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? > > >