with Si& AB'Z. Usingpoiri B as cenLer and lenglh of Si& BC as radiw R, draw an arc cuting Lbte BF aLPoirn C.

3. Usinlfoirt A as center and si& BC as radits R, &aw s second arc on the sarne side (of Une AB) asPoirt C.

4. Usingpoirt C as cenber and length of Si& AB as radiu Rl, draw a third rc inlersecling tlc second erc

at Point D.5. Conned Poinl A to Point B, B to C, C to D, srd D t'o A

(.)

(s- s)

t 5

( - r )

(r-e)

ADD. DRAWINGLINFS TANGE}.TT TO CTRCLFS

Frqn the certer of the circJc @oirt, O), draw a line pcsing tlrough Poirt A and e*ending to Point B'making the lengh of AB eqrul to OA'UsingPoirt O fr$, rrd Poud, B rr4 es ccnters with a radir.rs R eqr:al !o my lengh greater thrr AB inboth ipcrations, &av two arcs which intersect d Poinl C snd Point D'The line connerlingPoirt C toPoirtD (perpendicrrtar Uitectcr of Une OB) is tangentto Circle O at PointA

(Source: Relerence 5, Page 135)

1. Frcrn tlre certer of tlE cbcle @oint O), draw a line to Poirt P' Bisect Line OP md desigrrate the midpointas Point M

?. Using Poirt M as ceriler md the lengih of OM as radius, daw ert aa'c cttting ttr circle at Poirt Tl and T2'

3. LinePTl andPT2 reta-rgenttothe circle.(Sorrrce: Refcreoce 5, Page 135')

-)a \\ T ITA

2.

3 .

x

D

1B l

- - ] - - - - '

o

1 6

3i, L

LINE TANGENT TO TWO DIFTRENT CIRCLES ON TIIEINSIDE (CROSS.BE.TI:Draw Line Ol -O2 conneding the ocrtcr o-f the fir*.drcle (Circle l) !o tnc certer of the second circle(Circle 2). Erom the centers of both chcter, erect perpcndicular lirBs b tjne Ol-O2, irtcrsecting thecircurnferencc of Chcle I d Point A urd the circunfcr,ence of thc otlr circle (Circle 2) d Poirt B.ConnedPointAwithFointB by a brcken lirn intfnectingLine Ol-O2 atPointP.UsingPoirtP as e.corrtrnon point or.tsi& of bcth chctes, and thc proced:re fa &tcrnrining poirt oftangency of a line to a circle @roced:rt B above), locaie PoinL Tl on Circle I crd Poinl T2 on Circle 2.The line conneding Poirt Tl !o Point T2 is tangert bo both Circle I and Circle 2.

(Sorrrce: Rcfcreoce 5, Page 136)

L ,

3 .

4 .

t - 2 )

-l-I' l

+

' x,

.($zoz\lt

/ \,

:l

o l Y '

1 ?

\ .D.1 . Draw Line BD perpendiorlr to Lire ABC er Point B'z. Frcrn poirf, B -a *ine t!* givcn radir-s & &aw a shqt rc irtcrsgcti"g LiT Pg *?oirt o'3. Usingpoirt O as cer# ana-,"aius & &aw the desired circle passingtlrough Point B'

(Source: Rtfertnce 5' Pagc 135)

v

1 .7.

Draw Line BD perpendicular io Line ABC atPoint B'Conned point B to point p; Con$nrd. a perpenaicil".r bisecbcr GH of Une BP; desigrrate tk

irfersection

of Une BD rrd Line GH as Poirt O'3. UsingPoirt O as center and a radius equal to OB, &aw the &sired circle or rc'

(Source: Reference 4)

o'l

1 8

M. EIRCLFS INSIDE ^ P$ID OUTSIDEIRI.AiI'IGLES

l. Biscct Angle A by Line AD Qdcnding this lirr bqrcnd thc middls of the kimglc.Line BE intersecting LitE AD at Poirt O.

2. Draw Line FG ttrough Poirt O perpendictrlr !o Si& AB al Poinl If3. Usi4Poirt O as ccntcr rrd radiur equal to OH, draw the dcshed circle.(Sourcc: fufcrcnce 4)

Biscct Angle B by

CIRCI'MS CRIBING A C IRCLE .{ROIIND TRIANGLE ABC :

I. Draw a perpendicular bisecLcn (Jne DE) bo Side AB'Z. Draw a perpendictrlar bisectcr (line FFI) to Side BC intcrseding Line DE (first bisector) atPoint O.3. Usingpoirt O as cenber and OA(orOB) as radiu, drawthe desird cim:rrscribedcircle.(Source: Referarcc 4)

( r ) ( 2 ) ( l l

1 9

tt

IF. CIRCLES INSIDE A}TD OUTfIIDERECTA}IGLES:

L Draw the given si& AB zuchthalPoirt. A and Foirt B bcth lies on the circr'rnference of tbe circle'2. Frqn bothPoint A end Poirt B, craw one f irr .J pcrPendiculr t'o Side AB md intcrsccting thc circle

d

Poirt D and Poinl C.3. ConnectPoirrtAtoPointD,PoirtDtoPointC.6dPoirtC!oBincrdcrtocorrpldctlerectangle'

A#e

, c

(r)CR.CLE D{SIDE A SQUART/ CIRCLE .AROUND A SQUARE:l. Draw the given squa-e &sigrating each ccrner as Points A B, C and D, respectively; &aw

a diagonal

from Poini A to Point C.2. Draw a perpendicular bisectcr (Line FQ to Side AB irtersecting the dia3onal ilne AC at Point O' and

cr-dling Side AB at Poirrt G.3. UsingPoirt. O as certcr and OG as radius, &aw the irscribed circle' UsingPoirt O as cerdf and

OA as

radius, ,&aw the desired circurnscribing circle.(Source: Reference 4)

( r ) ( l l

r)D.

(z)FIitI

D c

.i){o

Xt,)bt 6A-t-

20

/\ rq, l--]'J

EG, HD(AGON INSIDE A EIRSI.,E/ AROUND A CIRCI-EHEKAGON IMTIDE A CIRCLE OF tu\DIUSR TACROSS CORI'IERS):l. Drar diarneter line AD across thc givm circle; AD cquals 2R RomPoint Aand using radir.rs \ &aw

rr rc a.dting ttr circlc at Poirt B and Point F'Z. Frcrn point, D and using the sarne radius R, &aw anothr rc o-ltingthe circle * Poirt, C andPoint E3. Draw Lines AB, BC, CD, DE, S, FGto co'npldc th hcxagon(Sourcc: Rcferencc d Pagc 130)

( r ) (z) (r)HEXAGON AROUND A CIRCLE OFRADruS R (ACROSS FLATS)l. Draw dirneter Line AD across tle given circle. Usiry radits R as length urd strting from Point A'

mark offpoints B, C, E, and F, at equal di$ances around the circurnference of the circle (six points totat).?., Frcrn the certer of tle circle @oinr O), draw radiating tines Q.ine OA' OB, -- OF), erch line extending

beyond the circr-trnference of the circle.3. Draw one perpendicula- line to each radialing lirr, making tlee pcrpendiculas tangent to the cide d.

Poirts A B, C, D, E & F.4. The intersectiorx of these perpendiculars are the cornrs of *te circumscribed hexapn

-X l /f-x-

II

, \I

\ ,)S, , \

\Y.- t

,fr X - t \(2-r)

E

ot.i

, - F T

II

. ,

/ \I

- ,C-

( r)

2 l

(r-4)

r.dl .

,

d ^ft crrfing the &cumfere*e of theDraw two diamders of the circle rrhich re perpendicular to each

other'

;,[5l#e, },s,k ff H**? *ilng the di *ance berween po inr p and po i nr A as rad ius,draw m rc cutingradius ON atPoirtX

3. Frmr point A *d;i"g tl* distare bctweenPoirt A and Point X a raditx, draw a

the circle at Poirlt B' DraT Line AB, -a *t ;ttttngth to daerrnine Points C' D'

circr-nnference of the circle'(Source: Refarmct 5, Pagc 130)

second rc cutingard E around the

I

( 1 )

l. Draw the diarretcr Lirrc Ali divide this dirrrter line into five (fcr Perfagon) equa'l segments'marking

&aw lhe c'ther four sides.(Source: Rcfercocc 5, Page 130)

( r ) ( , )

L

>;,/ \

\B(nishtSrde ) >\/ \

. D

,o"f

, / "" r /

c

/

-\___:-

G"fl-no")- .,

S\

D

OBLI qTE(45' Caveller gpe)

/

D ' -

25

A ISOMETRIC PROJECTIONIDRAWING:

l. In rr Isometric , the objec is prrojedcd srrh thd its prircipal elft' :. T:' -l:-:#.::5:t55pr.n. or fi3#il-ffft: fitii;oilo;'rl*- is sr rsorrrcrii unl r uie which i8 nct prallel to

nv isorndncr - -^ - l - -T - - . *e iaDlm

"*;; ffi;;'t*d hr* 'e,ry;*.;.*uJ io 'ny of tt e Isonrtric Axes | - t*f:,ltT-

;:^;ff il'#;";iifr;ffi;;:,;;;il;;;i;;."i""carewhereinaudistarresre0'8165tirrps tsrE lenglh/size. An isonraric drarring is &awn using crdinary scale, or full size'; :'"ffi ;"; #;X ;il#i ffi}i;;ffiJ;ffi"J ;r'{aq "'"T 1T*: If ,YHthaf are parallel !o any edge in thc nnrh,i-vi.* a*'h"g *iff Ue prallel, rcspectively, to thc ccrresponding

edge in the

isorndric drawingoltlfi*. strfrcs in isorrptric cm be &awn bv estabtishins q 1111.*T:ti' :lu:]flT"It*

tr,. i*.rJi";H::io esablish the obliqt'e pt-.' tr,. u,,o,'id ?t F "!tlq::li,::#l*:r'iliffiffi:$#? ;:,}Hffi;ffi; ffi;#"d.* *"*, uy the rse "rtrirute ororanerism:f h3;- rL^ ^. .a 6rrcl r f ret fJ. The Isornetric Axes may be plrced in [fty desired poeition; btt ttE angle bCween th a:

A ISOMETRIC DRAWING f.Cort'd,):iSOME:HC DRAWING OFNORMAL SURFACES -

(Source: Rcference 5, Pagl 600

alongcdges-.

-/

4,3o'

\t

) -

IT

!*\

IIr-l---T-

".'1

f

\

27

A ISOMETRIC DRAVJING (Conf d'):lsoMETRIc DRAWING oF INCLD{D SURFACES (I., M and }Q

(Source: Refcracl $ Pages 606 md 607)All rcllu:lcncrtr mlt

to thr rrln tdgc6b.cf

perelkltbo

I F:-_l' \ M I: \i;JT l.^.l1'F +

L

N-=-D*

\3 o '

43o-

l" +_

".a13etrg ber

ffiGoFoBLIQtIEsr'RFAcES(SrrrfacecreatedbypassingC\.dtingPlaneXYCt}noughPoints A. B and C)

OFFSET LOC ATION MEASUREMENT

H: E,

h28

A EOMETRIC DRAWING (Cort'd.):?, Objeds re more easily drawn by meru-of Isonrtric Box condrrrtjon This consids mainly in

enctosing th,e object in a rectangrlar box wtrose sides coincide rith the otternnst pointsrfaces of the object8. Since the only lirrs thal re true len$h in an isqnetric drawing are.the isonrtsic ares (and tlx lines

paratlel to these axes), NON-isorrrtric line will NOTbe hte lengih9. Angleswillprojecttsr"resi.agdlghetttheplamoftheangleisparalleltotheplarrofprojcctiorr. Since

the various surfmes of an object in ISOMEIRICS re uzually irclined to the plane of projection, angles gerrerally areNOTtn:esize. Incrdertosctoffanglesinisonrdrics, lirrrmerurerrrntsoftttcsidcsremadealongisorrrbiclirs.

10. If the general shape of tk object does rnt readily mrfcrm to the redrrgular fcrrrt il cm $.ill be dravmusing an inccrnpletc Isornetric Box

I l. Ornaes can be drawn in isornetric by rneans of a series of otfset rrnasrenterts. Enough points on thecr.nre should be establislred to rcuralely fix the path of the cr.rnte; t}r rrnre poirts used, t}r grealer th acoracy.Offset isornetric lines re then drawn from each poirt prallel to tlre isorrrtric axes.

(Sourcc: Rrfercace d Pags 608)

EOX CONSRI]CTIOII HTIrH

F---l

l-" -jT-

D

I

ISO}iEIRTC LINESI IT

Iq

---T----T-b

!_l_----r-c

\-\

(

f r

\ rl \

L__ -\\\\\\

: / -\ /

\\\

--t-M-

I t

,/r---7P,

, / l -I

, / ll -

l )4iI

N.r

\\\

A F Jr l ll ltl ^ I

FaBF

F(J

t-

82

F.1' conccpts:

planc (frontal or inclincO that is perpaulicular to a principal plmc will appcar in cdgcwhen profict.u

TT:tlm'ur edgc vicw of an obtiq'c pla$:,it mustbc projcltc-d orio an aruiliary planc; thcarxitrary vicw must b;;";;.t.i in , oirw dlrs.tionior finc o{ ti$Ji pri'ana to r t'c'lcngth

linc in the

oblique planc. Ttrc,;l;;.*ili* Irl will show trc obliquc planc in edgc-

c. If nonc of thc lincs in ttrc givcn planc appian.r tr.lcngth in ttrc principal vicws givcn, a

linc can bc con$rucred in thc g-vcl glane p*"rnriti a principal plane. This constuctcd linc will thcn appcar

m. f*gttt in the adjacentprincipal vicw'3' Proccdurc:

a horizontal Line cX in planc ABC in ttrc front vicw, and projcct it to [re top vicw'Linc CX is normal (trtrc lcngttr) in tht Top Vicw'

b. To o6ain the end view .ii;r,. cx, prolct qarallcl to Line cx from thc Top visw onto an

auxiliary pr*r. rn ,Jr,,rTr,;;,Jitty "i;;li ir'o* tt" tnini'* of Linc CX and trc cdge view ofPlane

ABC. ,(Rei Refererce 1 and 2)

Solution-

OivenH P

A rAr

/ r '/ l \

lB"I

I- l I

II

l l- l i

' u P

r - ( ' 6 .-l'L, "'

Edge Yleu

x,cr(Pv)

Cr#X.- - Horlrontal Lino

IIIr

/ n '\ l-

-- . l l

83

-A,

G.INCLIMD PI,ANE:1. ConccPts:

a. Thc normal vicw of a plane is a vicw in wtrich thc truc shapc.of ttrc planc is shown In ordcrto obtain sris normal ui.w, Ure planc muibc vicwcd in zuch a dircction thatthc linc of sig[t (or vicw dircction)is pcrpcndicular to thc cdgc vicw ofthe planc.

b. An in?tini6 ptan. oin .pp.rr in cdgc in one of lhc ptt"P4 views; thcrcfore' a singleauxiliary vicw projcctcd in a dircction pcrpcndicular to th-* cdgc vicw, will show thc truc shapc and sizc of ttrcinclined plane.

i. ncquiremcnt Givcn tk inclincd Planc ABC which is shown in cdgc in thc Top Vicw, draw thcnormal view of thc planc. (Procedure: From thc cdgc vicw ofPlane ABC in thc Top Vicw, project onto,anauriliary planc in the dircction perpcnrlicular to that cdgc vicw; ttus, thc nonnal view is obtained' Thisauxiliary vicw shows thc tue shryc ofPlanc ABC.)

(Source: Refaerrce l)

&,

TgP vlc.lr

Edgc

tt

Front Ylcu

Vlcn

t

Firet Auxllia4lVfcv

84

EPI.A}i[E:l- ConccPb:

a. To find thq tnrc shapc (normal vicm) of rr ob!{ui planc, two opcrations rc rcquircd: firstobtain a view in whicn't.bti$t; r*fio fo.* in erige; secon4 obtain e view in which thc surface

appesrs

in uuc shapc' b. Thc sovc two opcrations rc accompfishcd by drawing a first (or prrnary) aruiliuy vicw

and a sccond (or sccondary) auxiliary 19y'z. ncquircm.ni'iiiu.o ttn obliquc planc ABC, construct a vicw which will show the

planc in tue

shape (ard tue s2c).3. Procedurc:

a. In thc plane ABC, construct a horizontal Linc CX wbich will appcu true lcngth in the Top

vicw' b. consfiuct an atuiliary clcvation, from tln top vicw, with thc vicw dhcction parallcl to Linc

CX in nue length; thus, gjving an edge. view ofPlane ABC'c. construct a second aunliary view with the line of sight perpendicular to the edge view of

Plane ABC. This view will givc thc tuc shapc oflPlanc ABC' /

(Source: Refererrce 1)

Vtau (EdgeB;-- -...- ...-* B 1

Ccntral Viev

First &rx

X ,c t

Vlicw

Second Arr Vlsu (frue ShaPc)ALf

RPz

ldJacant

85

1. Rcquircmcnt Givcn thc Linc AB and 0rePlanc cDE in thc Top and Front vicws' locatc ttrc point

whcrc Linc AB Picrccs Planc CDE.I'ff#il''"diilit", an arniliarv showing thc p-1*t T cdsc' folc^PJT-*:1.T:.:.ott P wtrcrcun. e''ini;.Hil Jifrir?riTrffipr*i.-bE ?;itt} canbc piolctcdto thc principal vicws.

(Sor.nce Refereme l)

g t

P, (Pi'""i"1 P';n+)A t .

AulllarY Vfev

ldjaceat VievD 7

L4..

(Souree: Refcence l)

C\tLtlng ?ry

To obtain the piercing point by the cutting-plane me$9d' pa-ss a veftical. Plane MNOP (the cuttingptane) containing thl t';;AB",il.rgr,'u* 'igi'i'iir*' gLlg*t::i'f1fllll'i*; I*'-;*:*t#:ililjffiltfiltd;ii;'H,ffi;fiil, r"''fiffi;.;s;;r", is ir'i poi't ativiich Linc AB intersects Linexv.

85

6nrar Plase

METTIOD .. MCthOd IT}:t. Conccpt To obtain thc picrcing point using thc cuningplanc mctho4 a cuting plmc parallcl to any

of ttre principal plincs urd conaining.thc gilen lincf is pasleadlhouS F. Wg plurc---- --- r z. ncdutrcrn*t civ.n a ftrc ag andphnC cDg in two vicws (top and Front), find tttc Picrcing

PointP oflinc AB onPlanc CDE3. Procedure:

a. A vcrtical cuning planc containing Linc AB is passcd tnouglr Planc CDE, in thc Top Vicw'and cuts ttris plane along Linc XY.

-

b. Locite Line XY in thc FrontVicw by projcctionc. Linc AB will picrcc Line CDE somcwherc along Line XY; horrever,

cannot be locatcd in thc top vicw bccausc thc lincs coincidc.d. Thc piircing Point P can bc locatcd in thc tont vicw by tbawing Ijgc

point where it interseds with the projected Line XY'(Source: Refererrce l)

l g -

l(

rd\ c! ' , r ' t T

this picrcing point

AB and noting thc

^;i, l . /Dr

87

ce)BF

J.l' conccps:finc pcrpcndiculu to a planc is prpcrulicular tl try-gt-:**"t'lf'

b. Consequcnty, if in two .fi;;*t;;iC m fiot'i ti'oilpcrpcnOii'lu to a fiuc length linc

of rhc planc in onc vicw, andto anorhcrrnr-#ilil;;i.* pr*ff6. otfr.r uicw, thc rinc is pcrpcndicuru

to thc planc.t *tqt#:'*o

"rro, whcrcin a truc lcngh vicw of two diffcrcnt linos in thc planc'cxist

in both

views, constuct Line DA perpendic'lar to ph; ABi. procerhnc: consfrud Linc DA perpendictrlu to the

Euc lcngrh view of thc t.rir.irt^f finc AC ;'A; f.n Vf*' q9 pttp*Jt"far to the trui lcngth vicw of tre

frontal Line AB it uJd;;t"icw' Linc DA is pcrpcnql+tt toPlane ABC')

b. Given the Top anO noniueris of Plane ABC and Point P' corstnrct a line from Point P

perpendicular to pranc anc. br'0..0*., co*ru.t in boftr ui.*r trt. noti^ntur Lini cY and thc fronbl Line

BX in thc planc ABC. Draw Linc po prrp.r,[J., to thc truc lt;$h "ia;Frontal Linc Bx in thc front vicw;

and draw Line po pcrpendicular to thg *.i.t g,tt or tr,, rto-.iirr u;; .t in rhe Too view' Line Po is

perpendicutar to ptane'egc. To find ffr;.p"ft wh.er.e gt f:i"*q|*t pi"tt' Plane ABC' proceed in

accordance *itn pro..Ontt in ttttt*ining piircing p-oib el:t:eding pages;'

c. Givcn thc Top *O frouiui.*,-ofpt*" egC"*d"p';fit f' where Side AB is true length in

*e Top View, draw Linc po pirpcndi.'tar io pranc ABc. (proccdure: constnrct ar aux'iary view in the

dircction parallcl ,o t* es'rt'o*inq Utt c-d!c;cw.gf Pfu;;Bc'- ntt* Linc Po in thc Arx Vicw

perpendic'lar to ptanrigc. rin. po i, *.-r".,igtt *oi, p*"u.r-to trte auiliary projection plane; thus' it is

perpendicular to ,n, u*iti,'y projection rays' or lines' in the top view')Gef: Reference 1 and 2)

(e")(2")

FrI

( eu)

88

K. CoNSIRUCTION OF A PLANE TBOUffi A POINT AI.ID PIRPENDICULAR TO A LINE:

l. Conccpt Sincc a linc pcrpendicuh to a plane is perpcrdiculr to wcry lrt! io thaj planl--if ltrelincs uc &awn ttnough a givcn p-oini, botr pcrpmdiculu to thc given linc, thac two lincs will cstabfish thcrequircd planc pcrpcndicular to thc givcn linc'--r j. n.hd6cnt Givcn Linc AB and Point P in two (Iop and Fron$ vicws, constur't Planc NOPpassing tluougtrPointP and pcrpcndiculu j9 Linc AB'

3. proccdgr., Coottr.ihorizontal line PO prpcndiculr to Linc AB in the Top Vicw, thcn, con$ructfrontal Linc pN pcrpcndicular to Line AB in drc Front Vicw. Lincs Po andPN atablish thc rcquircd planc.

(Source:Refererne 1)

Bf

lu', l

\L

BF

89

a)t'/'

L.

l. ConccPts:a. A right circglu conc hr a circular basc with its axis palsing.tttr|uglr thc ccntcr of trc basc

urd pcrpndicular to thc planc of thc basc; ttrc altitrdc of trc corrc is cqual to trc lcngh of thc axis from thc

vcrtcx to its picrcing point in thc brc.b. Ttrc *rortcst distancc bctwcen a point and a planc is along thc linc pcrpcndicular to thc planc

and passing througlr thc point r t , __ __r:---c. A lini is pcrpcndiculu to a plane if the line appcars to bc pcrpcndicular to at lcast' two non'

puallel tnreJcngth lines in the plane-2. Rcquircmeng Constnrct a right circular conc givcn thc Top urd Front vicws of its vertcx Point A

and the planc surfacc (Planc CDE) ofits basc.3. Procedurc:

a. Con*ruct a true-length line in the Front View, from which first arxiliary view can be drawn

showing Planc CDE in edgc'b. fro* thc projcctcd location of thc vertex Point A in thc atlnliary vicw' drop a line

perpendicular to ptane cor Cn cOgO. This line is in its true length and is equal to ttfi attiuoe of the cone' The

'poiii *t .r. the axis pierccs im. bilg is also cstablishcd in ttris auxiliary vicw'

c. A second auxiliary view is next drawn perpendicular to tlre edge view of Plane CDE in the

first auxiliary vicw. In this second i*ili..y "i.*,

tfr. .it.,i*,basc, with maximum area fttat will fit insidc

hiangle cDE, can be cstablished- tre cxteni of the base area in both sides of the axis is projected back to thefint auiliary vicw.

d. All new points found in the first and second auxiliary vicws arc re-projected back to tiaFront View and the Top view, in order to draw the complete cone.

@ef: Refererrce 3 and4)

IROC. Ja&b:

-TTrp Ylcv(ctvc.a)

tr'rut Ylcv(clvcr)

90

tr \,7

| - ""

-.'I

A I,"'r F

Ef

--.-/ |-./-*

./- ID a

"a---'-- Rp

E 7

cF v-r

RPzPRoc. i tz

FRoc. 17t

91

CHAPTERVIIPRO BLEM.SOLVING

- - 1I u . . - ! '- . - - . i "

\ , - at

, ^ ; t '

. x l - ' l ; .\ t ('l^-^"r '

I

A.1. ConcePts:

a, Any two poirns which ar comrnon to both planes will daermine a $rai$rt line which lies in both

planes. and is tlr irierseclion of the two planes.b. A poin! corrunon to tlre two planes is deter.mined by Frnding *re poirt in which a lirc of on'e

plane

pierces tlre second plane (ocating the piercing poirt of a line on a plarr)' r hi-^ r,r\r.\x2. Requiremerts: Given two (Iop and Front) views of Plane ABC and Plane MNOP' ddermine ttre Line of

Intssection of tlre two Plarns.3. Procedue:

a. Determine the piccing point, R of Une Mp on Plane ABC; then, locde the piercing Poirt' s of Line

NO; bcth on tk Front View.b, pr-oject poirt. R and poirn S b tle other vieq and connecl Poirt R to Poiri S in both views' Une RS

is tle requned Lirre of Inta-section between Plane ABC and Plane lv{}{oP.c. By inspection, determirr visibility and show hidden lines accordingly.

(Source: Re lererrce l)

TOPPlaac l'lI{0P Plcrclng

Planc ABC)

' i , , ' t

' ; . i ' r ' ' i

BT

Br

FROXT

(,7M6

92

B.

l. ConccPts:* f atry planc cuts ttrougtr two ormorc prallcl plancs, thc lines of intcrscction made by each

planc with thc orting planc will form a sct ofpanllcl lincs;--- U. ft^rtty plane cuts ttnough two non-parallcl planes whosc dcfincd bourduics do NOT

achrally touch, thc lincs of intcrscction madc by cach planc with ttt crfiing plmc will, thcmsctvcs' mcd orint*r'.t ,tupoint in spacc which would bc commuto all tkcc plancs, if thch boundarics rrrcrc cxtcndcd zuchftrat ttrey will actnllY mect.

c. If two or morc prallcl plancs cutttrough two non-parallcl plancs, thc points wtrerc ftc lin91of intcrscction ftctwccn thc planci) mcct or intcrscct wrll dcfinc a stai$rt line common to both non-parallelplancs, hcnct, formthch ownlinc of intcrscctionr-----'

2. Requirement Given two non-parallel and non-touching Plang ABC andPlane DEtr'in the Top andFront Vicws. Dctcrminc the imaginary linc of intcncction of thc two plancs IF trc bor.nrduics of bottr plancsare sxtended such that thc two planes will mect or intcrse ct

3. Proccdures: '

a. Draw two parallel cutting planes (CP I AND CP 2) in cdge, or as a linc, in the Top View.(This is a vertical cutting plane, parallel horizontal cuuing plancs may !c drawn instea4 if daired).

b. Usin!^thc first cutting planc (CP 1), dstsrmins thc linc of intcrsection bchrcen Planc ABCand Cp l; then trre rne ?r intcrsection u.t*e.n Planc DEF and the same cutting plane- Draw thesc iines ofintcrsection in the front view, and locatc their point of interse ction as Point o.

c. Using thi sccond cutting planc (CP 2), delcrmine the second set of lincs of interscction(same procedurc as in 3iabove), and locate thePointP wherc this sccond sct of lines meet

d. Thc line coirnccting Point O to Point P is common to bothPlane ABC and DEF (that is, itforms their line of interscction), if thc bomdarie s of Planc DEF arc cxtendsd such that it will cut througlr PlaneABC.

(Ref: Refererrce 3 and 4)

(r")ToB Ylev

hrnt'Yl.cv

cPl

cP2

93

(r)fop Vlev

Front Yiew

(1" )

Trp Ylav

ErortYiew

(ra)Trg Ylcw

c? 2-

Bf

(r

.rt

FroatYl.ev

cr

94

c.

I / ' t r '

/Second. Axll1arY

Ba'

1. ConcePts:a- To ddcrminc 6c angfc bctwccn the line and ary pq., i! is ncccssuy to obbin a view hat

shows a normal visw ofthc linc an4 atttrc samc timc, thc cdgg vicw oftttc planc'b. dgr;A rorutioo to tbis problcm rcquires three aruiliary views; howcvsr, there ars two

approaches that canbc *utry thc planc: Edgc vicw - fs-Y* 'Ttlgc vicw'

Udr,; ttrc Linc: TL Vicw - Point Vicw ' TL Vicw2. Procedure:

a- Givcn a I ine DE and aPlanc ABC in two views $op gd-Fr9nQ.u. conrru.tttc fust and thc sccond arxiliary vicws to obtain thc normal vicw of Planc ABC.

Any vicw projcctcd ;;;.th;..ond aurliary vicw pcrpendiculu to Linc.DE will produce Linc DE in ruelcn'gth andplanc ABC in edgc, in onc vicw'

(SoLrrce: Reference l)

A L

RP. c2-Cantral VIeu

tr= t

\ ' . . . . . . . -

Dl

X, XATX; /- -'

D.

@ef; Refererrce I and 2)

Central Vlev-# (Top)

X

'- Dz

cf

\ /

1

l. ConcePts:a. Thc Linc of trtcncction (LoD ofnpo surfaccs is parallel, -and ggmmon' to both surthce s'b. Thc tlc angtc (or dihedral u.r".." i*o surfaccs can bc found by constucting an auiliary

view whcrcin the I-inc oflntcrscction upp.rrr'ri;;.t", itnd ui.w): or whcrc BOTH strfaccs appear in edgc'2. Rcquircmcnt Given planc ABC andplanc gdo in thc Top and Front vie*s' and Linc BC comm.n

to both planes. Determine hc dihe&al (anglo bctwecnthe two surfaccs.3. hoccdurcs

a. Using thc Top vicw as cenftal vicw, draw thc 6rst auxiliary vicw- with a linc of sigltt (Los)perpentliculu to Linc Bb which-is common; b;tl ptanes; graefY obtaining a tnre length view of Line BC'

b. Drawthe sccond auxifiary vi#p*.ff.f to thc uuc ttn-gtt of Linc BC (common linc)-inorder to obhin thc point (cnd) view of Linc 6c- rtt trrc anglc bctwccn ttre trruo surfaces will appear in thisvicw.

R?Z

2nd Ax Vlew

I'lrst Arx Yles

/

(Pv)

\,,

Bzcr

.d.laccnt.Tictr(Rcont)

95

!':.::1 .

tE. DETERMINING THE SIIORTEST DIITAI.ICE FROM A POINT TO A LINE:l. ConccpC Tte shorte$ distrncc from r point to a line is thc perpcndiculr distance from thc point to

trc linc. This pcrpcudicular can bc constructcd iq, at lcast, drcc Wap.2. Procc&trcs:

a Rccommcndcd (shown bclow): Construd a vicw \dth tF givcn linc in its tnrc lcngth (firstauiliarv view). 11om Point P, draw a pcrpcndiculr PrXr to thc linc. This pcrpndicular is thc forcshortencdvicw of thc shortcst distmcc bdrpccn Point p ud Linc AB. Thc fruc lengh of tris' distancc can be found bydrawing a sccond arniliary vicw in a dircction pcrpcndicular to hc Linc Pr&.

b. By rotation of the pcrpendicular PrX in Proccdurc 2a abovc to dctenninc is truc lcngth.Rotatc linc prXr to obtain tuc lengft PFIG. c. Another method of frrding the distance betr,wen apoint and a linc is to find ttrc normal view of thc plane containing bottr ttrc point md thc linc. In this nrthodihc givcn Linc AB and Point P arc corncctcd to formPlanc ABP; thc& thc first armliary vicw is conshuctc.dshoiing thc planc in edge. The- sccond auxiliary f qu*n showing a normal vicw of thc plutc; and in thisview, tie perpendicular distance fromP to Line AB is shownin nue length.

(R.ef: Reference 1 and 2)

RPaOentral 2rd turtltalJr Vlcv

{ P{ \-'\

RPrFtret lud.llary

Vtctlt

FJacent Ylcy

n

F.1. Concept To find the shorte$ distancc from a point to a planc' it is nccessary

only to construct a

vicw showing ttrc planc in edgc (c!gc vicw;f th; pr-0. rnr r.quitio oirt*.r is the perpcndiculu distancc

2' Proccdure :nstruct a horizontal Linc BX such that it will appcu in ruc.lengttr in thc

Top view'

b. Construct an a'xiliary view parallel to Line BXtfi;;i*gtft tfio* the top view)' so that

Plane ABc *t XIPffX^'ff; P1o1 rromPoinJ P1-prp'$i'."1*.1: tll-'^1'3,'^1:":":'.:i:i l$l#'i:*distancc from point p io thc ptanc ABC. P;;t d .* U. tocut,i ;;;;;;'fi";ting

a sccond auxiliuv view

perpendicular to the edgc view ofPlane ABC'(Source: Refererre 1)

from thc point to thc cdgc view of trc plme'

Oentral Vlev

+--

Ftrst '^&xtl larY

Fr tX\' / /

/ r hI 'I

I)----III

1-'

-l,.t l- uo,

BF

.6.tecent Y19v

/ 1 t

98

G. NTTTRMINING Tffi SHORTEST DISTAI.ICE BETTflEEN TWO SKE:W LINES:l. Conccpt The shortc$ distancc bctweco two skew lines will bc a linc wtrich is pcrpendicular to both

skcw lines.2. Mcthods:

. a. Onc mdrod of finding this distancc is to consfrud a vicw showing thc cnd (or point) vicwof onc of thc lincs, and drawing a linc from ttrc cnd vicw perpcndicular to ttr otrer linc. (Shown bclow).

b. Anottrer mcthod is to constuct a plane which contains one of tbc lincs and is parallcl to theothcr linc; and thcrl constructing thc cdgc vicw oftrc planc @roccdrc II).

c. Thc *rortest HORUONTAL disancc bctwcen two skcw lincs can be found b this ParallclPlane Metrod 2b above (Procedure H in ffsection).

(Source: Refererce l)

R.Pr

Ccrtral.(rop) YlavFhst Axtllary VC.cw

At

\rr ' YAzBz (Pv)

\ . / R?z

\ , /( L"_DlstalcrShortest(rr,)cL

Sccond. nrrtllary Vlcv

&Jacent(F:soat) vtcr

99

H.A].{D CD(With Second Method of Obtaining ShortestDistance XY) -- Parallel Plane Method:

1. ConcePlst '

a- Another way of determining the shortest distance bbtween two skew lines is to

consh:uct a plane, which contains one of the lines and is prallel to the other linel and then, find. tle

edge view of that plane. The perpendicular distance between the edge vie$t line of the plane and theothlr skew line wiil be the shortest distance (XY) between the skew lines'

b. If it is desired to fincl the shortest line corurecting the two- skew lirtes' a second

anxiliay view must be drawn to obtain a normal (TS) view of the pluttu and $e h.ue length view ofdre other skew line, taken in a direction perpendiculm to both lines ng and cD' The 4pa'ent

point

of crossing ofthe hvo lines represents the point.ri"r* of the shortest line (XY) behveen tlte hvo skervl ines.

c. The shortest HORIZONTAL (or vertical) tine between two skew lines can be foundby the parallel-pla1e metfiocl described above. The shortest HORZONTAL (vertical) distancebetween two skew lines can only be measured along a line parallel.to the horizontal (or vertical)plane. f}Ie lirre GH repesenting the shortest horizo*al line connecting ffre hryo skew lines can belocated i' a second auxiliary viav projected pa'allel to the horizontal ,"fotnt" plane line of thejirstauxilia-y vi"rv. ,fhe ap-parent point Jf ,rorring of the two lines repn'esents the point "i"I.o{.d"shortest H0RIZONTAL line GH behveen the two skew lines. Horizontal (or vertical) linesconnecting skeu, lines will be true length only in aview where the horizontal (or fontavprofile) planeis in edge view.

2. ,Procedures:

a- Gven hno skew lines AB and CD. Line DE is drawn parallel to Line AB' Lines

CD and DE define aplane containingLine CD and parallel to Line AB.b. A f6st auxiliary-vierv is dra m'showing the Plane- CDE.in edge' The-shortest

distance line Xy and the shortest HORZONTAL line GIi wiil be &envn in this viev/' after their

locations are established- ,c. Since ttre shortest horizontal distance line is required it is necessaryr to project a

second arxiliary viar parallel to the horizontal reference plane (RPl)' Ttt? apprent point of crgtp;;i;;, A;}ia io in ttre second arxiliary is the poinl (endjviar of the shortest HoRIzoNTALline GH connecting the two skew lines. The points G and H can be projected back to the firstauxilia-y and the given views. The line .o*u"tiog 6rrse tp.o points *uit 1nl9t horizontal in.theFront and Side Vi'ews. By similr procedure, the-shortest line at any required slope between twoskewed lines can be determined-

:- d. To find ttre shortest distance line between Skew Lines AB and CD, &a\'v anothersecond auxiliary vierv in a direction perpendiculr to Line AB a1rd *re edge vie\P of Plane CDEthereby obtaining the normal vierr (TS) of Plane CDE and ttre true lengfh view of Line AB' in orleview. The appa-ent point of crgssing of Lines AB and CD is trre point (end) view of the shortestdistance line betgeen the skew lines.

100

\.

pu.allel-plurc Meftod.of Determining Shortest Distmces Between Two Skew Lines: Shortest Line XY andShortcstHorizontal Linc GIH -

(Sor:rce: Refererne 1)

t-J ' i i

) Rpe' R.P, (H P)

td &u Vlcv-

eq (*e)

Ey

ct'

j"'-T

*

I r

'Ts

2.

2!d _Aullterr Vlal

1srrffirI|")

hil Arr Yl.evSffiffic)

E

$H(ev

K)"'F r \ ' r7 - \l . t ' \l t \l t \

l0 l

GIVEN PLANE (,ABCDE):l. ConcePts:

u. tf un outsidc line is consbuctcd parallcl to any tine in a given plane , ttrd linc will be parallclto the plane. l

b. Iftwo intersecting lines are consfiucted such that each of these constructcd lincs are prallelto two corrcsponding non-parallcl lin"es in thc givcn planc, thc conshucted lincs will fornr a ncw planc that ispuallel to ttrc givcn plane.

2. Mettrods:a. Using only the two given vicws, construct alinc PGthroughPointP and pardlcl to arry line

(I-ine AB) in the given Ft*i egCOfl thcn constuct a second line PH, also pasing ttrouglr Point P andparallel to a seconJ[ne Qinc BC) in the givcn plane ABCDE. ConnectPoint G toPoint H to complete the newPlane PGH,which is parzllel to Plane ABCDE. This method is shown bclow.

b. Anothcr method (not shown) is to construct a first atuiliary view showing thc givcn planeABCDE in edge. Tlren, draw the requfu'etl plane PGH in the auxiliary vierv, as a line (also in edge), parallel tothe edge view-of Plane ABCDE, r,d prrsir,g tlrrough Point P. Desigrate Poillt G ard H attywhele in lhe linerepre sJnting the cdge vicw of the new plane f Cn no3cctPoinls G and H to the given views.

(F-ef: F-efererrce 4)

tgz

tl

I. DRAVING A PLANE PRPEI{DISULAE To A p:rvEN ELANE: {Ncw}l. Conccpts:

&. A line drawn pcrpcndicttlu to a givcn plmc will bc pcrpcndiculr to wcry linc in ilratgivcn planc. Hcncc, if intwo adjaccntvicws lhc linc is &acm pcrpcndiculr to two NoN-puallcl tuc lcngttrlincs of thc planc, thd linc will bc pcrpendicular to 0tc givcn planc.

b. Iftwo diffcrcntlincs arc &awnin trrrc lcng& (onc cach in two adjaccot vicws), both lincspassing throug! a common point in sPale, and made perpendiculu to a line in a Wctr plane (not ncccssanllyin tuc lcng$r), a ncw planc dcfrrcd by thc two constudcd lincs is pcrpcndicular to ttrc givcn planc.

c.If Linc I is drawu pcrpcndicular to two NON-parallcl (mtcnccting) Lincs A and B in agiven plane; and Line 2, parallel to Line 1, is drarn also pcrpendicular to Lines A md B or arry two NON-parallel lincs in thc same planc, a ncw planc dcfincd by thc parallcl Lincs I and 2 will bc pcrpcndiculu to trcgivcn planc containing Lincs A and B.

2. Rcquircmcnt GivenPlane ABC andPoitttP in two adjaccntvicws. Ushg Conccpt la above,conskuctPlane NOP pcrpcndicular toPlanc ABC.

3. Procedurcs:a. Draw ncw Line BX such that it will appear tue length in the Top View From Point P,

draw Linc PO perpcndicular to, and intcnccting Linc BX atPoint O. PrqjcctPoint O to thc Front View.b. Draw Line BY sudr ttnt it will appear tuc length (IT-) in the Front View From Point P

also, draw Line PO pcrpendicular to Line BY intersccting the vcrtical projcction of Point O from tlre TopVicw.

c. Designatc Point N anywhac in ryace (for cxamplc, arqmtlcrc along thc horizontal LineBY orits cxtcnsion), andprojcctPointNtotheFrontView. Whcre trc projcction of Line N intersccts thctrue length Linc BY in the Front Vicw, c$ablish Point NF.

d. ConnectPoints P, O, and N in both views to establish Planc NOP

/"x

//j_=

lt 'PT-

_ 1 (

ToP VrE'hr

TNOlfI TTE1I

c7| . ', / - -f6I

ta..Q F

- ' -

103

K.

1. Concepts:a. Intcrscctions bctwccn a planc and a solid utilizcs thc advanced application of ttre basic

conccpts uscd in dctcrmining thflclcing Points made by the cdgcs of trc planc on thi'planc surfaces of thcSolid' The Line oflntersedion (LoD is the linc .onn..ting pierclng points on the same planc stufacc.b' A morc^advanccd application of ttrc iicrcinf-ioint and tiirc-of-intcrscction concepts

involves thc detennination of the linc of in&rscction bctween drc piane surfaccs oftwo intcrsecting solids.

- ?. Rcquircrnent Givcn thc Front Vicw and ttrc Right-Sidc View showing planc pllr{ e$::*Pl,!t) picrcing ryramid ABCDE, or the othcr way around iocatc the six picrcing pointr and the fourImes-ot-rntersection between ttre plane and thc plramid

3. Procedure:a' Using an edge (one edgc at a time) of Plane PLM Q in the Front View (so thd thepiercing poittts can be d^eterminert in ne nignt-Sidc View), determfure the location of tre first four piercingpoints on the sufaces

.oj tt. pyramid. Placi the cutting pi*.r onty utong cdges wlrich scem to cul at lcast,three edges of the solid- htrt, procedure, LinePL i! used first as thJcutting plane in edge view; thus,obtaining piercing points on surflaie s BCDE and ABE ortnc pyramia.

b. Using Linc QM ncxl piercing points on s'urfaccs BCDE (second) and ACD are located.c. By go.nstucting an auxitiary view wherein Plane PLM Q is in edge, all other possiblepiercing points made by Line s pr and QM on rframiO ABCDE can be locatcd. Sincc iine s pe and LM do

not intersect at least tlree edgcs of thepyramid, ihesc lines dogElpierce the plramid.d. Lines'of-Iderscction ktrrecnPlane PI-lvI Q i"A tf" plnu*iO ABCDE can bc dra*n byconnecting all picrcing points in thc same surfaces.

PROC. 1a &' t t

6,$rlliTl

Fnrt YLcy

lotI

./

' a ' ( -

. / \ . .

Kl. LINE OFII.ITERSECTIONBEI!ilIEENAPI,ANE AND A SOLID (Continuation):

P'e( Prhary rtrr Ylcv

PROCEDIIBE ]T:

\

\ \\ \' \

- \ A r

/ \ / - /

"-p::fL=

L. CONSTRUCTION OF A PLANE PERPENDICULAR TO A LINE IVHICH PIERCESTHE PLA}IE SOMEWHERE IN THE MDDLE

A Requirement:l. Btablish Plane ABCDE Front View, fom given Top Vierv, such that Line GH

will be perpendicularto the plane.2. Determine point in Plane ABCDE where Line GH pierces.3. Complete botr views showing correct visibility ofline GH.

B. Given:l. Plane ABCDE in the Top View.2. Point A ofPlane ABCDE; and Endpoints G and H ofline GH in the Front View.

C. Procedures:l. Draw Lirre A-l so that it will appetr bue length (TL) and perpendicula' to the

given Line GH in the Top Vierv. Line A-l is therefore horizontal in the Front View beingconstructed.

2. Since Side AB is horizontal in the Top Vierv (and therefore pa-allel to the frontalplane), it can be &arvn perpendicular to Line GH in the Front View.

3. Draur the second imaginary Line B-2 so that it will apea. hue length and parallelto Line A-1 in the Top View; hence, it must be horizontal in tre Front View and pa-allel to Line A-laJso.

4. ProjectPoint I (from Procedure l) and Point 2 vertically down to the Front View.Then, connect the hvo points extending this line in both directions beyond these points so as tointersect the vertical pojections of Point C and Point D fiom the Top View. Connect Point B toPoint C.

S.a)Since Side DE (in the Top View) is prallel to Line A-1 and Line B-2 (andpergrndicular to Line GH also), it must be bue length in this (Top) vievq and it must therefore bedrarn horizontally from Point D in tre Front View.

s.b)OR: - Draw the third imaginary Line C-3 horizontally in the Top View. Whenprojected dowmvard to the Front View, it must appear hue length and perpendicula' to Line G[I.Connect Point D to Point 3 extending this line to Point E

6. Connect Point E to Point A vatically in order to complete Plane ABCDE whichis now perpendicularto Line GIL

7. To locate theloint atwhich Line GH piercesPlane ABCDE, pass a cufting planein the Top View such that its (cuting plane's) edge view coincides with Line GII

8. Project the intersections of the cutting plane rvith Side BC (Point Y) and w'ithSide DE @oint X) to the Front View.

9. Connect Point X wi& Point Y in ffre Front View; and where Line X-Y intersectsLine GH (Point P), ftat intersection is tre Piercing Point ofline GH on Plane ABCDE.

10. Line GH pierces Plane ABCDE from bottom-left to top-riglrt for propervisibility, thereforg Line GP will be firlly visible in the Front View. Line PH will be cornpletelyvisible in the Top Via,v.

106

ILl.

GIVEN: l. Ptane ABCDE urllinc G'Hin trcTopVtcw';: i,.irtA of?lanc AgtogmdEn@oirts GandHintrc ltontVicw'

fn,

Ar*

nol

107

I,

PLANEPERPENDICIILAR TO A LINE ('Conf\:

REQIIIREMENT: 1. EstablishPlanc ABCDE Front vicw such ttrat Linc GII will bc

. fTgiTlT':1$lliliBcDE whcrc Linc Grr picrccs'2. Dctcrminc Pomt mSteps 1 - 4:

cT

-{! Lt\I \I

.

tB.

Stqps 4 - 7:

l l,1** I

llt l

t lI

.. Ir, IN\

ef

+Gp

109

U. PI-ANE PERPENDICIILAR TO A LINE (Conf d):

Steps 7 - 9:

x

1 1 0

C 6

Txr

I

iry.

AP PEND IX AS in gle-Strok e Goth ic Engineering letterings

(Source: Refererre 5, Chapter4)

Lcttcrings Bcforc thc Invcntion ofhinting:l. Old English Lcttcn -- Also adoptcd by cuty Gcrman printcrs2. Roman LetErs -- Uscd by cady Italians and replaccd Old Eoglish3 . Gothic l*ttcn -- Use in printcd tcxtbooks but inconccfiy rcfcrrcd to as Old EnglishModcrn Lcttcrings:t. Commcrcial (or modem) Gottric - Also callcd Sans-Scrif GotricZ. Singfc-Shoke (or En'gineering) Gothic -- Plainest and most lcgible style; cottld be executed with

single strokes of ordinary pen; also "sans serif'

Techniques in Achieving Good Lcttcring:l. Lcttcring is frcc-hand &awing not lcttcr ouiting nor wih thc usc of sfaight cdgcs or special

ins&uments.2. The six basic strokes are firndamcrtal to lettaing3. Good lcttering is always accomplishcd by conscious effort to improvc and nEvcr by adomatic

muscular movements4. The ability to make good lettering has little relationship to onds rwiting abilitg excellent

lettcrers arc often poor write6Standardization o f En ginee ring Lettering:1. Both Vertical and Inclined letters are standart but Vcrticd leucrs are more lcgible althoug!

hardcr to make.Z. If givcn frcc droicc, urly onc stylc (cithcr rrcrtical or inclincd) should appcu in auy drawing3- Guidelines are absolutely cssential for good lcttcring:

a) Ligfit horizontal guidclines arc neccssary to rcgulate thc hei$t oflcttcn.b) U$t vcrtical (or inclincd) guidelincs arc ncedcd to kccp thc lcftcrs uniforinty vcrtical

(or incltuted).Optical Illusions in Lcttcring: Some allowanccs rmrst bc madc for crrors in pcrccptions, or opticalillusions.l. Thc width ofthc stmdardlettcrHis normally madc lcss than its hci$rt to ctiminatc a squarc (or

squat) ElpearancaZ. firc numiral 8 is made naflowcr at thc top half to gve it an appcaratrc of sability. This is also

huc for C, G, B, K s, X 7-2,3, and 5; and also thc lcttcr E3. T1e width ofthe letterW in conbutto lcttcrH, is made grcatcrthanits hcight because the acute

anglcs of W gjves it a compresscd appcararrce.

v.

A1

4. If the horizontal (cenhalj *ot ",

ofthe letters B, E, F "ttq

H-a13 placecl at the exact mid-

height, these will apper to be below center due io optical illtnions; ttus, the shokesshould be drawn above the center.

V[ Guidelines for Capital Letters andNumerals1. In working rhawings, capital leters tre commonly 1/8" high rvith splce between lines of

' lettering from 3/5 to the futl height of the lefrers2. Lefiers within aword *u rp"riby eye estimate; that is, background areas of ry Fpu::t

between the letters should Ut uppo*i*ately equal Vertical guidelines ae NoT used tosP4ce the letters.

3. Vertical (and inclined) guidelines are used only as a part of a lefter grid, in order to.

establish the width of letters'

vm Vertical Capital l.etters and Numerals. Using a letter grid that is six (6) units high:I-etter ! and the numeral I havb NO width-l,etter W is 8 units wide (l-1/3 times ib height)'leters inTOM O-YAXY are 6 unit-rvide lettersAll other capital letters and numerals re 5 units wide'The letters re classified as either staight-1in, lrtto, (A q F, H,I, L, M, N, K T' V' W'\Y,Z),or curve&line letters (B, C, D, G,J, O, P, 9,.-R,1, U)' -Letters o, e, C and g; b"5g; on tiru rot .i*it, *iilt tltt right side of letters

B' D' P'

and R are semi-circlesThe lower portiols of letters J and U are segri-ellipses; while the lower pa-t of titenumeral ! is elliptical in shaPe.The numerals 2 and 3, and the le6er S, are alt based on the fig,n" E which is composedof a small prone ellipse on top of alarger Fone ellipse'

-

The numerals f and ! re based on the elliptical O (zero)'IK WholeNumbers and Fractions:

1. Fractions re twice the heigbt ofthe corresponding whole numbers..;,. firr r"r"r."tor md the denominator should be 3/i r hrdt ts the whole number in order

- to allow cler space between ttrem and &e faction br'3. Most commonly usej height for whole numbers is V8", and for. fractions is y4"'

X. Guidelines for Lower-Case lrtters:l. Lower-case lesers should have 0ree a four ba.sic horizontal guidelines "s

follows:a) Cap Line - Top limit for the ascendersb) Waist Line - Top limit for normal lettersc) Base Line - Lower limit for normal lettersd) Drop Line - Iowerlimitfor descenders (normally emined;

f,o13al AsccDderg Ilegccnders

IA

I

)(t

\

2. Thcrc arc 7 asccndcrlcttcrs: b' ( t b'L-t mdt1:. ifr.* '" j lowcr+asc lcttcrs crllcd dcsccndcrs bccausc thcirtails &op bclow thc basc line: & j,

p, q, andy.4. Tniilt.ibdwccnwaistlinc andBasclinc G+rydcnttg poutforluntu in a.lctto ertq may

vary from3/5 to 26'ofth. qp.o bdwccnfrc cairinc and thc Basc Linc, q bdwccn thc vaist

Lins and thc DroP Linc'Vcrtical Lower-Casc Lcters :i.

'

il; ,rr"po orurJ.rr to*cr-casc lcttcrs rc bascd on a rcpctition of thc circlc' ttp circulu 8rq ortt r ttraiittt linc, wittr somc minor variations'

z. The cen'al prrt {bii;; wuiri rior and Base Line) is ruuarg 2R the height of thc capiralLcttcrs.

3. nascd on a grid E units hidr (Cap f'!c to Drop Linc):a) Lcuers i and ! have NO Ydft:Ul Lcttcrs w and m arc 6 uniJs wi.dc'.j Letters j, 1, and ! ue 2 units widetf is 3 rnits)'O All othir lower-case lettcrs arc 4 trnits wide'

spacing of Lcttcrs and words (wi*r capital and Lower'casc lxscn):l. unifonnity in spacing letters is a oratt.t of equalizing spaces, or background areas, betrryeen the

)il.letcrs by e,ve. ,_ r-4__ _r-^,.tr

z. ffiialtil;d arcas, Nor trc dis.tgccs, betwcen lcttcn should bc approximatcty cqual

(Poor'Spac-tng)

Some combinations of lettets, such as:LT and VA, nuy be given a sliglrt overlap to secure goodspacing.

4. The widttr of a lettcr may bc dccrcased, L whcn itfollowed by the Capital I-ctter A'

6. The qpace bctrre en lincs of letterings is normally about 3/5 to the firll height of capital lcuers'

)iltr. Inclined Lrtterings. Inclined I-etters are idcntical to vertical Letcrs in all aspects D(CE?T thatthcirvertiear g,rLr?ii*r-uri in.tin.o'fonrard 68.2 dcgrcei, or 2 uoits forward for cvery 5 units ofhcight

)cv. construction of singlc-stokc Gothic vcrtical capitalI*tcrs, Lowcr-case Lctters' and Numcrals'(Shown Next Pagc)

(GooaSpaclng

Thc spaces betwean wortls are approximatcly equal to a capital Q,

AP PEND IX BDIMENSIONING

(Source: Refereme 5' ChaPtrr l3)

1. sizc Dc$cription: :ct, a drawing must also givc a complctcln addition to a-complctc SHAPE dcsaiption-of a1 obllsIZE descriptiory trrat"i; i ilillTai,n"ntion.io. Drawings-must

bc dimensioned so that production

personnel in witrety ,rp.r.t.o praces can_make mating parts th-a! wi' frt propcrty whcn asscmbrcd, or when

used as replacemcnt parts. rhc drawing should sh6w thc objcct in its complctcd condition and should

contain alt neccssary iilffiJ;;to utini"trr. drawing to its final completed objecl Do not give dimcnsions

to points or surfaccs that arc not acccssibtc to tfri workcr. Dimcnsions sho'ld not bc dupricatcd or

nrocrftuous. o"rv th;;'diln.ion, shourd bc givcn that arc nccdcd to producc and inspcct thc part cxactly

as intended bY thc desigrrer'

2. Lines Used in Dimensior-ring: - . --r:r r:-^ r--!. '

a. A Dimerxion Line is a thin' duk' solid line terminated by urowheads' which indicates the

direction and extcnt fiu oi*.orioo. In machinc drawing thc.dimcnsion linc is brokcn (usuar$ ncar the

middle) to provide *;pfi;;; f:^f: Oi**tion numtial The dimension line ncarest thc objcct outlinc

should be spaced "t

i.r!, r'inm (3ain) away. A' other para'el dimension lines sho'ld be at least 6mm(1/4in) aPart. (Ftg.I)

b.AnExtcnsionlineisathin,dark,solidlinethat."extends'fromapointonthedrawingtowhichtrre dimension '..rrr,

-ii* dirn.nrion rine meets ttrc extcnsion rines at right angres, except in speciar cases

shonn bclow. a grp ,i ru*t 1.5mm (l/6;) should 9:.!:t wherc thc cxtcnsion line wo,ld join the objcct

ourline . rhc cxtcnsifn"t'#rrr""rJ lr,itia about 3mm (r/8in) bryond thc outermost arrowhcad- (Fig' tr)c. A center Line is a thin, dark line .o*potid of altematc long and short dastret ard

is used to

represent axes of *;r*.il part-s anO,to denoti centers Center lines are common$ used as extension

lines in locating lr"l;;;J;Gi circular featurcs. whcn so used, the centcr linc crosscs ovcr other lincs of

the drawing wiurouiiulr. a ..nt.t line should alwrys end in a long dash-

I/V*/ \\ A*puo

\e"fe,,sL i n

f( r.s *^I e " rT:"'

| . D r n e r r s t a nhead L ine c"^te. l ' ,ne Uce.{ qsE_xfcxsic..n Li,.c

i on ie

FIGURE I 'Dimension Techniques

B I

F-r_3

FIGURE tr - Dimcnsioning in Spccial Cascs

FIGURE III -Placemcnt of Dimension and Ertcnsion Lincs

FICUREW - GrouPed Dimcnsions

( t ) Cornct

iD incnglon t o V l s i b l a L l n e s

( l u ) Poo r( l ^ ) cood

(r) trrooe

FJHGURE iltB ' Crossing Lincs

( t )Rigbt.

r-ftfrI --l

#(z)

t{roqg-lt+n-

92il.

| 6to4P-L

( !) corrrct (z) urons (4) wrons

( z ) vrone

r-F

I3. Pimension Fi8ures:a Dimension lefrers and numerals follon, the standards set in tre precediqg Appendix Ab. Irgibility should never be sacriliced by crorrding^ dimension figures into limited spaces.

Fo, ,*ry *ri ptoUtil, _apractical metrod can be used as shown in Figrre V below.(i) a"a tz) - rrtri.is enough space for dimension line and arrowheadiii - rt *r is room only for the numerals; rrorptreads are outside

ici- notr, arrowheads and figrres are placed outsideiSi*a (6) - Ottrer methods for different prob!9ms -c. Never rru"i udimension figure ove ary line of the draruing, but break the line if this

becomes necessary. plurc dimension figures outside asectioned aea, ifpossible. (Fig. vI)d. tn , group of parallel dimension lines, the numerals should be staggered, and not

stacked up one over the other' (Fig; Vtr)

4. DirectionsofDimensionFigures:Trvo systems of reading direction of dimension figures are sr'/ail$le, the prefen ed

UNIDIRECTIONAL system and the ALIGNED system' /

a hr the unidiiectional System, all dimeniion figures and notes are lettered horizontally on

the drarving sheet, and are read from the bottom of the drawingb. In the el,g,rlsvrirnl ar dimension figues are aligned with the dimension lines so that

they may read from [r, rigrit sicle of ttre sheel Dimension lines in this system should go! run in thedirections included in the shaded portion' (Hg VtrI)

5. Dimensioning Anqles:Angles * di*.nrioned, preferably, by means of an angle and a linear dimension' or by

nleans of coordin*". Jirororions'of the two legs of a riglrt hiangle. The coordinate dimensionnrethod is more suitable forwor* requiring ahigh degree of accuacy' (Fig IX),

In civil engineerring, slope iepresients the angle rvith the horizontal, whereas bafier is the

angle referrea to r,Jith th. ""*icJ.

Both are ex.pressedby making one member of the ratio equal tol=GG is similrto slope but is expressed in percentage ofriw per 100 feet of nrn-

In structurJ Or*"ingr, *gultt measurements are made by giving the rdio of "nm" to,,rise,,, r*ith the trrge.A*tniion Ueing 12 inches; these rigtrt hiangles being referred to as bevels'

, ^ * ]

]F(r) (4 )

FICLTRE V - Dimension Figures

- ] "

AF1Ir- [,..l F-3mnr h igh

B3

(r) (6 )

lfrongFIGURE VI ' Sectionlined Arcas

Brat OK

%ru

1.--"1( r )

(z)

EICURE VItr - Dimensioning SYstems

' r -4L

( . t ) Uniairect ional SYstcn

'Dimcnsionins enetl

( 1 )hoderred

(z)Poor

Y,'It -

t>'tII l . a

t- i') 3o'

U\

)r.

l . - ,+J(r) t l_Lt \7, t-T

(r)

(z) Altgnid' sJrsten

(6) s lopr

84

f lo*v

+I t_T_ || 2 5

$)rattcr

6.

n t lcr - ifor rq (dlrrl

,l

ton

drm

\ i

oneqri o,.

8.

i l

its

cel

I

t :I

I6. Dimensionine Arcs:A rit d; arc is dimensioned in the vie\ t in qftich its bue shape is shown by giving the

numeral denoting its radius preceded by 0re lefrer R The centers may be indicated by srnallcrosses to clari$' the dra*ing, but not for small or unirnportant radii. Crosses should not be shownfor undimensioned arcs. (Fig. X). Fon long radius, when the center {alls outsi& the avail$lespace, the dimension line is dram towrd the center; but a false center may be indicated and ttredimension line'Jogged" to iL

7. Dimensioninq Various Shapes: (Fig. )fl)a- e n'iangular prism is dimensioned by grving the height, widttr, and displacement of the

top edge in the front view, and the depth in the top viewb. A rectangular pyramid is dimensioned by giving the height in the front view, and the

climensions of the base and the centering of the vertex in the top view.c. A cone is dimensioned by giving its altitude and diameter of the base in the biangular

view. A frustum of a cone may be dimensioned by giving the vgrtical. angle and the diameter ofone ofthe bases. Another method is to give the lengh and the diameters of both ends in the Aontvierv.

8. Dimensioning Cylinders and Holes:

"

1'frr ;ght circula' cylinder is the next most common geomekic shape to prisms, and iscommonly seen as a shaft or ahole. The general method of dimensioning a cylinder is to give bothits diameter and its length in the rectangular view.

b. The use of a diagonal diameter in the circular view is not recommended except inspecial cases when clearness is gained thereby. The use of several diagonal diameters on the samecenter is definitelY conftsing-

c. 11, ladius of a cylinder should never be given since measuring tools,are designed tocheck diameters.

d. Small cylindrical holes are usually dimensioned by means of notes specifying thediameter and the depth, with or without manufacturing operations ryecifications.

R t o CcnterPaf se

FIGURE X - DimensioningArcs

Trua

f

c."t,l;-1"'./(6)

B5

EIGURE ){ ' Dimensioning Various Shapes( r )

nl( l ) ,Av

IA-_T_/t\ IA-lt l ll.-- --ll '

n-f- ---T- |

I l-L

-1 l..-l/rliA_Lt l ffi

l .I/r\

titF-frD\_-Tffl

a

FTGURE )OI - Dimersioning Cylinders( t )

( , )

t"_-- Jlrl_T

Lij_1_I

t---]:l,l

!I

FICURE )fltr - Dimensioningllocating Holes

( r)E-:i:ll-++f-fti

t-r

APPENDD( C

AA5) :

(REFERENCE: Refs 1 and

BASIC TA}{GENCIES:i. A Line Tangent t'o ooe Circle or Arc ( Shown in Chap II )'

r. Givsr: Arc wirh a de6ned ccrrler & a spccifiedgoint of lmpea:Y in thsl !Ic'

b.Giverr:Arcwitbadefinei l- .",*,4"b;ntNdr-,the.dcthroug!wbich|be|iremustpsss'2. A Iine Tangeiio to'o CL"Ls ( Shown in Cbap tr )'

r. Given: Two separate drct"s wrth dehrei centcs, & lbe required poin6 of taogeacy are on tbe same

side of the two circles'b. Givg.o: Two sepsra|e circlcs with &fmed ceilte's' & the requircd

points of tsngen.:y sfe on lhe

"PP"tL 'iao oft" two circles ( Shown in Chap tr )' r'

3' * T "t#: H-"Tl.f,il1Hti* point of uogency, & rhe spccifie.r radirs

of the required arc ( show:nin ChmII ).

b .Gi*" . .#"&the&f inedcerr terof lherequiredarc. .c. Given: Line, a point Nor

""

;;;;,hugh r*'hich the arc must pass, & the specified radirr of tbe

requhed erc (Ithrstrated )'.-d. Girno: Line wilb the sPcified point of targeocy, & a

point NoT on the lile ttroQ$ which tbe arc

must Pass ( Shown in ChaP tr )'4. An Arc with Spsified Radius Tmgenr.to aoother Atc'

t- Ai"it e specified point of tengency in the g*i i: 'b. Giveo: A point Nor on m ffi'rL-rhnoulh which rbe required

rc rnrst pass ( Ilhsttated )'

5' * o: "';il:: iT:t:ffi:#-eodicurar to each orher, & the specific,J radius

( lerge or small ) orthe

b. HH:t#: -:yr:[tih eilher an rcure or obtuse rrrgfe wilh each otber, & tbe specified radirs

of the required arc ( Illustrated )'c. ci";;;T*" ["es wkch are paralld to each other'

6. An Arc with SPcified Radius Tangent to two Circles or Arcs'

,_

"d;;;"r; d.1* rhst rrust be both outside tbe req.ired rc.

b. Gt*". i;;;;"iL" circhs rhd tmtst be both Irrside the required arc.

c. Givcn: Two separate .i".;;; one is Inside, *l'r" th" olber is ortside' the iequired arc

d. tTsH] arcs or circles,.one irsi& the orher, such rbar rhe rquired arc is ourside rhe inner, bui

Insi& the outer (Illuslral'ed )'7. Al arc or circle wiih Specified f'alius Taoeentto o Line '&aa

Al,c (ot Circle )'

"

. ail;t R"qitta rrc is ortsidc Gc givm arc' (Illuskated )'b . Gi;t ii"q"t"a arc is Inside the given rc (Illustrated )'

D. .ADVANCED TANGENCIES:l. Two Arcs forming a Cortimrous Curve Trogent to two

Lines' ^ '

,. Giverr: Trvo perallel lines, "

p"in,Ti*g*.y tp".aed on ge of tle lines" radi of tbe required rrcs

b. ffiI+*, parallel lines, ? ryi"r "f gcTq T5ffid on each of rhe lincs, rd rry desired point of

in8ecti'on betweeo the required ercs ( ilusu'ire{r .,c.Given:Twonon-prrallell ioes,apointoflangcncyoneacbof|helines,arrdaspccifiedradiusforlhe

dJof O" two required rcs ( Iltustrated )'.2.TrvoArcsfonningaCootinuorsCurveTaogcntt,o.ibfeeln&fsctingLines(Illuska'ed).

c1

B.

C.

3. Tracing ro Inegutar Curw with e Series of Tangent Arcs, lbe zucceeding Arc tangent to lhe preceding one( Ilhrsbrrcd ).

1. A Line Trngen{ to a Ellipser. Given: Point of targsrrsy oo tbe ellipse ( Illustalcd ).b. Given: A point NOT on tbe ellipoe tbroug! which lbe requhed tangeot line rmst pass.

5. A linc Tmgent to a Parabole"6. A liae Tangent to a $perbole

BB. OTHER TANGENCY-BASED CONSTRUCTIONS (REFERENCE: Refs 2 &,5):

DRAWING ISOMETRIC ELLIPSES1. True Lsomekic Ellipse2. OrtrFour-CentcrBllipse3. App,roximate Four-Center Ellipse ( Illu:rratd )4. Alternate Four-Center Ellipse

DRAW]NG INVOLUTESl. Involul,e of a Line ( Illushaled )?. Irwolute of e Triangle

3. lnvohrte ofa Square

4. Irwohrte of a Circle

CONTSTRUCTION OF THE SPIRAL OF ARCHIMEDES ( ILLUSTRATED )CC. LLUSTRATIVE TANGENCY CONSTRUCTIONS ( SOURCES: Refs 2 & 5 ):

I

A.

tI

A.CirclewithRadiusRpassingthroughPointPdndTangenttoLincAB(ScctionAAA3c):1 . Draw Line EE parallel to Line AB at a distance from Lrle,AB.equal to thc Radius

R'

2. using poinip Jt ."nt"r, and Radius R. draw an 8Ic such that it intersccts Linc EE at Point

O. a !- --r-- t^ a^+,

3 . Draw a perpendicular Linc oI q Li*-AB passing thrcugh foint o' h order to dctcrmine

the point of uneily _"i t"tlt X..

-Ilsing'Poinio as ienter, and Radius & draw the

t"quit l circle through Points P and X'

B. Circle with Ra.dius R Thngent to Arc o I - AB with Radius Rl' and passing through Point P

which is NOT in the Arc ( AAA'4b ):1 . using point ol of the given arc as denbr, & a radius R2

= Rt -Rrdraw Arc ol - cD in

the vicinitY of Point P'2.Usingpoi ' tp*cetlter 'arrdRadiusR.drawanotherArcP-FGintersectingArcol.CD

atPointo.CorurectPointsoandol-inordertolocatethepoin]o|unglncyatPointT..3. Using p;;t o ; centgr, ano n'adius R again' draw the required circle

passing illrough

Points P and T.

o l

I

II

^-{- t \

I

I

oX

c3

c .

!I

CIRCIE WTIH RADruS R TANGENT TO PERPENDICLII-AR LINES AB & BC( AAA'5A ):f . Urirts the in{ersectiotr point B r cerrler, and Radius ir, &aw an erc cutting Lire AB at Point Tl end Line IIC

rt pJint tZ. Points T1 ud T2 are the points of tangency of lhe requircd circte'2 . From Point Tl, and rsing Radius & &tw ; ;t i"ib" iua&ant fotmea by the fivo lines' From Poitt

T2' and

rsing rtso R.rdius R' &; e thhd arc intcrsecthg the secon{ arc tt Paint O'3 . Ushg Point O as ceats, urd Radius R, draw fhlrequircd circle passirrg lluough Point Tl and T2'

D. CIRCLE WT[I{ RADIUS R TA}IGENT TO TWO NON-PARALI-EL LINES AB AND CD:I . Draw Line EE parelld'to Une AB at a distance equel to R-edius P-2 . UsinS also a distance equsl to R, &aw eaother f,ir" CrC p-"n"I b li""-gD intssect;'g Line EE

at Point O'frd'foir,t q drew perpedbulrr lines to Line AB md Line CD sl Point T1 end Point T2'

3 . wih pofu,l O ag ccote< aod Radiu & &aw the ,.qoir.a "ir"t"

passing tbrougb points of trogeacy T1 aad T2'

t

c4

t )

E . ARC WITII RADIUS R TANGENT TO CIRCLE 01 WITTI RADIUS RI AND TO CIRCLE 02

wm{ RADrus Rr:wrrERE.cIRcLE or n oursIDE TIIE ARc wHILE cIRcLE 02 Isnqsog TI{E ARC ( AA'\6C ):

Froat the ceoter of circlc !f' 4"'*: 11:1.*'*f :i1i:Pl;H'I . IIH ff :ffi 3 :ffi-;:.; ;'g: it; i, *,'e.:l " *sl.*^" :3 ;P.trstr *l'.;'' lflHff;;;ttl'b.*.* pJ"o o -a oi,--J*t'"- Point o Ed 02, to loc're $c points oft-g*.Y at Tl nd T2'ff?it$,;;;;[, od Rrdirs R, &aw the reqriired arc passing throug! Points Tl aod T2'

I. CIRCI-E WITH RADruS R TA}iGENT TO ARC Oi 'RB \folTH RADruS Rl AND ARC 02 -

CD \!TrH RADruJnz, wrrnng ARc ol - RB Is