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Surface is smudged when two parameteric lines intersect on surface

When two intersecting lines are drawn (or when three or more lines
concur at/near a point) on a surface, the depicted surface becomes
smudged and shabby in appearance. Perhaps it is due to the conflict
resolution of differential patch occupation by trying to accomodate
both, however appears to  do inadequate justice to either, partial
overlaps seem inevitable and lines are jagged.

It is not known how the graphics algorithms are made, but looking at
what one sees, perhaps it  may be better to give the first appearance a
normal patch depiction and ignore the procedures for patching up the
patch on subsequent passes of the line on the surface.This option will
not only give clean surfaces, but also show clearer later drawn  lines
more continuously. 

Is there a ' see through ' option like in another system?
Best Regards,
Narasimham G.L.
Indian Space Research Orgn 695013

R=4 ; a=2 ;
'WRAPED RING >> circles radius R wrapped on cylinder surface
radius a ; u {u,0,Pi} is OK but (u,0,2Pi) smudges when two
 or more concurrent lines are attempted to be drawn on a 
surface '
ParametricPlot3D[{a Cos[R Sin[u]/a+al],
a Sin[R Sin[u]/a +al],
R Cos[u]} ,{u,0,Pi},{al,0,2Pi},
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