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From: steveb@eve.WV.TEK.COM
Newsgroups: comp.graphics
Subject: 3d Surface Patches
Message-ID: <5053@orca.WV.TEK.COM>
Date: 19 Oct 89 23:54:20 GMT
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Reply-To: steveb@eve.WV.TEK.COM ()
Organization: Tektronix, Inc., Wilsonville, OR
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To help clarify anything I might have said in my previous posting which is
unclear I offer the following e-mail dialog.  This is part of a discussion
between mathematician Mike Peters and myself regarding that posting.  If I
can clarify anything else, let me know.

Also, for the record, I can't remember who it was that didn't understand how
I could forward difference a rational polynomial, but:  The point is to convert
from 1 basis to another and through some minor contortions get something that
is evaluatable by forward differencing.  If that doesn't make sense please
let me know what you don't understand and I'll try to clarify it !!!

So... Here goes...

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To: steveb@eve.WV
Cc: mikep@tekirl.labs.TEK.COM
Subject: 
Date: 19 Oct 89 10:53:09 PDT (Thu)
From: mikep@tekirl.labs.TEK.COM

Steve,
	I scanned your posting. A couple of points, where did the order 2
differentiability come from and algebraic surfaces do not in general
have a rational parametric representation.

-MikeP


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From: steveb
Message-Id: <8910192346.AA12791@eve.WV.TEK.COM>
Date: Thu, 19 Oct 89 16:46:04 PDT
To: mikep@tekirl.labs.TEK.COM
Cc: steveb@eve.WV, mikep@tekirl.labs.TEK.COM, jt
Fcc: savebox
Subject: What did I mean ???

Mike:

1. I said order-2 (order minus 2) not order 2 differentiable.  Granted I did
   leave out the qualification that is must have no multiple internal knots.
   The reference comes from Tiller83 (IEEE CG&A pg 62 Sept 1983):

	" (P2) A B-spline curve of order k, having no multiple interior
	  knots, is k-2 differentiable."

2. OK, I admit it, I was unclear re: algebraic surfaces having rational
   polynomial representation.  Here's what I meant: (for example) a sphere
   can not be represented by an non-rational B-spline surface.  But, it
   can be represented either algebraically OR AS A RATIONAL B-SPLINE (NURB).
   That's what I meant.  Naturally, you are right.  I was unclear :-(.....

3. Also, for the record John T. asked me why I say rationality gives you
   more local control.  You can, I'm sure, argue with me on that matter too.
   Primarily because I didn't explain what I meant by "more".  So, for the
   record I consider the ablilty to move a curve closer to/farther from a
   control point by changing W an aspect of local control.  Since you get
   that with a rational curve not a non-rational one, I consider that "more
   local control".  Go ahead prove me wrong, you can, and you'll be right.
   Barsky will say that you can get the same functionality without
   rationality using beta splines.  And, of course he's right, also. (In fact
   he's extremely right).

See ya soon,

Steve


-------------------------------------------------------------------------------
FROM:        STEVEN C. BILOW --  Software Engineer,  Tektronix
EMAIL:	     steveb@orca.WV.TEK.COM     PHONE:  (503) 685-2463  
USMAIL:      P.O. Box 1000 61-028, Wilsonville, OR  97070-1000