ls of circles

8/8/2019 Ls of Circles

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M a th e m a t i c a l G e o lo g y , V o l . 2 4 , N o . 5 , 1 9 9 2

L e a s t S q u a r e s B e s t -F i t C i r c le s ( w i th A p p l i c a t i o n s t oM o h r ' s D i a g r a m ) 1

R ichard J. Lis le 2

P r o c e d u r e s a r e o u t l i n e d f o r t h e s e l e ct i o n o f a l e a s t s q u a r e s b e s t - fi t c ir c l e to d a t a p o i n t s d e f i n e d b yr e c t a n g u l a r C a r t e s i a n c o o r d i n a t e s . E q u a t i o n s a r e d e r i v e d t o a l l o w f i t ti n g o f c ir c l e s c e n t e r e d o n t h ex - a x i s a s w e l l a s o f f- a x is M o h r c i rc l es . T h e s e p r o c e d u r e s a r e a p p l i c a b l e t o th e e s t i m a t i o n o f s e c o n d -o r d e r t e n s o r s su c h a s s t r e s s a n d s t ra i n b y m e a n s o f M o h r ' s d i a g ra m ,

K E Y W O R D S : s t a t is t i c s , i m a g e a n a l y s i s , s t r a in , e s t i m a t i o n , s t r u c tu r a l g e o l o g y , t e n s o r s .

I N T R O D U C T I O NT he Mohr c i r c l e c ons t r uc t ion i s w ide ly u se d a s a me a ns o f r e p r e se n t ing s e c ond -ord e r tenso r quan t i t ie s such as s t r e ss and s t r a in a s we l l a s an iso t rop ic phy s ica lp r ope r t i e s , e . g . , m a gne t i c su sc e p t ib il i ty a nd the r ma l c ond uc t iv i ty ( N y e , 1972 ).T h e c ons t r uc t ion i s al so u se d f o r the e s t ima t ion o f te n so r s f r om the i r c om pone n t sme a su r e d in spe c i f i c d i r e c t ions . F o r e xa mple i n tw o- d ime ns iona l ge o log ic a ls t ra in a na ly s is ( Br a c e , 1961 ; J a e ge r , 1962 ; Ra m sa y , 1967) m e a su r e m e n t s o fa n g u l a r s h e a r a n d l o n g i tu d i n a l s tr ai n c o m p o n e n t s m a d e i n a n u m b e r o f d i re c t io n sin a p l a na r sl i c e t h r ough a de f o r m e d r oc k y i e ld po in t s ( x i , Y i ) o n a M o h r d i a g ra mw h ic h c a n be i n t e r p r e t e d in t e r ms o f p r inc ipa l s tr a in s . T h e c om pu ta t ion o f t hepr inc ipa l axes of the f in i te s t r a in e l l ipse and the i r d i rec t ions i s ach ieved by f i r s tf it ti ng a c i rc l e , t he M oh r C i r c l e , t o t he se da t a . T h i s no t e de sc r ibe s how a c i r c l ec a n be o b je c t ive ly c hose n in suc h c a se s by m e a ns o f l e a s t -squa r e s f i tt ing .

I n a p p l i ca t io n s i n v o l v i n g M o h r ' s d i a g r a m t h e p r o c e d u r e s d e sc r i b e d b e lo wof f e r a n a l t e r na t ive t o m e thod s w h ic h f ind d i r e c t ly t he be s t -f i t e l l ip se ( e . g . ,K a n a ga w a , 1990 ; E r s l e v a nd G e , 1990 ) . H ow e v e r , po t e n t ia l a pp l i c a t ions ar eno t l imi t e d to M oh r c i r c le s bu t i nc lude p r ob le m s in t he f i eld s o f r e m o te s e ns ing ,ima g e a na ly s i s , su r v e y ing , a r c he o logy , a nd w he r e ve r a be s t -f i t c i r c l e ha s t o bef i tt e d t o a s e t o f c o - p l a na r po in t s .

~ R e c e i v e d 1 7 J u l y 1 9 9 1 ; a c c e p t e d 6 N o v e m b e r 1 9 9 1 .2 D e p a r t m e n t o f G e o l o g y , U n i v e r s i t y o f W a l e s , C a r d i f f C F 1 3 Y E , W a l e s .

4 5 5

0882-8121/92/0700-0455506.50/1 1992 International Asso ciation o r M athematicalGeology


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456 LisleT H E B E S T - F I T C I R C L E C E N T E R E D O N T H E X - A X I S

T h e p r o b l e m i n q u e s t i o n i s i ll u s tr a te d i n F i g . 1 w h i c h s h o w s a M o h r d i a -g r a m o n w h i c h i s p l o t t e d t h e N d a t a p o i n t s ( x i , Y i w h e r e i = 1 t o N ) w h i c h a ret o b e u s e d t o e s ti m a t e a s y m m e t r i c a l s e c o n d - o r d e r te n s o r. F o r e x a m p l e , i f w ea r e e s t i m a t i n g t h e s t r e s s t e n s o r , t h e s e x i , Y i d a t a w i l l c o n s i s t o f th e n o r m a l a n ds h e a r s tr e ss c o m p o n e n t s r e s p e c ti v e ly m e a s u r e d o n v a r i a b l y -o r i e n t e d pl a n es . T h ee s t i m a t i o n o f th e p r i n c i p a l a x e s o f th e t e n s o r r e q u i r e s d r a w i n g a c i r c le w h i c hp a s s e s t h r o u g h t h e d a t a p o i n t s . F o r s y m m e t r i c a l t e n s o r s l i k e s t r e s s , t h e c e n t e ro f t h e M o h r c i r c l e i s c o n s t r a i n e d t o l ie o n t h e h o r i z o n t a l ( x) a x is o f t h e d i a g r a m .T h i s i m p l i e s t h a t t w o M o h r p o i n t s a r e s u f f i c ie n t f o r t h e c o n s t r u c t i o n o f th e c i rc l e .I n s i tu a t i o ns w h e r e m o r e d a t a a r e a v a i la b l e , t h e p r o b l e m is o v e r d e f i n e d a n d le a d st o th e p r a c t i c a l p r o b l e m o f c h o o s i n g t h e c i r c l e w h i c h b e s t f it s t h e d a t a .E q u a t i o n s f o r t h e c a l c u l a t i o n o f t h e b e s t- f it c ir c l e a r e d e r i v e d u s i n g t h es t a n d a rd l e a st -s q u a r e s a p p r o a c h ( e . g . , N e v i l le a n d K e n n e d y , 1 9 6 4 , p . 1 8 6 - 1 8 7 ) .T h e e q u a t i o n o f a c ir c l e o f ra d i u s r a n d c e n t e r a t d i s t a n c e c a l o n g t h e x -a x i sf r o m t h e o r i g i n i s :

( X - - C ) 2 + y 2 = r 2 ( 1 )U s i n g A = - 2 c a n d B - - w 2 = c 2 - r 2 ( s e e F i g . 1 ), E q . ( 1 ) c a n b e r e - w r i t t e n :

x 2 + Ax + y2 + B = 0 (2 )F o r a d a t a p o i n t ( x i , y i ) w h i c h d o e s n o t l i e e x a c t l y o n t h e c i r c l e , t h e f i g h t h a n ds i d e o f E q . ( 2 ) w i l l b e n o n - z e r o a n d e q u a l t o E i .

C o n s i d e r i n g a l l d a t a p o i n t s ( x i, y i ; i = 1 t o N ) , t h e le a s t s q u a r e s b e s t - f i tc i rc l e i s t h e o n e w h i c h m i n i m i z e s t h e s u m o f th e s q u a r e s o f t h e e r r o r t e rm s ,i . e . ,

m in Z (E~ ) = m in Z (x~ + Ax~ + y2 + B)2 (3 )T h e m i n i m u m o f ~ ( E ~ ) i s f o u n d b y pa r t ia l d i f fe r e n t i a ti o n o f E q . ( 3 ) w i t h r e s p e c tt o c i r c l e p a r a m e t e r s A a n d B a n d s e t t i n g t h e s e p a r t i a l d e r i v a t i v e s t o z e r o :

-5

' 1 / 3 ~ ' i " " \t X

Fig. 1. Fitting a circle w ith center on the x-axis todata xi, Yi. The data and results are sh ow n in Ta ble 1.


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Best-Fit Circles 457d Y ', ( E 2 ) / d A = Z ( x~ + A x 2 + xiy2~ + B x i ) = 0d E ( E 2 ) / d B = ( x 2 + A x i + y 2 + B ) = 0

T h e s e e q u a t i o n s s i m p l i f y to g i v e w h a t a r e k n o w n a s t h e n o r m a l e q u a t i o n s :A Z ( x p ) + B Z ( x i ) = - Z ( x 2 ) - Z ( x i y ~ )

A Z (X i ) "q- B = - Z (x 2 ) - Z ( y ~ )T h e s o l u ti o n to t h i s s y s t e m o f s i m u l t a n e o u s e q u a t io n s g i v e s e x p r e s s io n s f o r Aa n d B , p a r a m e t e r s f o r t he c i r c le . T h e s e c a n b e m a n i p u l a t e d t o g i v e t h e f o l lo w i n ge q u a t i o n s f o r t h e c e n t e r ( c ) a n d r a d i u s ( r ) f o r b e s t - f i t c i r c l e :

C

r I c 2 +E x a m p l e . T h e a b o v e e q u a t i o n s a r e e m p l o y e d t o c a l c u l a t e t h e b e s t c i r c l ef i tt e d t o t h e d a t a l is t e d in T a b l e 1 a n d r e p r e s e n t e d g r a p h i c a l l y i n F i g . 1 . T h e

c a l c u l a ti o n s a r e p e r f o r m e d b y m e a n s o f t h e B A S I C p r o g r a m l i s t e d i n A p p e n d i x

Tab le 1. Sample DataData in Fig. 1 Data in Fig. 2(on-axis circle) (off-axis circle)

i XI YI XI YI1 4 . 6 - 1 . 9 4.6 3.32 6.0 -3 .8 6.0 1.43 10.8 -1 .7 10.8 3.54 10.7 -2 .5 10.7 2.75 10.8 1.9 10.8 7.16 6.7 4.0 6.7 9.2

Best-fit circleCenter 7.6 0.0Radius 3.9Roo t-mean-square deviation as percentage of radius:5.9%

7.63.9

5.9%

5.2


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458 L i s l e

A. T h e b e s t - f i t c i r c l e h a s c e n t e r , c = 7 .6 a n d r a d iu s , r = 3 .9 . T h e r o o t - m e a n -s q u a r e d e v i a t i o n o f t h e p o i n t s a w a y f r o m t h i s c i r c le ( e x p r e s s e d a s a p e r c e n t a g eo f t h e r ad i u s ) i s 5 . 9 % .

B E S T - F I T O F F -A X I S C I R C L ET h e n e e d t o c o n s i d e r c i r c le s w i th c e n t e r s l o c a te d a w a y f r o m t h e x - a x is

a r i s e s in t wo s i t u a t io n s . F i r s t , o f f - a x i s M o h r c ir c l e s a re e m p l o y e d to r e p r e se n ta s y m m e t r i c a l s e c o n d - o r d e r t e n s o r s q u an t it ie s ( D e P a o r a n d M e a n s , 1 9 84 ; r e -v i e w e d b y Me a n s , 1 9 9 0 ) . Se c o n d , c e r t a i n g r a p h i c a l t e c h n i q u e s f o r f in i te s t r ai na n a l y s i s ( R a g a n , 1 9 8 3 , p . 1 9 0 - 1 9 1 ; L i s l e a n d R a g a n , 1 9 88 ) in v o l v e t h e c o n -s t r u c t i o n o f t h e c i r c l e w i t h o u t a p r i o r i k n o w l e d g e o f th e p o s i t io n o f th e o r i g ino f t h e M o h r d i a g r a m ; t h e c e n t e r o f th e c i r c l e b e i n g u s e d t o f ix th e x - a x i s .

T h e e q u a t i o n o f a c i r c l e o f r a d i u s r a n d c e n t e r a t c , d ( F i g . 2 ) i s(x - c ) 2 + (y - d ) 2 = r 2

U sing b = w 2 = c 2 + d 2 - r 2 ( F i g . 2 ) , t h i s b e c o m e sx 2 - 2 c x + y 2 _ 2 y d + b = 0

Fo r a c t u a l d a t a p o i n t s ( xi, y i ) , e r r o r t e r m s E i w i l l a p p e a r o n t h e r i g h t - h a n d s i d ewh i c h , a s b e f o r e , a r e t o b e su b j e c t e d to l e a s t -sq u a r e s m i n i m a l i z a t i o n . Pa r t i ald i f f e r e n t ia t i o n s o f t h e e q u a t i o n f o r ~ ( E 2 ) w i t h r e sp e c t t o c i r c l e v a r i a b l e s c , d ,a n d b , r e s p e c t i v e l y , a n d e q u a t i n g t o z e r o y i e l d :

d ~ ( E 2 i ) / d c = ~ ( 2 x ~ c + 2 x iY i d - - x i b - x 3 - x i Y 2 )= 0

d Z ( E 2 ) / d d = Z ( 2 x i y i c + 2 y Z d - y ~ b - x Z y i - y 3 )= 0

d Z ( E 2 ) / d b = ~ ( - 2 x i c - 2 y i d + b + x 2 + y 2 )= 0

Y .1 0

I k

5 10 Fig. 2. Fitting an off-axis Mohr circle to data pointson the Mo hr diagram. The data and results are shownin Table 1.


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Best-Fit Circles 459a n d g i v e t h e n o r m a l e q u a t i o n s :

c ~ ( 2 x ~ ) + d ~ ( 2 x i y i ) + b ~ ( - x i ) = ~ ( x ~ + x i y 2 )C ~] (2x iY i ) q'- d E ( 2 y 2 ) + b ~ ( - Y i ) --- ~ ] ( x 2 y i + y 3 )

c Z ( - x i ) + d " ~ ( - y i ) + b N = Z - ~(xi 2 + y ~ )T h e s o l u t i o n o f th i s s y s t e m o f e q u a t i o n s a l l o w s c a l c u l a t io n o f c, d , a n d b

K ( D F - E 2 ) - L ( B F - E C ) + M ( B E - D C )C = A D F + 2 B C E - A E 2 - F B 2 - D C 2

- K ( B F - C E ) + L ( A F - C 2) - M ( A E - B C )d =A D F + 2 B C E - A E 2 - F B 2 - D C 2

K ( B E - D C ) - L ( A E - B C ) + M ( A D - B 2)b = A D F + 2 B C E - A E 2 - F B 2 - D C 2w h e r e A = 2 E ( x ~ ) , B = 2 E ( x i Y i ) , C = - Z ( x i ), D = 2 Z ( y ~ ) , E =

X 2 1- E ( Y i ) , F = 1 N , K ~ ( x~ + i Y i ) , L = ~ ( x 2 iy i + y 3 ) , a n d M =( x 2 + y 2 ) . T h e r a d i u s o f t h e b e s t - f it c i r c l e i s f o u n d f r o m r = x / ( c 2 + d 2 - b ) .

E x a m p l e . T h e b e s t - f i t c i r c l e i s f o u n d f o r t h e d a t a p o i n t s i n F i g . 2 ( l i s t e di n T a b l e 1 ) . T h e e q u a t i o n s a b o v e f o r t h e c a l c u l a t i o n o f t h e b e s t -f it c i r c l e a r ei n c o r p o ra t e d in th e c o m p u t e r p r o g r a m i n A p p e n d i x B . T h e c i r cl e s o f o u n d h a sa c e n t e r a t c = 7 . 6 a n d d = 5 . 2 a n d a r a d i u s o f 3 . 9 . T h e p r o g r a m a l s o d e t e r m i n e st h e r o o t -m e a n - s q u a r e d e v i a t i o n o f th e p o i n ts f r o m t h e ci r cl e t o b e 5 . 9 % o f t h er a d i u s .

A P P E N D I X A . L I S T I N G O F A G W - B A S I C P R O G R A M T O F I N DT H E B E S T - F I T C I R C L E C E N T E R E D O N T H E X - A X I S

1 0 C O L O R 4 , 7 , 3 : C L S2 0 P R I N T " P R O G R A M M O H R "30 P R I N T " = = = = = = = = = = = = = = ~ = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = ~ = = = = = = = = = = = = = = = = = = = = = = =4 0 P R I N T " F i n d s a l e a s t - s q u a r e s b e s t f i t M o h r c i r c l e t o a n u m b e r o f p o i n t s x , y "5 0 P R I N T " . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 0 D I M X ( 1 0 0 0 ) , Y ( 1 0 0 0 )7 0 R E M d a t a i n p u t8 0 I N P U T " N a m e o f i n p u t f i l e w i t h x , y d a t a " ; D $9 0 O P E N D $ F O R I N P U T A S ~I1 0 0 C O U N T =I1 1 0 I F E O F ( I } T H E N G O T O 1 5 01 2 0 I N P U T4 ~ , X ( C O U N T ) , Y ( C O U N T )1 3 0 C O U N T = C O U N T + I1 4 0 G O T O 1 1 01 5 0 N = C O U N T - I1 6 0 F O R I =I T O N1 70 T L = T L + X ( I ) ^ 2 + Y ( I ) ^ 21 8 0 T M = T M + X ( I )1 90 T R = T R + X ( I ) ^ 3 + X ( I ) * Y ( I ) ~ 22 0 0 B R = B R + X ( I ) ^ 22 0 5 T Y = T Y + Y ( I )


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460 Lisle2 1 0 N E X T I2 2 0 P R I N T2 3 0 P R I N T2 4 0 C E N T R E = . 5 * ( ( T L * T M - N * T R ) / ( T M ^ 2 - N * B R ) )2 5 0 R A D I U S = S Q R ( C E N T R E ^ 2 + ( T R - 2 * C E N T R E * B R ) / T M )2 6 0 P R I N T " C e n t r e o f M o h r c i r c l e a t " ; C E N T R E ; " , 0 . 0 " ; " R a d i u s = " ; R A D I U S2 7 0 P R I N T " . . . . . . . . . . . . . . . . . . . . . . "3 4 0 R E M r o o t m e a n s q u a r e v a r i a t i o n e x p r e s s e d a s % o f r a d i u s3 5 0 F O R I = I T O N3 6 0 D I S T C E N T = S Q R ( ( X ( I ) - C E N T R E ) ^ 2 + Y ( I ) ^ 2 )3 90 T S V A R = T S V A R + ( D I S T C E N T - R A D I U S ) ^ 24 0 0 N E X T I4 1 0 R M S V = S Q R ( T S V A R / N ) / R A D I U S * I O 04 20 P R I N T " R o o t M e a n S q u a r e D e v i a t i o n o f p o i n t s f r o m c i r c l e " : P R I N T "e x p r e s s e d a s % o f r a d i u s = " ; R M S V ; " % "

A P P E N D I X B . L I S T I N G O F A G W - B A S I C P R O G R A M T O F IN DT H E B E S T F IT O F F - A X IS C I R C L E

1 0 C O L O R 4 , 7 , 3 : C L S2 0 P R I N T " P R O G R A M M O H R O F F "3 0 P R I N T " = = = . . . . . . . . . . . . . . . . . . . . = = = = = . . . . . . . = . . . . = . . . . . = . . . . . . . . . = = = = . . . . . . = . . . .= = = = = = = = "4 0 P R I N T " F i n d s a l e a s t - s q u a r e s b e s t f i t M o h r c i r c l e t h r o u g h p o i n t s x , y "5 0 P R I N T " T h e c i r c l e h a s r a d i u s r a n d a c e n t r e o f f t h e x - a x i s a t x = c , y = d "7 0 P R I N T " . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 0 DI M X ( 1 0 0 0 ) , Y ( 1 0 0 0 )9 0 R E M d a t a i n p u t1 0 0 I N P U T " N a m e o f i n p u t f i l e w i t h x , y d a t a " ; D $1 1 0 O P E N D $ F O R I N P U T A S ~I1 2 0 C O U N T = I1 3 0 I F E O F ( 1 ) T H E N G O T O 1 7 01 40 I N P U T ~ , X ( C O U N T ) , Y ( C O U N T )1 5 0 C O U N T = C O U N T + I1 6 0 G O T O 1 3 01 7 0 N = C O U N T - I1 80 A = 0 : B = 0 : C = 0 : D = 0 : E = 0 : F = 0 : K = 0 : L = 0 : M = 01 9 0 F O R I = I T O N2 0 0 A = A + 2 * X ( I ) ^ 22 10 B = B + 2 * X ( I ) * Y ( I )2 2 0 C = C - X ( I )2 2 5 D = D + 2 * Y ( I ) ^ 22 3 0 E = E - Y ( I )2 4 0 F = F + . 52 50 K = K + X ( I ) ^ 3 + X ( I ) * Y ( I ) ^ 22 60 L = L + X ( I ) ^ 2 * Y ( I ) + Y ( I ) ^ 32 70 M = M - . 5 * ( X ( I ) ^ 2 + Y ( I ) ^ 2 )2 8 0 N E X T I2 9 0 R E M c o e f f s o f e l l i p s e e q n ( a c , b c , c c )3 0 0 D E T = A * D * F + 2 * B * C * E - A * E * E - F * B * B - D * C * C3 1 0 C C = ( K * ( D * F - E * E ) - L * ( B * F - E * C ) + M * ( B * E - D * C ) ) / D E T3 2 0 D C = ( - K * ( B * F - C * E ) + L * ( A * F - C * C ) - M * ( A * E - B * C ) ) / D E T3 30 B C = ( K * ( B * E - D * C ) - L * ( A * E - B * C ) + M * ( A * D - B * B ) ) / D E T3 3 5 R = S Q R ( C C * C C + D C * D C - B C )3 40 P R I N T " . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 5 0 P R I N T : P R I N T " B e s t f i t M o h r C i r c l e "3 60 P R I N T " . . . . . . . . . . . . . . . . . . . . "3 7 0 P R I N T " c = " ; C C ; " d = " ; D C ; " r = " ; R3 8 0 R E M r o o t m e a n s q u a r e v a r i a t i o n e x p r e s s e d a s % o f r a d i u s3 9 0 F O R I = I T O N4 00 D I S T C E N T = S Q R ( ( X ( I ) - C C ) ^ 2 + ( Y ( I ) - D C ) ^ 2 )4 10 T S V A R = T S V A R + ( D I S T C E N T - R ) 24 2 0 N E X T I4 3 0 R M S V = S Q R ( T S V A R / N ) / R * I 0 04 4 0 P R I N T " R o o t M e a n S q u a r e D e v i a t i o n o f p o i n t s f r o m c i r c l e " : P R I N T " e x p re s s e d a s % o f r a d i u s = " ; R M S V ; " % "


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A C K N O W L E D G M E N T SThe author i s gra teful to Dec lan De Paor and an anon ymo us r eviewer for

the i r useful sugges t ions for improving the manuscr ipt .

R E F E R E N C E SBrace, W. F., 1961, Mohr Construction in the Analysis of Large Geologic Strain: Bull. Geol. Soc.

Am., v. 72, p. 1059-1080.De Paor, D. G., and Means, W. D., 1984, Mohr Circles of the First and Second Kind and Their

Use to Represent Tensor Operations: J. Struct. Geol., v. 6, p. 693-701.Erslev, E. A., and Ge, H., 1990, Least-Squares Center-to-Center and Mean Object Ellipse Fabric

Analysis: J. Struct. Geol., v. 12, p. 1047-1060.Jaeger, J. C., 1962, Elasticity, Fracture and Flow: Methuen.Kanagawa, K., 1990, Automated Strain Analysis Using an Image Analysis System: J. Struct. Geol.,v. 12, p. 139-143.Lisle, R. J., and Ragan, D. M., 1988, Strain Determination from Three Measured Stretches--A

Simple Mohr Circle Solution: J. Struct. Geol., v. 10, p. 905-906.Means, W. D., 1990, Kinematics, Stress, Deformation and Material Behaviour: J. Struct. Geol.,

v. 12, p. 953-971.Neville, A. M., and Kennedy, J. B., 1964, Basic Statistical Methods for Engineers and Scientists:

International.Nye, J. F. , 1972, Physical Properties of Crystals: Oxford.Ragan, D. M., 1983, Structural Geology: An Introduction to Geometrical Techniques: Wiley, New

York.Ramsay, J. G. , 1967, Folding and Fracturing of Rocks: McGraw-Hill.

ls of circles

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