r A )A112 565 ARMY MODBILITY EQUIPMENT RESEARCH AND DEVELOPMENT COMM--ETC F/6 15/7NEW RADIUS OF CURVATURE FORMULAE AND THEIR APPLICATIONS. (U)SEP al M H FRIEDMAN

UNCLASSIFIED MERADCOM-2336 NL

11.0 22

111 11L

NAFINAL HUHILAU Of b[ANDAULKJ, IA,3 A

[AD

Report 2336

wNEW RADIUS OF CURVATURE FORMULAE

AND THEIR APPLICATIONS

by

Melvin H. Friedman, PhD

September 1981

Approved for public release; distribution nlimited.

U.S. ARMY MOBILITY EQUIPMENT

RESEARCH AND DEVELOPMENT COMMAND , Y;' ,. 11111 FORT BELVOIR, VIRGINIA

I"

Destroy this report when it is no longer needed.Do not return it to the originator.

The citation in this report of trade names ofcommercially available products does not constituteofficial endorsement or approval of the use of suchproducts.

UNCLASSIFIEDSECURITY CLASSIFICATION 01; THIS PAGE (**he. Data En.wrd)

REPOT DCUMNTATON AGEREAD INSTRUCTIONSREPOT DCUMNTATON AGEBEFORE COMPLETING FORM

1. REPORT NUMBER 2.GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

23304. TITLE (srid Sobtill.) S. TYPE OF REPORT a PERIOD COVERED

NEW RADIUS 01F (URVATURI FORMULAE AND)ThI 1R A PIICAI0N S 6. PERFORM4ING ORG. REPORT NUMBER

7. AUTHOR(&) 6. CONTRACT OR GRANT NUMBER(-)

Melvin 11. Friedman, Pi)

9. PERFORMING ORGANIZATION NAME AND ADDRESS to. PROGRAM ELEMENT. PROJECT, TASK(AREA & WORK UNIT NUMBERS

('ountermine Lab-, Mine D~etection Division-, I)RIM [-N 1):US Armly Mobility EquI~ipmen~ft Research and DevelopmentCommand, Fort Belvoir. Viryinia 22060

11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

Septemlber 19~81I13 NUMBER OFPAGES

5414. MONITORING AGENCY NAME a ADDRESS~i'j diffeetI from Commt,mfirin Off,,*.) 15. SECURITY CLASS. (of this report)

Unclassified

IS.. DECLASSIFICATION'DOWNGRADING

SCHEDULE

16. DISTRIBUTION STATEMENT (.1 thl. Report)

Approved for public release; distribuitionl unlimited.

17. DISTRIBUTION STATEMENT (of the astract entered In Black 20. it different fromt Report)

ISl. SUPPLEMENTARY NOTES

IS. KEY WORDS (Continue on reverse side it necesary and Identify by block numrber)

Radius Of' CUrvature Cylindrical coordinates Road ban kingCurvature Spherical coordinates Aircraf't nianeuverabilit\Measurement ApplicationsSpace (Curves Mline detectionD~iscrete points Land navig'ation _______________

20. ABSTRACT (Contin"~ ow, reverse side It necessary end idennrty by block nuawher,

The shape of' a surface influences reflectance !cprIes 1-v ore, shape is o1W t'actoraniong mnany which determines how difficult it is to -,.-.e-' a surf'ace-laid mlinle frTom tile air.The shape of* thle surface canl be locally characterized jsjig the radiuis of curvature conc:ept.A survey of' thle literature showed no systematic radius of' curvature treatment wh6ich~i dis-played explicit formulae for calculating radius of' curvature in three dimiensions and "~hichemphasiz.ed applications of' this concept. In this report a new and physically nll~t i~ateddefinition for radius of curvatunre is given and t'roi this definlition thle well-knowkn and

DD IJAN 73 1473 EIINO161 OSLTE UNCI ASS!11111)

i SECURITY CLASSIFICATION OF THIS PAGE (Wh. V~eto Fnt,.,.,f

UNC'LASSIFI I)SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered)

(Block 20. (Continued))

some hitherto unknown radius of curvature formulae are derived. The problem of com-puting radius of curvature in three dimensions for curves known at discrete points is solvedand applied to the problem of measuring one parameter characterizing enemy aircraftmaneuverability from radar data. Other applications include: the design of a device formeasuring road or railroad track radius of curvature, the design of a device for checkinghighspeed roads for proper banking and a technique for the use of radius of curvatureconcepts in the design of a land navigational system.

U TN(LASSI F II)ii SECURITY CL ASSIrICATION OF THIS PAGFWhen nlr. 0ntr.ed)

t vircle ,imil diIereitt 1'reeit the itial eheiuttiii 11i hr rait if- eel cttr attire -- i ~'ll. Thle frie%%

enlelittti i- iiel Ito qertie Ihe li-tliall tvme liiteit-ieeril 1herretlav leer raeli- eel curt, at ft'#- io

eaitrle-tiain mtid pttelar ceeoreltiott. lititil jearatinetric leetitimla leer tli raitItit- eel ettt

Natire in if t~s4e elimittttetl eartetim t eereitiate - %*tevtt t al-i ederit~eil. ed-t tite, %eeter

b irmit tit I licit? jIt s iva I atrgtiitenI- are iiel Ifeo -It m~ thatit I t, eetesr Ifll ittItit leero rt I tti- -I

etir' attire t I aitl lee'r -I eave eiir% e-. I.% it- g)1* tlii- re-til t it di %t II k t,% it kitetiat i\I pt-

teen-. beer the aveeleraitegit if( I % eheIt )1I inI i tree eItri teiia o) emi i lati e -tetIll..

Pit railtetric fi irititlat, Ifor I f-rateIi tz. o el jr attfi re i if rectalit iilr. cN Iiierica I. i fIIilt i i

eeeordiiiate are ilvri~vil. Tihtee feertulae are dheft ii-d to) ,erike seii iler thet raelitt-

eel' eir attre ini Ihe~v vitorlifiate Ntett lhellI (f t i lt h arialee are kttemi it it- i-Ittetice

Oft the thirdi variable mide akee fir tlt- ea-e v% lhere the arialve are ~ittpleill\ related Ife

1'iteh tether. the fpte' tieen 4el* eeeiijeitiftg the raditi- eel iriattre feer -jedee etirs e le'iitee

4eill- at eli-crete 1 ,9iiit i- ceeeideered. ande ant evllivit leerititila -ilitalt leer tititerical

elttilion fittder them- veettijtimu.t i deri~eeI. The rvetilt. (of' Ihi, v crk art, dihei ajppliede it

lotir plereleti :

~ It. fiittiete t eel elte araiteter eltattatterizitiu! ettiitm% aircraft tteattei~rajieIit

I'riete radar data.

If. lThf ede-14ig ,it aeetee leeor (of-trittt e rahjit- eel eurmatire leer reed- for railre,ael

track.

v. te .el-,I~ ee a01 ele I if -v leer c Iiet-, -kn Iii -J ileeeeI reeai I- Ieer lineeIeer beitikiiu,.

ed. \ techiqute leer (lte Iee--ilee tt-, eel raeliti- eel etr~attir cee11111411 ) tt tile III,- -- It eel' a

ltand flid igat itital - tit

Cv.'

CONI N US

St~i. 11011 I iltI'

s(Al MAR 1\1

11 1 L S I R I R) \S

I\ I R( It WCIION

I.Iii I ro ductio

I )eri t imi of RadiIu (ius o i rk III k~cclml~iiHIdr 0 Ai c

.\cct or Rkcprrcn t lut ionQ

5 wcc:ili/autio to Sp ' ck ( uirn. 1()IRcprsceutatioi n m Polar ( o ;rI ics IIkla~~tion~ip \\ itu .\Itcruiitc I Ictuiul I

S. I riuLic fo'r tlic RaiuiN of (iirxaturc iII I IiTCC I IIIuuCnHuou0 Il) , Raius (io' ( uirxdaturc for ti irc, K nt )\\ ii IiuiplicitIl\I

1I \1 1 1 l(\ I I )Ns

I 2. i i I It inc oI R\Iu I r( )d I rdck

1-4 111, tuihr I mid i\ Niici, I ion ii I

I I SII)\s

A\IlI' NI I\ sI \1\1 \R~ () 1 RI sI I I s

I LLUJSTRATIONS

Ii~'ireTitle Page

Def~cinitioni for Raiis ol (uuiturkc I 'sed in H us Rc'porl

L'1C. I ri /I int) 11 Of\\ C I I-K1Ow \%I( ~iuCItrica I RcstiIt

2 Mea,,iriiu, Railroad Itrack I Iori/ontal RaIdius ol ( urx\aturc 24

4 1h li roper 1kmik inu for a I evcl ( nrvd Road 2o

DQ\c icc 1'or NICaSUring' ( )I i1111uin AuItoMoblc spcctl on l2

Bikcd. Curved Road

PrincipIc of' \a\ ic-ation S\ stem tsiNM Radius of ( unaturc 28

I'le Sign for Rad ius ofl ('urvat nire 3 1

NE;W II..III S OF (:1H~.IVII HE FOl-MI LAE A~N1) 'I'llItU AIPPIIA'fl( N!

%%[Ili fi- 4,1 iiiiir-!at tir l'e. rt -warcher-P ati rttr ittlttzttil and~~'a -1- m .O*mJn' li

ar i\\ v Sn litu. e cti m Ion. I;ui, mici,n .s if% . il 1 S4 r C 'rva, ( a I i ll - Own i- Ili il-

t r I). ilil mn ' i l I' ciu U /atr ti ' v iron k A t' '( c(nt un htiL14 114 ap'tiab '4, ii.l 11 iv

(. S.ir lnt' . illt eara/, i n'(,h. -iIr;'rin rtli b a iii I' ill flit\ m,) P.' / ill

ill .Im 'iv ~~ Too N al, theS ra iu 4 In tuecone tdw 7t 8).rtoha ot

Si I- ji ld Rr I ) Nok kh i ,)' l w I /a, 111,, )I 'i Ina /G*r, Ih StI 843I t,, 11, I I

1 ). 1t~ r v .n i ppi :I! h R ti t~li r~. ui, Il , ('uiarur 5k ;Pl iv. bn rfleatl x'' h I ;fi ?

S, wit c.r rht I It.tia Mjru 3,N' 3if ('(tnU I )siti ,r-~ ., I U.mum I

4 .S. olc'.a n,lat ali Wr' Cirmal, 11x lit C , miii it.' t vp Ipn 1w h ,i t 'f i

%I/ .I. ctim n and (Ir . Hmi er i iiflila t(. tI'n't .1 ml~u(f' .1 t/i ', fr h 'n i, Rlai ' Ci .,tdr

ot /,wk at v Gm'rr ai,/ (ni ,a, SurfaI.s. Ot . I mg I Q.~k \N6. 8ta 'c,m is il I.Im''x./ NK 2

1) \1,6rat . I )1upi Ilmhimmi / Irk inding Radi,, o i \ re f Sk% ait N Iri ( a' C 'i'i U'

VU (ari r ,foh ,% fC i ind Cri a i Ah ll'. WIit a 'tomt i'. ''Iih Radiu -j' I ( ua li h (Al. \2. Ih i i t! 1)

8 1I. i, . I1c1riml Ppib, Ii( rM i h S ,iiti ,,t 4l aCrd /~ n I m, (~/ 1911 'n I m ti .''ha.

qI /%~,r Cldl in. it. Ia i Ma" i ad .Mrcntrmisti .1i SM I'll%1nr t~m , 1 r. th. 147zi WI N'Ya i

13 ,r B,w Spt jicrmli I r ( I r i l ih ii .'. u .' I c, a h iu pm* un ta JZ'xi't 11 .111 /i 1 1', hT' Ti , 1l. ,1

14 1, dt% di. I \i, %and / R- A.lmi . I trmI ' p i ,,I ir,: t ip, 'i , I/i I ' S1 P,, ; I, r' Ci'ii I A" 1.

1) Si lp'%, lie . I t I sp',1 ~ ~ a I ~~I ppf n iarm it 1 I ? arpi'In'r '' ml..' 'Irat ihiim. I-CPrmm-;':, ai ,;dv ili n

I.'iremn ti . N I lqtifft PIj..J v~~j 11loai a./ N(,-0 ) ,I

12 k , ( I i lie n 1 1 M b c ipt , m i 1R r /lj a u j i ; m 1 lri - U

Tht, 111111 111 il i- ill milic arl'ort, i- -Ili I'l illo-- it%% 11, 1 n I ... 1-111-6.41 filific- that it*(- liflic

it l'i I 1 .4 - I i I if I milw- hal I 'i I I I Im -r, I I I i I I I ( I I -I I I . .% .-4 1 . ( , I I I --cilliell1k. file- viliplia'i- ill Iffill, in' k A h in if a, 01 mi A mlil Alm 6 a i v - ralitil-1,411-114. 1114-1114,4k. I.m. flit- lllq, t Pill't. fllv v rvk (Oil ill..

r, 0 111 1 r i I I - I , I I I I I-v t 11.1.1 it Ill- I f I I

I i I I I I I i I I I I n , I - I - i N t I I radial it Ili

I H it I 1 4 r I - i I i I I I I t I 14 - r I I I I r 4 1 11 , r I I 1 4 - I I I i I I i -i I I I Ill] I la rvi I I r-

I it )I it I I I I I I I I it- I it I it - I I rfil( -4

F lit, 4 '4 1 11. 1 t I of rat I i i i, I I f I i i r% it I I i rv mii I I ri I% v i i v I'll I i I I I i i i ri i I - I r I I - I I I I I I I I I q - - I I I

I r I I i i I i I I I it - rt I I - I - i i I i i -i I I fro q it r t I i i I I I I I it - i i I - I - . t I i I - I I I , i -I i i I 1 4 1(' fit I 11

I to - r I I I I - I I i I I - \ i I I I - t I I I I it- I - I I - I t 1-4 1 f I I , I '. I I I - I I I . - ( . t 4 ( I

I It - I I I I - I ) I ) , - I i I I t i I 114 1 11 i I I - , i \ i I i I i I I I I I t i I it 0 il l\v -al ill .1 c.I I I I i

midila\ %% Ill ill--- tell- it Iraffic hu-111 I i 1 4 -it it i I -,v \% I I Iv I I -I rN i i i it - I I vo t if i 1'r, if I i A i I I I

I'dr. I'll,, 11-4111_p-l 11,11111 till- 11-40111 lol- 1-cal, it" %% c.lkc.1 tofiwl Ctittle' frfoll) Ill#. ol-Ilivic. it Ow lilli'll \\,[-I -it I-t.( I I I or. I lit-4. od)-vr\ atiml- if rv lllli lllari/cd 1,\ : till- I 111 it it ic -1 -lit I j, c- u i \ , t I

-1 ri Ill u v-I it Ili,. )nr call ilk, 1 4 11 I c r\ t- I I I M I lit- al 111,11-vilt .11 lincti.l. I (tit- 'fill t- .4 '11

ill d i Ift-1-4-fit I*(- I'll I i I I \ ith flit- -h a p t. Ott I I it , I I I. I It t . : f f 4 1 1 , to -4 -1 - --- i , , I

I I ruc it I I I I'l rc 111 i it 11 wt vi. %% If i It, 'I I ill -v- %% i 111 l I it rl I I - I I r\ ill I I rt - l_ i \ v a 111 J I It.1 I I q -. I I cI I

I I io movir. \\ livii flic car- -1 a ri Ili it it I\ v. cl I if I I i i I --, I I I I. ml _. Iv- I it .1 \% ce-1 I fill- 0 it .- I. r% cr. I h,

rvl Ivoi or. if if, I t lic -, im-cv. , It#- ca 11 1 )1 ) O-rlo 4, 1 bal I I it, 1 .4 it 14 1 it it il- %% hich In I I - I I of. -jt i-1 ir, I i i it

rf-ficci it oil, frmii it flat qobj4kci It I Ilillv iw, I I mc'- . I- ill-4. 'I ri 11 -- rill . \% Ili Iv I il. of [,lit wil-

hich tit ii t I it- iit i4it,4 I fri )ill it 1.111-\ I., I q.4 Im I - \ 1 11 qvc( it I 'It i I I , - i flit I , I I I I If -t. ), -

ri if . Tliv I* I lit, Iv I It I-, if I -, If too I I i 1 41 11 w clim I.\ , I I r I . I I I - I I 11, 11 - . I I I I I I I _, r I I

i - t lic c4 im lit it Ili 1'4 1 r ill@, re-fli-4.1 it I I I I I I Alinc ilit I I I I to. v\ 1- 1 It I I 14 - I I I I-1,11 "r.

I'lic pvr (Pii %% it,, millf'-froilli Ow airmii\ likc a 11 mini-I ha\--, mm -\%%iml-hit-141-likv -m-lacc- 4i that 1*['(Iffl I I I I , I I I %%M11d 1 14 , 4 It'W4.11-41 to i I I I it)..

re-flo-climi- fri till a var I I I I r i I 1 .1 Illidda ill I I I 11 I I I I I I I . I I r-o I I - 411-i I I I 1 0 1

Olt It-- lit it 40bli av I liv jlvr i Ill %% lit I Ollo I I I I I I I I It -f I I I I -I I I . But hi m CIO Olt, 11114'. ill. i I I

vilil fill- illf,1114-111-c Oil' Ila p c ()It liv d f-14-4.1 ;illil it I I I a -1111 al-1--la ill Ill i I I I - I I I 1 -111 1 it - a I I

1 a1 1 i tiu u -u r i t h z Iir - t i t I tile tI I r II I I I II 4 1i I a ~- t I I It III I~~u it II

veuuur tu,1 uueu cal10 1 -(I 1 11 r% liii 1,Ih1' 1 l t I lea ii- I l It I , iII 1 ll ~ , ii

% l fr rvaif ill iii i ii 1 lh I r i14 1 ii u i tI i I2 1 i '11 1 i iir- ~ . ,i~-ir

i I ir i- ui-ur I I I I I e. h r (1rt 1 1( 11 t II-I I I j f

i. vil I ie irru m it ltxt. ile rI.,ti ah-iliii 1c ilm ti I rr~ liitrr iirl,d ,i 1-mp

rarr~trr I - ll ilt a ii tIiuI I ;it I~ l Iiti i r % v -ta1rt - I I Ii c thus i it i t i it

%tIrri iZ 'iii Ow r~h m i ii li ii. h aviiir tlrnrurrt tIl tli iuiti ti 1llilru \ an12111i,,-t~

\ Si. adr i / .i h( II, .J11/o . I lirtcr i 1-1. Irrlti lr al or ila \ o nt 11. a \al e

Stt

lIl

Ni 1fit t i . \1 S21l k %ss dI I kh'i i . dil i '110 . 1111 1sc Al '7 1 1\ IlX ild S 1' 1Il II

\V W i nc, I )l r 1, ,h al ati llt ii l \ uh-tilssii 1. kt . I 2 a11id(o I1 1I

A.. i421.1 S1111a d R I ,1!L ,'n t 'l ,O ~ ,w a la rl a il n -,

t IIII'III II1 4 1~ 11 1, I r II I l_ a 111. -m ll._ I 1 1 .1

I ,II 1- -It_ __ _ _ __ _ _ I f) - ila t 1 I, I IL

-11 -f ii

1-4 1 11 a I I I I 1..

ri,, r, - il tO, .Perpen1di.uiar is-ecior , of ithe t-wo c -h m le I t1, oin t0. , her10,1

\In 1. - v% .I' , , I,, - i- A\ I- A Ila, I- , I-Ht- A "ilcl.''t A~ t A-1.-It-h l' ri I I i _ , , -t' r. i t'ItI1 l A' 111 A, oll ,' A f., A I- < I A. I I A, ,I d I A ,,LI ~ I

%\, ll A II,\ .Ir A I. I , . 1, : I i, 1-, ill cl, t.. u Ah i it III, IA I , \%t l', ic l it, I I -A . ., .I ' I, I - , +

I. h l , th at i t- i ', - . - a - A l\ I I \ , 1,A, i I I I I

A, , -l,, - ll t-I I A . it II ,l . , A% - ,-I .. 1 1 it ' 11., , ,\ - - i \ .l +tNi I

I A .111A, I A rtd t~I -1'\S ,A-t I t : l , ) II I -r ,,r \ ,. I- , A( , \r. r. - ,,' , ,\ A I:- IIIl -,.114if A' A,' v ,, H .- , 1 l ,i I \ I I lA A- tt11 + I I a. \1,,- At A- All , A~l I <,. - A~ I I,, ,' A

\\ t - i,1 t b it IIA I A ,_A,A IAA l l' , +] , A hr II , , 1, , 11 It - r I ll t ,, IIII A i l ,, I I L " I ' %< I, l r," I

I Ii i ,, i I l h l , A AA - '% 1 I r \I ,, i l A I II II ilI, I IAA, Ii I

Figure 1. Definition for radius of curvature used in this reportP _,P and P, are three adjacent points oil i Curve C. The distance 's front P to P' cquals Ow,

lance from P to P,. Perpendicular bisectors of these two chords ineet at it point 0. The, radlu,

curvature for the curve C ait the p)oint P iS do~fined as the distance, from 0 to P Il t~he limit ,approaches zero.

1 , t I I T A t I d

, A', I, fl~ l ,~ I c ,lh , t: ' t: A IIt~ cl~ JI,{ i I.II I I % I I

Itl~~~It icuri tlit' It-Irt-i itIi III #-riI i II i t dit 1 tti it itll f li I tIt -I i i i t rttl(dlltl. i lt'

2.i i fie, i n of I BI 1 ,pr iit vi If (ir Patagrap i . Iearn a tr (e i(r- if I tt . II 111 411I-

lIlti .13)1 111 'tr% liil ti'il t' tiilt-ap 3ti it p r-"r p Ii til i'~ t Cr111 Pho% iI tha fli t- pa ertll l it Itr r t t

'til oti ill- rdl i tii c liai r- Ilii lwirtjl i littliitilt-ka ltt r ame-tlt'Idr itsl t ittj ftilt O rw

%%ie t r - k m i i i m f0 fp r g i I -1)%i i t e iiI tIIIo

I a iii if -ii % a tir lt t lt- re- irl t_-rv - i h t lt-11, 1il dtlilit t'l . m d 1 1,1 d

%s hichI van lit, rv's rittull in the formi

\ + 1112 . \ 2 +4 1112.% 2. I-

I O

III m (0111

_X\ I e 4.4 rit i atc- IV ) I a proac a litiI i i- ii eI ih tl ,I II h

,' ' \ ' l \ " a I ,' 'i - ui - ,'J Il \,u'' \ii. ' .l l , ,, l ,., l , . r , . l , u Iti ,l

P .1 11 \ 41 114 \l' '. 1 q ill ll

I1 i11 I-0 17 ).I I -i ,I vl I'lllabJ: IhI l, il - 4, i-..'r- Ihlla Ih* ,li il ,'.,-il',' ill - I. ,li\ ilh',l I,

-I I l 1114i 1 ill f~ ' th ' -,','e ,.lI 1h ,'..14 11 i ,. ,'In - 1 l i I I ll 1 l il I 1 - 112 1 1~ o l 1 It'll ' ,\ ' l l , , i-

I I I'lII I l , l'i l i ll d ,,, n~ Ihzi l \ ' 1l, 1,,*, U~ 1- ,u 1 a 11 t I - 'I~ q r. ' I th,.I I

1, i li , ' h ( -\ I Ofl ll i uI l , if t ;I it l I , I , i hl i t r v t .I l~ lilll v - )I i t l' I I 2 l' 1 i 1 4l' t h ,

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I,'r iI .... ,,

\\ \ \

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ill'' ' lll lli ll" ( ';l ll] l}{ . ;1 . \ - i| *', tl~l i,,l- ( li' ll il' '\1 1"+--+,1 i l t ,'l,11111

\" ' 'SI11

1,141 114 rI i ti tai t' o ct i tire p)i ftiiiid f rom

SI'- ' + I N 101~

'. I'nrm .i riv Re eejr l fit I itII i Ol. I I 411I it. _,I\ t-I I ji I raI I I'Iri I Iir II I

M I~

A ~ ~ ~ ~ ~ ~ ~ ~ d Ix I x cr e 41v f i f-rtIiti t u l r o l-v-.Tl h l I .4

itIi 'l l, i I 1-1I -1. a Ie t f- titi If iti N c%% (i C a ii ttt f T ,o4 r - 1 4itia

dt

(it I v

(13

I.~ector tRelremellutinl. Fin tilt defiuillion of radiiu.. tt cirattir* gi~vtj ill thii

relpin-t. it i. aipijarewIittat the radii illi iumallre i. at jeoiert., -I' tilt, tilre aittile andi iittt

4ofi tite' cetrdiiiate -%-tevm. iliti-. Ilii radliiu-. telt' virat imre at -talar anti, tie' %% , oui exp-,e it

to Ili- rep1 re,eultale inj term- o4 -ernie' ca~lar cenultnatimti 4f %i'etetr,. It'it radiiu. 'vctetr r I-

anthre i j are unit %ecetor aheiig lte x and % axi. thien the'ilt. ~ etr =tie

v(t r xl+ tl.t

and tilt- diie'itratjimi % ecterit i- Z tl i

lii.' niviralee idi I 1 t lotik. like v ule 11t Ift ile iiiutIItI tltilt-' vloittit % 4.4tqtr %% Ii iii

tie tiemIiiiia~lttr (1i 11 ht look- like tule z colipt,:eu tltert- tit etililtc - rI11. 4t4 tOwt lteierrtitu

allt %aeiit.% %tttp,. A~ itli tile lill i t' iulatimil t )6 ait I7 itili i tf 0. tb'rifit'tIil hatt'tia-

tuuimi It t1anl lbe %% rilii i tihe (irii

Inii etillueu I H tihe ueiuuultraltr anti Ideuiouiiuatiir art, Il,ii ar-. Thiu.. t'tliiatit i 181

i e xpe\1 iiillIN tlliit 1) ai m.ealar imi~teltlm oi' toiordluuate -''i'i rai-lalttim-. rotla-

t oll-. icr bIt* tit cttorlinate i.At ,ei tti~.

5. Gecaizttm (n Sepace ( un es. NoIvmIIe thi v( Iti it I it I gt 14 , rii ti

curovjtulre (if a enrile o'ttifiine I t) the Ixo plate ill term. of tile, nagnitide o4 ile z im~t

iteuu 4f at %eetur. Sup~pose now that the euurne is no longer infillee 1,e tilt' 1x.% t jelin'. III

that eui-e. the veineralizatiin oef equiatiouu I 15i

r( t M I( t ,i + It )j + At Ik

R~ealizie that lilt- raeiiii. oii' ctiur tuure i- olefuinet a%- Ohw Iimiiin_ 1eetititei (it' threr litoilt- auth

thlat thiree 1 ettiit- dlerine at 1 i . jimsi. iii at cimerdiuiiatt' - \-te'11 *iiuaiei rotaed .4t that

the \ and i ,at inl tilt pilanue ti' lt- inn e. e'ehultitn t 113 tt , Inl Ii ill le %auth. illu.

tiiere i- li edl tee t~, t41ee ietautet 4-mierdiuiate bvil' -'u e'a ie ii.' aetla ttll ail 'lt .414.

aee'e rt r ftll'am ll eiii'l wt't'ter- ililect thee'1 Ow ri I inl atiel rtttatt'ti e't-florthinlt - %elu-

Ihiui-. f'eiuat iel t 181 i- % .thit I'll -pacit, c'urt'-.

10

0. Repres.ejtaitiuii iii "oi'tr ( oordinait,'. III pol ar coo rdiniates. tilt- rad iii- %tctor

Ar = r 1)4

I. Itre it is it imiit %ecteor ill tilt, directionc of r. All1 three of the quanictiies ill eimatiol 41 19calt Ille timlioclt of a' ,iitetion i,.f limeit. Differenitiatinig equation I ) m ith respiect to timle

tielt. ejfitv is f4okllcdl:31 35

=r =r ar + r6a 1101 O

[it equeatuoi 120(4 &i,, i uit %ctlor iii tilt, ulirecicc of hnert eing~ 6. i c eecraitmc xeeteor

a = =r = r O4a 4 rW + 2r64 aiP

Fromt tiecatloll 12M4

+= r. + r 24.

Irom ipte1 1iioft 421) and 12l))

it Y 4 rzr r + 2;64~4 k

hT.m. fromi vthlat ion 1I 8)

(r + r2 2 p1/21216' r 2 Irh ( rW+ 2r6f 22ii

Ititatiei 4221 is i poar tcorciccatt accitltg (of' vhecatiticc (I 141 andli .ctutalv it' r acd 0 art.1,11c14tiercs of tittle. To sholif the ret'eialblente. Of etjciatucn 4224. 1% thit is not 6%viiu iti an Ngot' tilt- ried' reft-recctt. it cacn lov aiciedt to at iitral liarauettric oftc~'catec t a ur.

Sm H i n r twl Iuei on is: ,n-'s~ 9 t

K. . S Im ll 11, h mN. dilson k ~le ( 16 1 1.

Si IP~ I IIin Iit- paarll ic~I re I rqe41I i I)Il 1t (1 i irv It in Ifi v nqatiui 122 1 %ItAld- it a a

r i. klo%% ii a at I'lii t i' 0:

r = 0). 1231

lit Ihat sd-s. volaio 1II23) sdll be jaramteterizi551 ciomens~ictoik If% Ow~ reilat ipil-hii

0 = 1: 12lio

r =rIO i = rlt I1. 1211o

Fljimtii 12 tit) and 12 11)) arr viipleldIN equIN Avl 14 l equuhi 1231. lit vijlultiml (21.1

1(111141 lit hhissight sdit, it a iencral Ilarallctr rather than a- at times. khthigh ill udri~ iticipldili 122). 1 % i- tllillIlt 4dit it lime1. it' 5511 rc% iqN%- the sicrix atioll tt equlilim 1221 it

i'- allhsareml that il i- ill ifast apicale ts' anl pariirwllri/.alnim id' r muid 0. Fnim lpla)h

t on- (2 11:

r 7 r

1251

lit coisliii 125; fit t refer t411is siferviitIiit Iif it i I I h re-1 ,- -1tj i tme ams I )rintu- re ',r If uhf-I

fereuitianuui %ill re-lustI 141 0. ( -6wl eiulatuoll (251. euj(im ion (221 hbcopllf-

p r' 2 + r Q~iIr" rir _

%%111 h ic i. i alternae 4 eri . at if silt 4-1 p l m(I i.

%. -inilar to-hiflite #-.tit be, awde is) sirip Owe apiplicale tisritiua l'r ths ca-s

%hiere 0 =O~r(. The rc-tilt i- giush ill this apiduii a- equllimsl ( N

7. . hu1t ioltmiij witll] Alliruiuite ITrvaic ti. he4l-E alllIllteldIi lIA

ait l I I I sim his sIrgsi \ frospillif hitIi iv ,v "elsittricaIIis rjmI l ll i isjrthoi Ct rats1hi- tif* 4-iii attir

11oris,r rI il I i am I I t#e 4-1~5lsh )i\cfmtI tl-ftlitI if ll ds rat Il st so'qtir\ at I Ire 11,1 11la(sl- It m ait i

12)) -Ils. dial tist~u ssNji itiimi. are tsjiikaitst.

#) 12T j

i-r " -l

copi~ dim-c it opi imi12 Ti % ith I ir- 1 rIl r* 11a it' It 1 i

iii 1111. t co m- i -itiI

r \ r i*I r hr iltllil

,A 1N B4 X*I 4jto I). ; ,1tkt.

i: N r-i. *i: \ i-, = r: .( r ('\ nl). 281ll

4 lit \li4 th A.hBnllR

r \ ir r ri r l- r r - rI l

i , ii iIi ti1 i l Iri il I 1 r lh 1i r irn

If:~~ ~ r ri rf m rr r l' r'i: '' - 1 . ,2 1

0il l - 111,_ Il i il ui ii'm 1 8 . 41 1 1 1 1 'l l il l i 18 (li 1- .h , l i t rl. *,ifi l . l It, 1i ..

I k l

It. Firi t mi114. for d le Ba11(i i. (of ( irat re ini Thiree Dimii i*I. '~ii

iooil iilltiiii I I I %aliii iiir it -pace emir% v i\ 1 irv-v4I inl rectimi-tiar iiiiiriiiiatt vali lif-

I i ll % 'i i.. 'Ilicti. I lii r~i iu- 4i it'ciro a imrt i. I'mniid Irofill I~~:

+~ + ; + ) I"I 4 ( III(, 42 +(/ \/4 +4 *4 4

iiijj% I,, uuliijiij. M-ipii. that ftti _,,iiraizahli- ill' equlimii 12l)4 allot tipiaitiuiI 121

am+ r A,+ (2411

am 4 m~ '~+ IrJ + 2fh + 7 k. 1211

I'livii. flii paiammi. liri miila fio fit, radijl 44 crtiiri in i, Ii . iiirical iiuordiiriit- 4"1il

+ a rO &^t, + rO -[in (It.:

AN I-f 'r0 2

(1 '% +n1 4Y + 2rl* - ro* -il 0 4-,, 01 in,

+ (r)-ill 0 -+ 2 0 -ill 0 + 2rik; vll- HIa

In hv ui.mtmlum I W4 - ii-efiji idiiin~ anii / .are kniim a- tiimi- fit 61ll.

% = % :/ = 12l'

It ,i ailpl 1iif tuirimiia %aliui ill tii-a-v. tv 1 iru- ciimiiml 129 plia"14-irli I. file

1iiN filll-

k1 P

th I. lhvaproria teti XA. l . mil o l.)) radiit'ofl arel-ftiif cant het toin' ftrmiat' are1-,I t- it re a I , i teill tkentd I t tr ia it iatt are anei c Ii -tiri ti i.' iivde t viitieii

\10)l eul i an ilit 11 r ing epail I X 2

Iiigiila ii t\12)iiliti i ) - i-til %%e ilieu o t'dtnate,r~i ' r- .0.an ,.ii art knil litio a-

ti,1 ttui aria lt I i ') t.aII, i t ; q I.i it,,et Ijti I (li eI ittr \ 12).

9. ltillad u of ( r Eqatn mI\ fTor' I X . and (~ 10IQ). I% hlicht ar tit'ii it tiiiiofe 'un ti1 til

t'iltatut- ar e tunemi a t nti~nct (iritit' I t rhe thidcor ite. ae iuri- ttt hIt' tliit tae ill-

t'A1toit'idol~ 4 i ''for otitittf file %arualtit". i'truilat arte noli for tOe ratini- ofiiii atilre

thu attirt' realize that lt(e ttttali liite'reiiil dii -- vi It

dI" 1\+3th 1

l,4 1uatitIll 1311) vut- __ a- at fiulictiitt ttf i~ .trI It . fltifft'r' riutui eq ili 130t i-,aii

1. ll li- -1111011-1 il i f llcl ol o \ M il

dI 4 I IN I I d tI d\

SIIIt lf a v k l % tIlm lv p Ill 13 11 a d (31.1h11dii- I1 %jti .c lII.

%it444 '. dI vti % v4 d44 r144% tii it444 I v i4I l~ i .oll ; 4 con I I. l i6 111-( 1 1 1 1411 *1411 .1 V, 1,1

"11124111111 t1'- andi4441

Il %

.. :111 j

% \m 0. 1 .14 rl~44 irc k l%it nI 1.4111 441 4'Ir% 4111 b 13). 1 4 13 11 I l, I 1 ll 111 Il\. 0 4 1114 Il144w

1.111141 l44

I I

0 ~ ~ ~ r 1 1 1 I I r I I I - e m lvl I a i- '

tn IrIi i t. s z ti1iu Iiit r,4liti tit 1 flIIit 'i irlit i ' /1 ve VI' x ~ l tit 'i Irtlt I Il...1 I

F F FV

y z y z

G G G Cy z y z

(it' ~ ~ ~ ~ i I ,I fII f II ffi 1i I i dvl

le I I o II i Ii 11 37 t IitI I II II t ' Iit it dIIi IIII14 1 I ti

IF F F F

C;G G Gdr7 r &

Fr z d Fr F7Th

Gr G zG r cz

Hit, ,alt it'Iltill i- lc li-i' ilvl 'Aiiil ii il tiir11 and tit1. i-i U t iili t

I , i II III (IIII1 * 11 litattir ri1 1, I u t le.e III II 4-iI I 011rI 1 w I-IIrI

kll~l\Il i illic llis

I to - aI ,r l -I I 1 I I I -4-4 1 - NS Ia I I i l io'1

I i Ii~it 1 111 A i tti IllI trlI~ I I I i 1- 1 111 i i t i dll 1 0 1 1, I,( I II I Iv II

Iit 21 1 jit rj t iiiia tirI ',l~ itl I~ 21 i l I) i a l t-q r i t I li i t i I I I t I I i \2 1 2 I

Ili thi. paaura:,, . it me'i..: I-, :;.n ,pv Io th-r dlK'i II\ II

1%i t t il. v it i111) 2i i l flrr jl.11.j i- r v 1(il - I l. 11-i i Ia 14 ol O

I~ It I. i -I__ Ii ~ I

_______ " t ( l|l \ II " \ IIl l

iii hi I I d 'I)I I I.. I 1 1.11 11,

I I I I...i,. 1 ll ( I I i III , I ' 1 ,ill, 'li IlI IIl w 1,ii 'i l

12~ ~S ll 'E 1* 11 I * ~tiI I I I i Iij 1- 1.If0

r- I i l | i I n. Il

'111 O li 111 11 1 1 1 1 11 1 lil.l IIII J I

ItI

r. lm - llii a 1 0 1- : *1ic pa '14111 111il la l '

fI 4 1 ii '.. I it, I IAr I 144441 4' ille1thhh;14 lii Irj, : ii'.~I'~ i til .i.~i~

'/1' 1. 111 13 1.1(-l 0 1.1 - % i flill4 m~ . r11 . lfl- 4 r~Ill i ll11r i111.-f 1I

Ih.I/,tlI I I ~% I IIgl% h I mlllilt. hIIlflhihI. if lj2 4 -. I t, I Ig I % 1 4 l- It

ir ~ ~ ~ ~ ~ ~ ~ ~ ' I -Ittit-11 1 v f.11i 41i

i--mr 2. O

PI

A CFigure 2. Generalization of a well-known geometrical result.

AC is a diameter of a circle with arc APC. P is an arbitrary point on the circumference of the circleand if v, and v, go from P to A and from P to C then a well-known geometrical result asserts thatthe vectors v, and v are orthogonal and this enables the radius of the circle to be readily computedin terms of ,v, 'and ' : . In paragraph 10, a formula for the radius of the circle is found when v,and v, are vectors from P to arbitrary points on the circumference of the circle, and in section 12the results of paragraph 10 are specialized to the case where v, v .

OIe- %112I 11 a 11.

Ow1 Ina l- ri l. 114 I' I I an411 a11 ircraf i i Ow' I 1144 ll'' '4'.-o l * 1' 4 .1 -f, iin A 1 11 Ito I a O Ii I -c

f'im 4i i' I 144 1 , Ina11 144 .'41f11'1 l Ii I' I--Iv hi-' 141 I4' I41 I.I11 I \ I Il41 li , 11i4 I I hr li' 111 4-

114 r li I lrI4I l4' i cnroo ii re% tll.'Ill i r\ 111 II " I1II4'I\ I if4 4114 I itIIiII a - '' a .4 1 1 1 '41

.1,1c.14' n villiii' l 44 14 14iI ''4r '()1 al 144 .411.1 r;I 411-1111ki% i' I li 441 1114' 1 4' i l'4'I '4 li44

I11 I r I i- 4 4'14 1 r 1 II I). itl Ii, 1 4 Ii"1I4 111'.'+'4 I I 114 i I 4. I I I* 1 111. I I ' I -I r* 4 * I( ar - 1 1 1

it11 IfI il I ''

I 4 1 14 4 1'41j411 11 v I4I IIlI I~ -II ll 1141 I .I 41 l it I I I4kI. I r a " i I I I iI . 111 I'1if l I4 I4' 1 4 1 144 1 '4r1it 1it14,Il I - ug14 11 u I I , II:

Il I I't4 ', *I I42 I I' 4I'II I ,I I .1 44 ,1 1.1 1412 i 11411444 I41 \4 Iii I44. 11 -1I~

ra1111 ano i' t - i'iit 441 4 %%1' IIl' ii% i.'th lu' 14'l% 'l lae . 4 11 1141. t d l itI. I.\4I'll 1, 11410

'0141 >>444. ll \lI(Rd, p,4'l11 -1,414%-.

"Il-%w il- (f 111llfold Tr-ic . O~l - a vailIrltiict- l

AFigure 3. Measuring railroad track horizontal radius of curvature.

AD represents a top view of a railraod track section. Points B, P, anid C are point on the track

(mrounted on wheels) which can move. BP and PC represent rigid rods of equal length v I oined

at P so that they can rotate about a vertical axis through P. Then, the horizontal radius of curvature

is determined by v, and the angle between the vectors v( and v-.

T II v\iii-i I~ 11,1 1 1i i, r iI 1i c h 'l

-1r a i~ i 11 i4. 4 il6 1- 1 1 11-1 61 d I r i I II U11ili i 3 11 )

17 NO rdil, 1.

r) -I Il

% I

a 4. l i -IcI I I1 11 44c1i It5. - 11211 Iik l a Ir Ii- h --i /l4 .1111 11 r . I I s cmI i i I I ii ick -m

B

P mY2

E tan E - mg -

mg my2

A C

Figure 4. The proper banking for a level curved road.

P represents the center of mass of an automobile moving with speed v perpendicular to the plane

of the paper. AB represents a road surface banked at an angle 0 with respect to the horizontal

AC. The road is curved with radius of curvature p. The net effective force acting on the automobile

has two components: a vertical component mg and a horizontal component mv /p.. The diagram

shows that when the condition tan v) /(pg) is satisfied that the effective force acting on the

automobile is perpendicular to the road surface.

f'ic ie er alicicielIniiig tiii- i- -keiet iii Figiirc 5. W.i I. andc III) all

rvjirv-'eii rivid rotd, lit pi nted ii "i i fIR- 1. aw I ad -10, (d e 1tilt .fit . a D I) a 1cci-

teeot tit' ee iial t legth %% itt roed III) andie. ;lt-e. eeliiitt c iji tlc I##- eillt toie a iuccI

%% iditt or omeii iiiti.4er iiitilctpe cf'it aLaine %cIcitti. Reod, W. IW( . midi III) ia~v hile- cirittcc

-cc tha~t flt-.% vani Ait reete ireei t aeeit a crtival jiui Jecaetd at piiii If. leclir lotcijcte

toilled il at heceli . t. . aiie 1) art, reqiiircil Ioe cc1 eralt, dii d-eie \ jucejei lecateit at

potl \. It. ati ii A aln, thiei~l roadi -i vae acti onie keeje- tile, fillci tilt- I aiiilleui

bille %iicil ctitiiic- til citit,o cit' the ctir\e v %%Ii t tile jeerecu at 0 kicp I ti-lie o ici cIi tie

I Pitei tjnc I ilje 11 deitI I#I it1 in-,id ci Iftil -ilr% v ;ti(ict at-cP keele iIit re itII I appjrocivi i iat I1,1 wcrutenIic Ieear It Iit i elir% v. [iurthermoicre. ait ieeir peelccji keel, thee reed- it

It -it1111a iirtP jjei li itvIi % ci Itcat. 'I' Ide\ icev ha it %% P ili itd i ite-itrlil it i-tfritii -Iit - at It.

)ii-i t uetirvcI t- ie i a Ige ieet-%ccii tileit roed If X t atd it ff:( . The ve-ilie of tii and~e vqult

III . it, ill equaitioni j.)Al). The either itilrunieit med-lire. the aligle teet~evii fill and fit-

hoerizonitalt n cor~i te ti flit, an 9 clefiiiect ini I'i'gire 4. A-inail jercgramuinllce

eciiii11iiiter take- at- ilIIf11 II#- iet-11ioth c fi t rice WI. the angie WCf. ad li , tile v 0.i~t Thnci

tile- radiui- ccf ct urdirepi coimpeutced froent eqioiii (50) and tileit, illi -heed-4 t'recll tilt-

reviltili el Pt, tilt (h". It, tile, I%%ee iighe arc.feet d tilinltieat\ and eeiliitiiiioi-t\ ill-

ti a prccralltulilce calcultatoer ii- tile- eii ice i- miiied iatiecui tile- ellre. tilati aii eepercteir at

It cvan ceiiiouiil[% moneiitoer (h ejtio liiir .jee coelljileilII i t- I ierq iera Ili inaIct-

eCitlliultcer aind cbeck that it i!. ceul-tault % ittilli certalin Iilit-

'I'itii Itea itIItiitrilcl iini ili re -s 5i- it 1-cc dfit 1ite'it lice 4 trllie- ind rmlnrct track.

C/Ih

h

AB

hh

Figure 5. Device for measuring optimum automobile speed on a banked, curved road.

The device can determine the speed of an automobile such that the effective force acting oil et witl

be perpendicular to the road surface.

I .".eiimir am. 11d Ni 4sgtioul S.. t-1mi. I it. r'lifili- (it4 11ir% .iiir. i~i..,al I-,- i I

4 1ru t -ii I I~ Ii ll iiimild 11,1%i c l Iiiil 51 tI I ildt i .

V

O V IVy

Vx

xFigure 6. Principle of navigation system using radius of curvature.

By knowing the radius of curvature pAt) and the speed vU) for all time anid the initial angle .,. the

angle .is known for all time (see paragraph 11). Knowing 00t and (. W the x and y components

of velocity are known for all time as the figure above show. By numerically integrating v,(t) and1(

v (tW the position of the vehicle is known for all time.

I~ %r ii cft i 'lcc c q ul icil X t-t llc lil-,- Jlcc il l,-c . c14- -l- tewl d 1 wi .rc-I ll fc- c ill i c u l. tc l -

he w i-tic iliii 4 l i i- lIt 11c s4.1 clf I lIt al .111 - 4 1t 11 'It c - ( it o Itc du t I. x li c .ll il.it. I u.t

dii ci t-li- - mic ll. Ii l itt c ll id t ic aindic il ici 6i 1-11111 i ll- ut Icll. i.-l i ll.

It IIccIN Ii It rciiiiit It I a Iii iciic c citlicl~\ i~cici- I i

tctili~~~t~l.. ni Ictii lit Iit it Ic Ii Ii cii I Ici. j 1i111illcul atit I cititiil cllc-

-I I it ifc iilil- I c I -I Ic~iil~ I t I Illi \ilt Iii , -tic ii- o. '. riIl tI , .

I i f I ti l i-c -r i- - , 4I. - - I I i i , . I II - r i I .- I l i 1 I -t ~ 4 - ar %%i I )1- I %S I, I

-I: a I Is ilf III %%I I al ii t 4a .4 111 - I I I. 1 I I i 1 I - -1 ill m id it I I- I I d i I. c t It I.- i p il k

Vill 110 1 . , I chill Ii he ic-.ll c I Jill- cIcc ill it h \- ll-(1 li il r.ia i 1c - il'

I I t I ItI k I i a t t Ili I I I I i I I I I . 11ith tI

'4% il il -' ll O wt rt''.' rii il l it ' n i t'v palm . %ril f li l u ll111f a lt

<(-Io fi - ,t-ti-k i , m Ii ) t o I it)% ~i;Ird im )%Iit -,% IIA' tII tnr- lINf a - .I II

ill Ill Fhe it 14-"t 'r ti ll- illd 'ittth I row14 %% t)vl al'kiim ia ati illta ivt o 111' the11 it-N 0,

tilt'I 11.0 -I I r It 'ttttl ti I l4- I lt .I ic n ' i1 4 111 ' .11 i 4' .4lllI% I I I1--44 1 all ti I -- f- '11:-1Ir

itr I11 11 I1 1 i iII i r r Ir II 0 %% I t itt 1 1 1 14 rN(i

Y

x|

Figure 7. 1he sign for radius of curvature.If one imagines a vehicle travelling along a curve, those points where the steering wheel is turned

toward the left correspond to positive radius of curvature while those points where the steerinq

wheel is turned toward the right correspond to negative radius of curvature. The reason for this

is that in counter-clockwise motion s and 0 increase together while for clockwise motion increasing

s is associated with decreasing

\ If -. '- 0111: 621,1

1 0,-

m~~~~~r,~~ ~ C)im l t~ l..11l m l -. iltit\ .11 1h

I\. I W\v lIJ !,lHI ,

11. A1 . 1 [Ilu iii In .. IIh ,' r qi III- 4.I . i I-- I, .u ; t nr, -t i.I Ilzit-III ---I\ ,Il I I I , v , _ ilq l .l l,

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\ il ii I l r i - I I r, .4' ili I I tI II II I , I \ i . l nI t \ ' I Il -- i i I 4 t11 1 1i 1, il t'r d l -, . ,,

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k ilitI i .I. t i l i -,- I I l I I I i i I r I -I _, , -i li- r ,- li l , l l .. , l r , -

Ir j Il .- III III III, r i f - ., I I'l r l l l 1 ~ 'l l l ' l 14 1 o- 1111- I , ' I t I I I lit, I I i, I I I la . -1 -on I I ll .li l ,

I i l~ l ll I\ , i r l ,. I II ; I Ih ,- I i,,,, 14 1 1 li1r h i t-l I " I I i i ta1- i. - _l 'l, i , I.

\ -' r, h ,I I h , lrli- , I II t lII I)Itil' l 1it tI I , I , 1 1 1 -ll ,,i-- , I _I I t .I i , l , i- , i -

,l I ," < l ,, i l I\ Ih , I ,% itlll li. h ,j .1i 1 .4 I I I )I t , l -i Id I ,I i i r il- . 1 r ,; ( i hI-0 r i ill I I l I I I "

t I " 'if i l I+ l' 1 1 1111 1 1 it Ii "' Is I I . 4 1 I', i f h I 1, v l I I I - I t i \ l t

' I 1 1\ l i l h . .. i l il "

l i t ,I -,r i iil lu . I IlI I I , le I Il I \S i -I 1 1 I I.'i ,' lh -l l-, %k i.

11iai 111,i i,' 1li ri Il ict o i- k nt'i + l l -< l i ,', if)' Irr ,, I I X, I ll , 1i I, -1 - ,,2 1 1 1 1 1I l

I \ ic 11 ( i 11 ll i t I , I I(- , i fi %tl i i -, k hi iv i f i it1, 1 - i l i . r- I i , I m i . I Ii t i l -I % Ii l Ii % I ,h

I (ln i Iili I \,2l'll 1i f-l I I, - ri h4 1 ht ri ~ Id I 1.II'. l i l ll, i I IsrIlt -I i .. , l l -. 1 ... I~l

(lit (1 , - i i 1 Ilit W 1 (il '\ , I t ki , %%If f ll lI , - i . , . ,l l i , l \. Ii i, 1, 01, 1i ). Ill -\, ,,

Il il 1 , i m i I I 1i i I to, riftl Il i l I lir \ ilI rI I , Il i . I , if ,i I i i , - . l i I . .. il , t I l I I~ l 1,,I

-ill ti h rlihl ti I , i ,i i i on I ~i \ l l i .ii l, ,i -Il ill l,-i I, I,, tl i l , i il l ,

%at *iir Ili lurev qlimiitmi . SiiarI % it i- conssiysllrsl that 4l. ttt 1 .1 a cmIirepir-ciliteu a a -valar vsiiiiiationi Ill %ectior-. I-'4r vxali. tOw iiuimiir t it 4pjialls.;

I 21 -_ 7 B~in hius can the denomiiiatt'r III- v~prsv.uh o- a -, Am- .fidm'ii-

tutul 'i %ectusr-. It' 1sjuuatim I k2(ittiilth loe. v'jsro-,-e a- aI *-alar souithuinii -o 'I i-

thiil il 11;ii Htoili \2 11 ialzit ho e hruket ulurecth\ andi -1tri,Il irsrflkrI~ t,,im vq ii.itl i I\2

34

SU~M RY 01 RI-SLS

V-I I~ o dimenisional ctmrs

kit

I. 1 II6 f 01ti r k'F d rC '-i% CII Il k ( I ITlCH

I 4 Cl h\

_____2 _ A 3

d 35

ti iio (11 uLr\% oj% tn p.trInitri, ll\ h~\ i r( I1 0( 11 I1 h n

W2

i)

(r r r -r I ~

rl Ill m~k %u~ 'ir \rrI a I Ivn

H3

Idt~f 1 ) 10r~ 11 ~ Liuc IidI I\ t I. \ .

4\2 ~)3/2

1,

Al

qk \tj of , cu V i4 i h/ \ \)2 3

+

4

Ikj1Ujjj()fl tl CtIB~ 1,N -i h% ( t l

[quation oft ClirXL' 1%O. 11 Mrar icI,iII h\ r -ri i i. 4) o4 i /

(I. 1", 20ria 1/odi , N,,.: l o

r '.rT

* r

dl

_. I qtI.ItIOI ll O I r\ c .'i\Ll h\ 44 /44rI, / >- /4r). Ilhcn

-j 4 I r'

44'

-4- / , )4 -

414(J r4' 4- j)' /' rl)'/"-4 -q 4/"' 4 r4)':/') 2 1r Ir~4l' -' yI)" 4 .]/'4)

d4j

3 . F 44,4 ioII ol &U r L' Zi~cll CI r\ rU 1 . / ,, 4 I' h 1 K

4r '2 ± r" - /' ) 2

44r'/' r/")' 4 (r-','' It"' r)/'4: + 14!'' r4r ?r': 4:

Jr

4444'' 4 'T'II' 4* II"' l14: )2 44 (r'" I' 4111' (II" r "'II ' !

3 \'-'

rr

SphL'rikcaI coordwidtc\.

I. I(IlItio1 Ot 'Lll-%C 2i~t'l paraintricailfy b r ri t i. Om - d PH) ~ I 1 k-r

r2 + r' +~ - Q n2 ))

(IHr iH I r( )2 + -t- ( A.rq si 0) + tArL/ 8rt)'

%% 1icre

r ) 2I r -2 i (4

( -r() +i ro' m 2o~H44~ 2i4 II

I J J 1 1 0Idt~ W rL I' IIii 4 =o4) r j. r . Iic

i + 1 24 -4- r2 C' 2 iE

Bro I m N IH Cr0' C + (( r ' Nill (4 + T AO' I 11

h .' +-2''Nh i l 0ri' cos (

.3. (1 L10101 11 I % c~ii h%~ r r( 0. (4 l ). Ilicri

(t r + r , IH 0

Bhr("' mI~ I r) , + (r \r(,' irm 01- \r i 'i

39)

H r' r'Ninl)~ U

( r0'' NUi) f Nil) -'r t 2r ' t ' o

qu lm () ul~irw c ,'icii h\ r r( (.

( r' r 0' 2 r ,1III

fiHr Nin rln')-' r' \I Nili ) +I ,.\rU' Hi i

.\ r"' rU r inr

H r0"' + r'' r in )s.U

I r' i i ol' ,

Imf i ric n reir 'ulo n o~t canres. ~ilI I ll

I r 1iI

;)IOMlhcrc I C______ I".

Spacct CIurxcs.

I Cartc,,ian kcoorklinalc',. I qtiation of' cur~v deliiicd h\ i1IL'r\L, ion of \1Urt,,

1 ) 0 \kith 'dirtacc (vx 0. 1 lici

F F F Fx z y x

G G G Gx X z dz _ y x \2

dx F F dx F Fy z y z

G G G Gy z y z

I (jtl. tiof (A-22) v, '~ t h ) t COiiIMC 111 /iiI . I liii n 1) ii m tcd from

2. d ~ i~ri~aI coorkiaIics. I (I uiI ion ofI u~rwL dc I nct I 1\ it!~n Ic oI I of "k I n dc

Itni', p I ) \%i I rI i cc ( i rP(/ 0f. I ict

dO F F de F F

Ge z Gr G

G G r

I 11H ,i l t \2. (A ~i tL' ((I o ()(l' r" andf/ I fzin i iN L)Imptilkd Iniiii

41

3. Splicrical oo~1rdinmLt-, I qualtitoi1 oI tir% C LiI illd b\ inltersc timl N) 'Urt'dcCL

F F Fr F

G CG G Gdr _ 0 r 0r

d 9 F F ~ d C- I Fr

r r

I qa ion \i24) arc iused to compult r" and l'. ichnkii ~Ofp t~troll

CkLf ti10tio(A 18I).

V4. D~iscrete representation ot curves.

i, e , r, .uid r. he r~idiuN % cLtor, it) poiniN onl the curve with r,,o witc to tiuc lid-

dieC Ot' I t he R tire fOiIN. I hmII dLIC1 e1 11I, r, jilf r, )

I'l

It v, then redfuces to

A (

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I S 1 [I,,AN411m)k44

No. Copies Addressee No. Copies Addressee

I President C, Matl Tech Lab, DRDMI-VUS Arrmy Armor and Engineer D)ir, Prod A&T Directorate.

Board DRDMIE-T

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ATTN: ATAA-SL (Tech L L')White Sands Missile Range..,M Department of the Navy

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I Naval lraming Iquipnient (enter

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4o

DATE