The Circle Spline routine is currently being used for generating both two and three dimensional spline curves. It was developed for use in ESCHER, a mesh generating routine written to provide a computationally simple and efficient method for building meshes along curved surfaces. Circle Spline is a parametric linear blending spline. Because many computerized machining operations involve circular shapes, the Circle Spline is well suited for both the design and manufacturing processes and shows promise as an alternative to the spline methods currently supported by the Initial Graphics Specification (IGES)

Dolin, R. M.

Jaeger, D. L.

English

NASA Technical Reports Server

EXAMINATION OF THE CIRCLE SPLINE ROUTINE 
Ronald M. Dolin and Dwight L. Jaeger  
Los Alamos National  Laboratory 
Albuquerque, New Mexico 
ABSTRACT 
The C i r c l e  Spl ine  r o u t i n e  was developed by D. L. Jaeger  and is c u r r e n t l y  be ing  
used a t  the Los Alamos National  Laboratory f o r  gene ra t ing  both two- and three- 
dimensional s p l i n e  curves.  I t  was developed f o r  use i n  ESCHER, a mesh gene ra t ing  
r o u t i n e  w r i t t e n  by W. R. Oakes, to  provide a computat ional ly simple and e f f i c i e n t  
method f o r  bu i ld ing  meshes a long  curved sur faces .  C i r c l e  Sp l ine  i s  a paramet r ic  
l i n e a r  blending sp l ine .  Because many computerized machining ope ra t ions  involve  
c i r c u l a r  shapes, the C i r c l e  Spl ine  i s  we l l  s u i t e d  f o r  both the des ign  and manu- 
f a c t u r i n g  processes  and shows promise as a n  a l t e r n a t i v e  t o  the s p l i n e  methods 
c u r r e n t l y  supported by the I n i t i a l  Graphics Exchange S p e c i f i c a t i o n  (IGES). 
C i r c l e  S p l i n e  cons t ruc t s  a s p l i n e  by gene ra t ing  a s e r i e s  of c i r c u l a r  a r c s ,  each 
of which passes  through three  success ive  da t a  p o i n t s  (Pi-1, P i ,  P i + l ) ,  where i = 
2,. . . ,n and n = the number of da ta  poin ts .  Thus, f o r  n da t a  p o i n t s ,  n-2 a r c s  a r e  
cons t ruc ted .  These segmented a r c s  a r e  then blended toge ther ,  us ing  a l i n e a r  
blending func t ion ,  to  r ep re sen t  the d e s i r e d  curve. C i r c l e  Sp l ines  need only have 
the da t a  p o i n t  coord ina tes  spec i f i ed .  Because i t  i s  both a .two- and three- 
dimensional r o u t i n e ,  the three  coord ina tes  must always be s p e c i f i e d ,  even f o r  two- 
dimensional a p p l i c a t i o n s ,  i n  which case  the t h i r d  coord ina te  i s  s e t  t o  zero. The 
s p l i n e  passes  through a l l  the da ta  p o i n t s  and these  po in t s  do n o t  need t o  be evenly  
spaced. C i r c l e  Sp l ine  does n o t  r e q u i r e  f i r s t  o r  second d e r i v a t i v e  c o n t i n u i t y ,  b u t  
i n s t ead  has a d i f f e r e n t  approach f o r  determining a e s t h e t i c  q u a l i t i e s  such a s  
smoothness and f a i r n e s s .  
The r o u t i n e  genera tes  the s p l i n e  by taking th ree  success ive  d a t a  p o i n t s  (Pi-1, 
P i ,  P i + l ) ,  c a l c u l a t i n g  the parametr ic  l o c a t i o n  of a c i r c l e  c e n t e r  R ,  which pas ses  
through the p o i n t s ,  and cons t ruc t ing  a n  a r c ,  of r a d i u s  R ,  through t h e s e  po in t s .  
With t h i s  information,  the curved a r c  l e n g t h  between d a G  p o i n t s  can be c a l c u l a t e d .  
The va lue  of the  curved a r c  l eng th  f o r  each i n t e r v a l  i s  then s to red .  The r o u t i n e  
then increments a long  the curve one da t a  p o i n t  and r epea t s  the process  f o r  d a t a  
p o i n t s  ( P i ,  P i + l ,  Pi+2). With the except ion  of the two end p o i n t  p a i r s  ( P i ,  P i + l )  
and (Pn-1, Pn),  each p a i r  of po in ts  ( P i ,  P i+ l )  i = 2,...,n-2, w i l l  have two a r c s  
pass ing  through it. This  means t h a t  each i n t e r v a l  ( i ,  i + l ) ,  i = 2, .  . .n-2, has  two 
va lues  f o r  the curved a r c  l eng th  (one f o r  each a r c  t h a t  passes  through i t ) .  The 
rou t ine  determines which of these two curved a r c  l eng ths  i s  l a r g e r  and r e t a i n s  t h a t  
value. 
Once a l l  the  a r c  lengths  a r e  c a l c u l a t e d ,  the t o t a l  a r c  l e n g t h  and paramet r ic  
pos i t i on  of each d a t a  p o i n t  a r e  ca l cu la t ed .  The r o u t i n e  uses  a r a t i o n a l  parameter i-  
za t ion .  I f  the  sum of the curved a r c  l eng ths  i s  S ,  the parametr ic  va lue  of each 
da t a  po in t  becomes the r a t i o  of the s p l i n e  l e n g t h  from the s t a r t  of the curve t o  t h e  
da t a  p o i n t  d iv ided  by S. The parameter "u" w i l l  then range from 0 < u < 1. 
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https://ntrs.nasa.gov/search.jsp?R=19850020261 2020-03-22T19:37:13+00:00Z
The use of the l a r g e r  curved a r c  l eng th  wi th in  each i n t e r v a l  was based on 
C i r c l e  Sp l ines  implementation i n  ESCHER. Be t t e r  mesh p o i n t  spacing is achieved 
using t h i s  scheme. A b e t t e r  parameter iza t ion  may be achieved by using a  func t ion  of 
the two a r c  lengths .  I n  any event ,  using the curved a r c  l eng ths  to c a l c u l a t e  the 
t o t a l  s p l i n e  l eng th  no doubt l eads  to  a  b e t t e r  parameter iza t ion  than the chorda l  a r c  
l eng th  method used by rnos t s p l i n e  rout ines .  
The two a r c  segments passing through each p a i r  of d a t a  po in t s  a r e  blended i n t o  
a  s i n g l e  curve. The user  s p e c i f i e s  the number of subpoin ts  t h a t  w i l l  be used f o r  
the blending opera t ion .  These subpoints  a r e  evenly spaced paramet r ica l ly  a long  the  
curve. The code can a l s o  be used so t h a t  the subpoints  a r e  weighted, e.g., l e f t ,  
r i g h t ,  o r  c e n t e r  weighted. An automatic  scheme whereby the subpoints  a r e  c l u s t e r e d  
i n  a r eas  of varying curva ture  may produce a  smoother blend. 
The da t a  p o i n t s  and the subpoints  rnay be coa l e scen t  a t  p o i n t s  a long the  curve 
without  a f f e c t i n g  the sp l ine .  Because the da t a  po in t s  a r e  allowed to  be unevenly 
spaced, the number of subpoints  t h a t  f a l l  wi th in  a  da t a  p o i n t  i n t e r v a l  may vary. 
The number of subpoin ts  wi th in  an  i n t e r v a l  r ep re sen t s  the number a v a i l a b l e  f o r  a  
p a r t i c u l a r  blending opera t ion.  
Let  a r c - i  r ep re sen t  the a r c  passing through po in t s  (Pi-1, P i ,  P i + l ) ,  and 
arc- i+ l  r e p r e s e n t  the a r c  passing through poin ts  ( P i ,  P i + l ,  Pi+2). Both a r c s  w i l l  
pass  through po in t s  P i  and Pi+l .  For the  purpose of example, suppose 10 subpoin ts  
l i e  wi th in  t h i s  i n t e r v a l .  These 10 po in t s  w i l l  be evenly spaced along the i n t e r -  
va l .  The rou t ine  w i l l  determine how f a r  each subpoint  i s  from poin ts  P i  and P i+ l  
and based on t h i s  information w i l l  c a l c u l a t e  the subpoin ts  p o s i t i o n  r e l a t i v e  t o  the  
two a rc s .  The c l o s e r  a  subpoint  is  to  P i ,  the c l o s e r  i t  w i l l  be posi t ioned t o  
a rc- i .  As the subpoint  placement approaches P i+l ,  t h e i r  p o s i t i o n s  w i l l  move c l o s e r  
to  arc- i+l .  A t  the s t a r t  of a n  i n t e r v a l  the blended curve i s  tangent to a r c - i ,  a t  
the end of the i n t e r v a l  i t  is tangent  t o  a rc- i+ l ,  i n  the  c e n t e r  of the i n t e r v a l  each 
a r c  w i l l  have a n  equal  in f luence  on the subpoint  pos i t i on ing ,  and the p o s i t i o n  of 
the blended s p l i n e  w i l l  be i n  the geometric c e n t e r  of the p lane  defined by the  two 
a r c s .  
I f  Pb equals  the parametr ic  va lue  of a  subpoint  w i t h i n  the  i n t e r v a l  P i  - < Pb - < 
Pi+ l ,  i t s  p o s i t i o n  w i l l  be determined using the fol lowing equa t ions  
Rat io  = (Pb - p i )  / ( ~ i + l  - p i )  
Ratml = 1 - Rat io  
The value of the X-coordinate is  thus,  
XX(i) = R a t m l  * X(i)  + Ratio * X(i+l )  
where X(i)  i s  the X-coordinate of P i  and X(i+l)  i s  the  X-coordinate of P i+l .  YY(i) 
and ZZ(i) a r c  ca l cu la t ed  s imi l a r ly .  Using t h i s  scheme i n t u i t i v e l y  suggests  t h a t  t he  
blended curve can have, a t  most, one i n f l e c t i o n  p o i n t  per  i n t e r v a l .  
Spl ine  smoothness can be measured by examining how w e l l  t h e  c i r c l e  c e n t e r s  
f o r  each a r c  r e l a t e .  Because curva ture  is  the inve r se  of the  r ad ius ,  when the  
curva ture  is  zero,  R w i l l  approach in£ i n i t y .  Thus, the R s  provide an e x c e l l e n t  way 
of moni t o r ing  the curva ture .  Many manufactured i tems c o n t a i n  c i r c u l a r  shapes,  f o r  
example, ho le s ,  rounded edges, e t c . ;  thus ,  the C i r c l e  Sp l ine  i s  w e l l  s u i t e d  f o r  
Computer Aided Manufacturing (CAM). 
The C i r c l e  Sp l ine  is  computat ional ly e f f i c i e n t  and easy to  understand. I t  was 
r e c e n t l y  compared t o  the Wilson-Fowler s p l i n e  i n  two dimensions; i t  compared favor-  
a b l y  f o r  most a p p l i c a t i o n s ,  hut  a s  might be expected,  pa tho log ica l  c a s e s  e x i s t e d  
where each r o u t i n e  produced b e t t e r  r e s u l t s .  C i r c l e  Spl ine  can be used i n  both two 
and th ree  dimensions and i t s  a p p l i c a b i l i t y  to  the design and manufacturing process  
makes i t s  f u t u r e  u se  promising. 


Examination of the Circle Spline Routine

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