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From: neeman@s5.csrd.uiuc.edu (Henry J. Neeman)
Newsgroups: comp.graphics
Subject: Re: newell plane equation
Summary: too many points
Keywords: graphics, plane eqation
Message-ID: <1991Mar7.203947.27550@csrd.uiuc.edu>
Date: 7 Mar 91 20:39:47 GMT
References: <9283@exodus.Eng.Sun.COM>
Sender: news@csrd.uiuc.edu (news)
Organization: Univ of Illinois, Center for Supercomputing R&D, Urbana, IL
Lines: 39

In article <9283@exodus.Eng.Sun.COM>
stephenj@deblil.Eng.Sun.COM (Stephen Johnson) writes:
>I have been using the Newell technique for computing the plane
>equation in the form Ax + By + Cz + D = 0
>and I discovered that the plane defined by the following points is
>computing incorrectly:  (2,2,0), (12,2,0), (2,2,3), (12, 2, 3)
>In the loops: if (i == n) j = 1 else j = i + 1
>A = SUM(y[i] - y[j]) * (z[i] + z[j]) for i = 1 to N
>B = SUM(z[i] - z[j]) * (x[i] + x[j]) for i = 1 to N
>C = SUM(x[i] - x[j]) * (y[i] + y[j]) for i = 1 to N
>D = (A * x[0] + B * y[0] + C * z[0])

Actually, I think D = -(Ax[k] + By[k] + Cz[k]), for any k in 1..N.

>Unfortunately, with the previously mentioned points:
	[A=B=C=0]
>Oops, by inspection we can see that the equation for this plane is
>y = -2.  So, what's wrong?  Has someone noticed this before and fixed
>it.

Actually, by inspection we see that y = 2, but I guess that's a typo.

Anyway, your problem is that you're overdetermining your plane.  A plane
can be described by three points, and I'm guessing that Newell's technique
takes advantage of that.  Let's try it:

(2,2,0), (12,2,0), (2,2,3)
A=(2- 2)(0+ 0)+(2- 2)( 0+3)+(2-2)(3+0)=    0*0+   0*3 +0*3=  0+ 0+ 0= *   0 *
B=(0- 0)(2+12)+(0- 3)(12+2)+(3-0)(2+2)=0(-14) +(-3)*14+3*4=  0-42+12= * -30 *
C=(2-12)(2+ 2)+(12-2)( 2+2)+(2-2)(2+2)=(-10)*4+  10*4 +0*4=-40+40+ 0= *   0 *

So, we have 0*x-30*y+0*z=-D.  Plugging in (2,2,0), we get -30*2=-D or D=60.
In other words, -30 * y + 60 = 0, or y = 2.

I think it's safe to assume that we'll get the same results for any combination
of three of your four points.

					Henry Neeman
					neeman@csrd.uiuc.edu