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More about algebraic simplification

• To: mathgroup at smc.vnet.net
• Subject: [mg110494] More about algebraic simplification
• From: "S. B. Gray" <stevebg at ROADRUNNER.COM>
• Date: Mon, 21 Jun 2010 02:10:57 -0400 (EDT)
• Reply-to: stevebg at ROADRUNNER.COM

In response to someone who objected to my statement that replacement 
rules are not adequate for what I want to do, I will state the problem 
more completely without getting into irrelevant details.

I have n points in R3 in general position. Through all (n choose 3) sets 
of 3 points I construct a plane and a circle. The plane computation 
results in three parameters for each plane, a[i],b[i], and c[i], each a 
simple but not trivial function of x,y,z of the 3 points it goes 
through. The plane is now described by 3 parameters instead of the 9 for 
the 3 points. Each circle is described, after a moderately complex 
computation, as center, radius, and unit normal, each also a function of 
x,y,z of the 3 points. So the circle is now described by 7 parameters, 
each with a clear geometric meaning that can be verified using 
Graphics3D, rather than the original 9.

To avoid extreme complexity in the following steps, I do not let Mathematica 
reduce these parameters to the initial x,y,z's, but define new symbolic 
variables describing the plane and circle. Next, I want to find the line 
that is the intersection of two of the planes, z=x*zm + zd and y=x*ym + 
yd, where the four new parameters are functions of the 6 parameters 
describing the 2 planes. Finally I want to compute the two (or zero) 
intersections of that line and the circle on one of the 2 planes, in 
terms of ym,yd,zm,zd and the 7 circle parameters.

If I let Mathematica express these intersections in terms of the x,y,z's of the 
six points defining the two planes, the result is a horribly complicated 
expression that is not only incomprehensible but involves repeated 
evaluation of quite a few subexpressions. It makes much more sense to 
define new variables at each step and derive symbolic expressions for 
them in terms of the results of the previous step.

My questions are:

1. If I do everything symbolically, what is the best method of 
preventing Mathematica from expressing the final answer in terms of the initial 
ones?

2. If substitution rules allow this to be done, how to do it?

3. Having read a few responses to my first, crude, statement of this 
problem, I still do not see a clear answer.

4. It would be nice for Mathematica to first, be able to prevent full evaluation 
of the final answer in terms of the initial parameters, and, second, for 
Mathematica to find suitable intermediate symbolic variables by itself, both to 
create comprehensible expressions and to reduce duplicate evaluations.

5. Obviously computations by many Mathematica users involve a similar number of 
separate steps where the same type of help would be useful, so I wonder 
if WR is planning to address this.

Steve Gray