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hyperbola-formulas.wxm
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/* [wxMaxima batch file version 1] [ DO NOT EDIT BY HAND! ]*/
/* [ Created with wxMaxima version 12.01.0 ] */
/* [wxMaxima: comment start ]
Load packages.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
load("lrats")$
load("format")$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Swap left and right hand sides of the equation.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
swapsides(eqs) := map(lambda([u],rhs(u)=lhs(u)), eqs)$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Back substitute common subexpressions into an expression.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
backsubst(eqs,expr) := subst(swapsides(eqs),subst(swapsides(-eqs),expr))$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Print a formula with 2D display turned off.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
print1d(expr) := block([display2d], display2d: false, print(expr))$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Check values to test calculations.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
checkvalues: [n=1000000,m1n=3,m1d=2,m0n=4,m0d=3,x01=839,y01=1190];
checkvalues: [n=13500,m1n=2,m1d=1,m0n=1,m0d=1,x01=96,y01=136];
checkvalues: [n=119,m1n=2,m1d=1,m0n=1,m0d=1,x01=9,y01=12];
checkvalues: [n=70,m1n=2,m1d=1,m0n=1,m0d=1,x01=7,y01=9];
checkvalues: [n=868,m1n=2,m1d=1,m0n=1,m0d=1,x01=25,y01=33];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
y:y(x);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
hyperbola: A*x^2+B*x*y+C*y^2+D*x+E*y+F=0;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
slope1: solve(subst(-1,diff(y,x),diff(hyperbola,x)),y)[1];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
slope2: subst(slope1, hyperbola);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
slope3: solve(slope2, x)[1];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
print1d(slope3)$
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
h0: y = n / x;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
tdef: t = -4*A*C^2+(B^2+4*A*B-4*A^2)*C-B^3+A*B^2;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
slope4: ratsubst(lhs(tdef), rhs(tdef), slope3);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
udef: u = -B*C+B^2-A*B;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
slope5: ratsubst(lhs(udef), rhs(udef), slope4);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
vdef: v=C^2 + (A-B)*C;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
slope6: ratsubst(lhs(vdef),rhs(vdef),slope5);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
[tdef, udef, vdef, slope6];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
remvalue(y)$
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
hyperbola2: A*x^2 + B*x*y + C*y^2 + D*x + E*y + F = 0;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
skewx: rat(expand(subst(x-y+r,x,hyperbola2)),x,y);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
skewy: rat(expand(subst(y-x+r,y,hyperbola2)),x,y);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
xlate1: [x = xold, y = yold];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
xlate2: solve(subst([y = y - x + r1], xlate1),[x,y])[1];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
xlate2: solve(subst([x = x - y + r2], xlate2),[x,y])[1];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
top: y = -m0*(x-x2)+y2;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
left: y = -m1*x + r1;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
topleft: solve([top,left],[x,y])[1];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
subst(topleft[1], left);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
bottom: y = -m0*x + r0;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
solve([left,bottom],[x,y])[1];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
m0: 1$
m1: 2$
m2: 3/2$
r0: 13$
r1: 18$
r2: 31/2$
x0: 6$
x1: 4$
x2: 5$
n: 44$
H(x) := n/x$
L0(x) := -m0*x+r0$
L1(x) := -m1*x+r1$
L2(x) := -m2*x+r2$
wxplot2d([H,L0,L1,L2],[x,x1,x0],
[gnuplot_preamble,
"set grid; set xtics 1; set ytics 1;"]),
wxplot_size=[400,400];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
kill(n,x0,y0,m0,r0,x1,y1,m1,r0,x2,y2,m2,r2)$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Transformations between xy and uv coordinate systems.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
L0: (y-y01)/(x-x01) = -m0n/m0d;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
block([tmp], tmp: L0*part(L0,1,2)*part(L0,2,1,2), tmp-rhs(tmp));
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
xy: [x = x01 + u*m0d - v*m1d,y = y01 - u*m0n + v*m1n];
print1d(xy)$
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
uvdenom: m0d*m1n-m0n*m1d;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
ev(uvdenom, checkvalues);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
uv: subst([uvdenom=1],ratsimp(solve(xy,[u,v])[1]));
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Format the solution as a list of equations of sums of fractions of polynomials in m.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
uv2: format(uv, %list, %eq, %p(m0n, m0d, m1n, m1d));
print1d(uv2)$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Start again with newly acquired experience with Maxima.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
depends(y,x)$
hyperbola: A*x^2+B*x*y+C*y^2+D*x+E*y+F=0;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
slope: solve(diff(hyperbola,x),diff(y,x))[1];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
diagonal: solve(subst(-1,diff(y,x),slope),y)[1];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
h: x*y=n;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
huv: psubst(xy,h);
print1d(huv)$
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
diag: v=u;
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Find intersection of uv-based h with the diagonal u=v.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
huv2: format(subst([diag],huv),%eq,%prod,%p(u));
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
m01def: [m01d=m1d-m0d,m01n=m1n-m0n];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
huv3: backsubst(m01def,huv2);
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Take the second solution with the subtraction of the radical.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
udiag: solve(huv3,u)[2];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
radicand: part(udiag,2,1,1,1);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
format(radicand,%p(m01d),%p(m01n));
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Looks like (a-b)^2. Complete the square.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
complete: (m01d*y01+m01n*x01)^2;
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Formula for the intersection between u=v and the hyperbola.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
udiag2: subst([radicand=complete+ratsimp(radicand-complete)],udiag);
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Special case for m01d zero.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
udiag3: solve(subst([m01d=0],huv3),[u])[1];
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Special case for m01n zero.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
udiag4: solve(subst([m01n=0],huv3),[u])[1];
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Formula for the value of u at which the slope of the uv-transformed hyperbola is -1.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
depends(v,u)$
depends(u,v)$
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
dvdu: solve(diff(huv,u),diff(v,u))[1];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
dvdu2: ratsimp(solve(subst([diff(v,u)=-1],dvdu),v)[1]);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
udiag5: solve(subst(dvdu2,huv),u)[2];
vdiag5: solve(subst(solve(dvdu2,u)[1],huv),v)[2];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
checkcoord: ev([udiag5,vdiag5],checkvalues),numer;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
udiag6: format(udiag5,%rhs,%frac,%poly(x01,y01),%factor);
vdiag6: format(vdiag5,%rhs,%frac,%poly(x01,y01),%factor);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
m01def: [m2d=m1d+m0d,m2n=m1n+m0n];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
udiag7: subst(swapsides(m01def),udiag6);
vdiag7: subst(swapsides(m01def),vdiag6);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
m01nddef: [m2nd=m2d*m2n];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
udiag8: lratsubst(swapsides(m01nddef), udiag7);
vdiag8: lratsubst(swapsides(m01nddef), vdiag7);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
udiag9: format(udiag8,%rhs,%frac[%p(m2nd,m2nd*n),%f]);
vdiag9: format(vdiag8,%rhs,%frac[%p(m2nd,m2nd*n),%f]);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
ufactor: part(udiag9,2,1,2,1);
vfactor: part(vdiag9,2,1,2,1);
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Determinant.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
m01xdef: [m01x=m0d*m1n-m0n*m1d];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
udiag10: subst([ufactor=format(expand(ufactor+m01x-subst(m01xdef,m01x)),%f+m01x)],udiag9);
vdiag10: subst([vfactor=format(expand(vfactor+m01x-subst(m01xdef,m01x)),%f+m01x)],vdiag9);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
udiag11: subst(swapsides(m01xdef),subst(swapsides(m01def),udiag10));
vdiag11: subst(swapsides(m01xdef),subst(swapsides(m01def),vdiag10));
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
After common subexpression elimination.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
uvdiag12: append(m01def,m01nddef,m01xdef,[udiag11,vdiag11])$
map('print,uvdiag12)$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Check.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
check8: ratsimp(subst(append([udiag11,vdiag11],m01nddef,m01xdef,m01def),huv));
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
checkvalues;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
ev(huv,checkvalues);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
ev(uvdiag12,checkvalues);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
check: solve(ev(uvdiag12,checkvalues),[u,v,m2d,m2n,m2nd,m01x])[1];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
check,numer;
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Formula for v in terms of u.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
coeffs(lhs(huv-rhs(huv)),v);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
u2v1: solve(huv,v)[1];
v2u1: solve(huv,u)[1];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
uradicand: part(u2v1,2,1,1,1,1);
vradicand: part(v2u1,2,1,1,1,1);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
format(uradicand,%p(u),%f);
format(vradicand,%p(v),%f);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
mxy0def: [mxy0=m0d*y01+m0n*x01];
mxy1def: [mxy1=m1d*y01+m1n*x01];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
ucomplete: rhs(mxy1def[1])^2;
vcomplete: rhs(mxy0def[1])^2;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
uradicand2: format(expand(uradicand-ucomplete),%p(u),%f)+ucomplete;
vradicand2: format(expand(vradicand-vcomplete),%p(v),%f)+vcomplete;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
uradicand3: subst(swapsides(append(m01xdef,mxy1def)),uradicand2);
vradicand3: subst(swapsides(append(m01xdef,mxy0def)),vradicand2);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
urest: part(uradicand3,3);
vrest: part(vradicand3,3);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
uradicand4: factor(uradicand3-urest)+urest;
vradicand4: factor(vradicand3-vrest)+vrest;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
u2v2: subst([uradicand=uradicand4],u2v1);
v2u2: subst([vradicand=vradicand4],v2u1);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
u2v3: subst([m01x=1],u2v2);
v2u3: subst([m01x=1],v2u2);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
uvdefs: [m01s=m0d*m1n+m0n*m1d,m0nd=m0n*m0d,m1nd=m1n*m1d,mxy0d=m0d*y01-m0n*x01,mxy1d=-m1d*y01+m1n*x01];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
u2v4: lratsubst(swapsides(uvdefs),u2v3);
v2u4: lratsubst(swapsides(uvdefs),v2u3);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
u2v5: format(u2v4,%rhs,%frac[%sum[%expr[%expr[%f-4*m1nd*n]]]]);
v2u5: format(v2u4,%rhs,%frac[%sum[%expr[%expr[%f-4*m0nd*n]]]]);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
uvformulas: append(uvdefs,mxy0def,mxy1def,[u2v5,v2u5])$
map('print,uvformulas)$
map('print1d,uvformulas)$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Check calculations.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
utest: [u=1];
vtest: [v=1];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
u2v5;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
ucheck: subst(append(checkvalues,utest),append(uvdefs,mxy0def,mxy1def,[u2v5]));
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
solve(ucheck,[m01s,m0nd,m1nd,mxy0d,mxy1d,mxy0,mxy1,v])[1];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
vcheck: subst(append(checkvalues,vtest),append(uvdefs,mxy0def,mxy1def,[v2u5]));
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
vcheck2: solve(vcheck,[m01s,m0nd,m1nd,mxy0d,mxy1d,mxy0,mxy1,u])[1];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
vcheck2,numer;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
expand(subst(subst(append(m01xdef,mxy0def,mxy1def),u2v2),huv));
expand(subst(subst(append(m01xdef,mxy0def,mxy1def),v2u2),huv));
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
udiag12: subst(-swapsides(mxy1def),udiag11);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
uvdiagcrit: append(m01def,m01nddef,m01xdef,mxy1def,[u=floor(rhs(udiag12)),v=floor(rhs(u2v2))])$
map('print,uvdiagcrit)$
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
uvdiagcrit: append(m01def,m01nddef,mxy1def,subst([m01x=1],[u=floor(rhs(udiag12)),v=floor(rhs(u2v2))]))$
map('print,uvdiagcrit)$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Equation of a line tangent to the hyperbola with -slope = 1.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
tangent1: ratsimp(udiag5+vdiag5);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
tangent2: format(tangent1,%rhs,%frac,%p(x01,y01),%expr,%f);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
tangent3: lratsubst(swapsides(m01nddef),backsubst(m01xdef,backsubst(m01def,ratsimp(backsubst(m01def,tangent2)))));
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
tangent4: subst([m01x=1],solve(tangent3,[v])[1]);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
udiag13: subst([m01x=1],udiag12);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
mxy2def: [mxy2=m2n*x01+m2d*y01];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
uvdiagcrit2: append(m01def,m01nddef,mxy1def,mxy2def,[format(udiag13,%f-mxy1),lratsubst(swapsides(mxy2def),tangent4)])$
map('print,uvdiagcrit2)$
map('print1d,uvdiagcrit2)$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Final CSE.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
csedef: [mxy2=m2d*y01+m2n*x01];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
udiag14: format(lratsubst(swapsides(csedef),udiag13),%rhs,%frac,%p(sqrt(m2nd*n)));
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
tangent5: lratsubst(swapsides(csedef),tangent4);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
uvdiagcrit3: append(m01def,m01nddef,mxy1def,csedef,[udiag14,tangent5])$
map('print,uvdiagcrit3)$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Check.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
check1: solve(ev(uvdiagcrit3,checkvalues),[u,v,m2d,m2n,m2nd,mxy1,mxy2,sqrt])[1];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
check2: check1,numer;
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Characterize m01x; this is called the determinant.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
MX(m1,m0) := denom(m0)*num(m1)-num(m0)*denom(m1)$
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
mlist: [m1n/m1d,(m1n+m0n)/(m1d+m0d),m0n/m0d];
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Proof that after insertion of the mediant, the two new intervals have the same determinant as the full interval.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
expand(MX(mlist[1],mlist[2])=1)-(MX(mlist[1],mlist[3])=1);
expand(MX(mlist[2],mlist[3])=1)-(MX(mlist[1],mlist[3])=1);
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Alternative formula for v in terms of u.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
au2v3: format(huv-lhs(huv),%rhs,%p(v));
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
au2vsubst: [a/m1n=x01+m0d*u,b/m1d=y01-m0n*u,m1nd=m1n*m1d];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
au2vdef: format(solve(au2vsubst,[a,b,m1nd])[1],%list,%eq,%p(m1n,m1d));
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
au2v4: lratsubst(swapsides(au2vsubst),backsubst(au2vdef,au2v3));
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
au2v5: solve(au2v4,v)[1];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
radical: part(au2v5,2,1,1,1,1);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
au2v6: subst([radical=format(radical,%f-4*m1nd*n)],au2v5);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
au2vformulaa: append(u2vdef,[au2v6])$
map('print,au2vformulaa)$
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Check.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
autest: u=1;
avtest: subst([autest],ev(ev(au2v6,u2vdef),checkvalues));
vtestfloor: v=floor(rhs(avtest));
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
expand(ev(ev(huv,[autest,avtest]),checkvalues));
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
check1: ev(ev(xy,[autest,avtest]),checkvalues);
floor(check1);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
subst(append([autest],checkvalues),u2vdef);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
expand(ev(lhs(h),check1) <= rhs(ev(h,checkvalues)));
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Experiment.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
u2v3;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
expand(((u+1)+mxy1)^2-(u+mxy1)^2);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
rad[0] = mxy1^2-4*m1nd;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
rad[i] = rad[i-1]+2*u+c1;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
v[i]=(sqrt(rad[i])+c3*u+c4)/c5;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
ev(solve(isolate(format(expand(rhs(huv)-huv),%p(u)),u),[u])[1]);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
term: coeffs(lhs(rhs(huv)-huv),u)[3][1];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
format(format(expand(term)-rhs(mxy0def[1])-rhs(m01xdef[1]*v),%p(v),%f)+mxy0+1*v,%p(v),%f);
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Effect of changes in UV on XY.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
xy;
xy_i: subst([x=x[i],y=y[i],u=u[i],v=v[i]],xy);
xy_j: subst([x=x[j],y=y[j],u=u[j],v=v[j]],xy);
vars_ij: [x[j],y[j],u[i],v[i],u[j],v[j]];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
xy_uincr: rest(solve(append(xy_i,xy_j,[u[j]=u[i]+1,v[j]=v[i]]),vars_ij)[1],-4);
xy_vincr: rest(solve(append(xy_i,xy_j,[u[j]=u[i],v[j]=v[i]+1]),vars_ij)[1],-4);
xy_duv: rest(solve(append(xy_i,xy_j,[u[j]=u[i]+du,v[j]=v[i]+dv]),vars_ij)[1],-4);
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Formulas for changes in UV as a function of XY coordinates.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
lratsubst([uvdenom=1],solve(xy_duv,[du,dv])[1]);
/* [wxMaxima: input end ] */
/* [wxMaxima: comment start ]
Formula for changes in R as function of changes in UV.
[wxMaxima: comment end ] */
/* [wxMaxima: input start ] */
L0: y=-m0n/m0d*x+r[0];
L0_new: y=-m0n/m0d*x+r[new];
L1: y=-m1n/m1d*x+r[0];
L1_new: y=-m1n/m1d*x+r[new];
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
w_eqs: append([
solve(subst([x=x[0],y=y[0]],L0),r[0])[1],
solve(subst([x=x[new],y=y[new]],L0_new),r[new])[1]
], subst([x[i]=x[0],y[i]=y[0],x[j]=x[new],y[j]=y[new]], xy_vincr));
h_eqs: append([
solve(subst([x=x[0],y=y[0]],L1),r[0])[1],
solve(subst([x=x[new],y=y[new]],L1_new),r[new])[1]
], subst([x[i]=x[0],y[i]=y[0],x[j]=x[new],y[j]=y[new]], xy_uincr));
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
w_eqs2: rest(solve(w_eqs,[r[0],r[new],x[new],y[new]])[1],-2);
h_eqs2: rest(solve(h_eqs,[r[0],r[new],x[new],y[new]])[1],-2);
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
w_eq3: expand(solve(subst([uvdenom=1],ratsimp(w_eqs2[2]-w_eqs2[1])),r[new]));
h_eq3: expand(solve(subst([uvdenom=1],ratsimp(h_eqs2[2]-h_eqs2[1])),r[new]));
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
uvdenom;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
radcan(r - radcan(sqrt(2)*(sqrt(2)*r - r)))/r;
/* [wxMaxima: input end ] */
/* [wxMaxima: input start ] */
solve(1=diff(rhs(solve(h,y)[1]),x,2),x)[3],numer;
/* [wxMaxima: input end ] */
/* Maxima can't load/batch files which end with a comment! */
"Created with wxMaxima"$