Fig ure 3.14: Locating J’oints Relative to the T -R B S s . T h e positio n of a link p o in t P/ relative to a L-RBS is d e term in e d by e x a m in i n g the angle between the n o rm al to the V -R B S trian g le / corresponding to the 1 - R B S M.AP and the n o rm al to the tria n g le formed by P/ and the base b of
I.
I he d ' -RBS M.‘\ P is used di rectly to d e t e r m i n e the posit i on of P/ relative t o the T- R BSs by e x a m i n i n g I he angle bet ween the n or ma l to t he V - R B S t r ia n g l e t c o r r e s p o n d i n g to the L-RBS .MAP and the n or mal to the t r i a n g l e f o r me d by 1^/ and the ba s e b of t (see h'ignie 3.11). T h e t r a n s v e r s e region in which P/ resides is d e t e r m i n e d by c o mb i n i n g this r esult with the location of the M.AP (i.e., in which axi s it lies).
T h e pr oces s for the V - R B S is less s t r a i g h t f o r w a r d . Al th o u gh for p o i n t s t h a t a r e l ocat ed over or to ei t her side of the t e s s e l l a t e d V - R B S t he AI AP r e p r e s e n t s t he s u r f a c e well, for poi nt s locat ed at or beyond its e nd s t he M A P wdll a l m o s t a l w a y s be t he e n d - m o s t s u r fa ce t riangle, giving a flat s u r f ac e u nder all c ir c u ms t a n c e s .
T h e V - R B S t essellation p r oc e s s g u a r a n t e e s t h a t t he e n d - m o s t ])air of a d j a c e n t t r i a n gles f rom ei t her end of t he V - R B S ( referred t o as ' ' e n d - t r i a n g l es ” ) will have t hei r b a s e s in different a xe s (see l-dgure 3.15). If t he M.AP is n o t on e of these t r i a ng l e s it is in the cen- Iral, well-defined pni t of t he \ - R BS. In thi s c as e the M.Al^ is used directly t o d e t e r m i n e w h e t h e r P/ lies a bo v e or below t he V - R B S by cal c ul a t ing t he d o t p r o d u c t of t he no r ma l t o the M.AP t r i an g l e a nd the ve c t o r f r om P/ to t he a p e x of t h a t t r i a n gl e (see F i g u r e 3.15). If the M A P is one of the e n d - t r i a n g l e s then the s ur f ac e n or ma l is c a lc ul a te d using the i nt er pol at ion met h o d below (see also F i g u r e 3.16). N o t e t h a t the s a m e m e t h o d is used a t both e nds of the \ ' - R B S tessellation.
( ' h n p l e r 3 . C l a s s i f y i n g t h e H a n d e d n e s s o f S n p e r s e c o n d a r y S t r u c t u r e A d o t i f s 59
E nd-triangles
U se in terp o la te d
n orm al
U se n orm al
to MAP '
E nd-triangles
Figiiie 3.15: [.o rating I’oiiits [{elative to the V-R BS. if tiie M A P is an end-triaiigle an in te rp o la tio n m eth od is used to d eterm in e the location of a link point (see te x t). O therw ise a link p o in t P/ is located by e x a m in in g the dot p ro d u c t of the two n orm als shown.
1. 1 w o t r i angl es ( 0 and
I
2)
a r e f o r me d f rom t he f our p o i n t s t h a t c o m p r i s e t he vert i ces of the e nd - tr ia n g l e s , t he s h a r e d e d g e bei ng t h a t be t we e n the end axi s p o i n t s o f ei t her a xis (see Fig tire 3.16). T h e i r tinit n o r m a l s {iii a nd 7^2) and areas® a re c a l c u l a t edas sh o wn , ddie r atio I'a of t he a r e a of 0 to the s u m of t he a r e a s of 0 a nd 0 is also cal cu l a t ed .
2. P/ is p r oj e c t e d o n t o the v ec t o r v t h a t p a s s e s t h r o u g h t he apices of a nd / ; (in t h a t o r de r ), giving a po i n t I T on v (see Figtire 3.16).
3. If lies beyond the a| )ex of / % or T then the n o r m a l t o the o th e r t r i an gl e is used as the s ur fa c e n or ma l (i.e. no i nt e r p o l a t i o n is d o n e in t hi s case).
1. If 1\, lies bet ween t he api ce s of T and I2 t hen t he rat i o (0 < r < 1) of t he d i s t a n c e
f iom the a p e x of I2 to l \ , , t o the t o t a l d i s t a n c e be t we e n t he api ces gives t he vari abl e
by which t he n or ma l s t o a nd 6 2 aae inter])olated.
5. /• is wei ght ed so that the c e n t r e of the i nt er po l a t i o n , wh e re /’ = 0.5, is shi f t e d t o 7^ (i.(\ it is shift I'd t o w a r d s tin' s ma l l er t r i a ngl e ) , ddie i nt er po l a t i o n is ‘' s t r e t c h e d ” on one side and " c o m p r e s s e d " on the ot h e r t o a c c o m m o d a t e t hi s shift (see F i g u r e 3.16). 6. d he i n t e r po l a t e d n or ma l n is cal ctilated as follows (see F i g u r e 3.16):
(a) ddie t or si on angle (^) b e t we en 7?i a nd 77.2 is c al cu l a t ed .
(b) .4 unit vect or 77,3 is f o r m e d f rom t he c r o s s p r o d u c t of 77,1 a n d v so t h a t 77.], v
and 77,3 form o r t h o g o n a l axes.
(c) 77 is c a l c u l a t ed using t he following ec]nation:
0 0
77. = 7 7 1 co s ( ( 1 - cos 7-7r)-) + 77.3 sin ((1 - cos?' 7r ) - ) (3.3)
^d'he area of a triangle is given lyy the length of its normal vector divided by two [Bowyer and VVoodwark 198.1]- Where the normal is gi\ en l.w the cross product of two of its side vectors.