Re: Straightforward factoring and simplification
- To: mathgroup at smc.vnet.net
- Subject: [mg111855] Re: Straightforward factoring and simplification
- From: Christopher Arthur <aarthur at tx.rr.com>
- Date: Sun, 15 Aug 2010 07:39:10 -0400 (EDT)
Follow-up question:
AES,
Try the ColorFunction option, but you have to reset the LightSource also.
AES a =E9crit :
> Some time back there was a post in this group about how to convert the
> expression
>
> (Eq_1) a*b + a*c - 2*a*d + b*c - 2*b*d - 2*c*d + 3*d^2
>
> into
>
> (Eq_2) (a-d)*(b-d) + (a-d)*(c-d) + (b-d)*(c-d)
>
> to which I made an absurd response. My meds are working better now, so=
> I hope the following will make sense.
>
> Although I'm not attempting to answer the original question, I did want=
> to understand better the symmetry of these expressions. So, suppose we=
> scale a, b and c to d, and rename them as x, y, z. The expression then=
> obviously has a zero at x==y==z==1, so shift the origin of coordinates to=
> that point (suffix 's') and get
>
> f==(x-1)(y-1)+(x-1)(z-1)+(y-1)(z-1);
> fs==f/.{x->xs+1,y->ys+1,z->zs+1};
> fs//Expand
>
> xs ys+xs zs+ys zs
>
> This has an obvious axis of symmetry, so rotate the axes into alignment=
> with it (suffix 'r') and get
>
> {xs,ys,zs}==RotationMatrix[{{0,0,1},{1,1,1}}] .{xr,yr,zr};
> fr==fs;fr//Expand
>
> -xr^2/2 - yr^2/2 + zr^2
>
> Note sure what a geometer would call that, but it's obviously a figure
> of rotation about an axis (ellipsoid of rotation about an imaginary
> axis?). Confirm this by making a couple of plots:
>
> x==Sin[theta]Sin[phi];y==Sin[theta]Cos[phi];z==Cos[theta];
> fs==x y +x z+y z;
> fr==z^2 - 0.5(x^2+y^2);
> ps==Graphics[SphericalPlot3D[fs,theta,phi]];
> pr==Graphics[SphericalPlot3D[fr,theta,phi]];
> GraphicsRow[{ps,pr}]
>
> or by mapping f onto the surface or spheres of varying diameter:
>
> Manipulate[
> x==a Sin[theta]Sin[phi];y==a Sin[theta]Cos[phi];z==a Cos[theta];
> f==z^2-0.5(x^2+y^2);
> Show[Graphics[SphericalPlot3D[a+ b f,theta,phi]]],
> {{a,1},0,5},{{b,0.1},0,1}]
>
> [and in the process learn that Mathematica allows spheres to have
> negative radii.]
>
> Follow-up question: Suppose you want to plot a unit sphere with
> SphericalPlot3D in which different areas or patches on the surface have=
> different colors. depending on the theta, phi (or x,y,z) values. Or, a=
> SphericalPlot3D[r[theta,phi], theta, phi] for which positive values of r=
> give a red surface and negative values a blue surface. How might one do=
> that in some simple way?
>
>
>