算額（その361）

算額（その361）

山形県山形市旅篭町 湯殿山神社
山形算額勝負-湯殿山神社を目指せ-
https://www.sci.yamagata-u.ac.jp/wasan/pdf/20180711SSEP.pdf

正方形の内部に 2 本の斜線と大円，中円，小円，甲円，乙円がある。中円の直径が与えられたときに甲円の直径を求めよ。

大円の半径と中心座標を r0, (0, r0)
中円の半径と中心座標を r1, (0, r1)
小円の半径と中心座標を r2, (x2, y2)
甲円の半径と中心座標を r3, (x3, y3) および (x5, y5)
乙円の半径と中心座標を r4, (r0 - r4, r4)
正方形の一辺の長さは 2r0 である。斜線と正方形の下辺の交点の x 座標を a とする。

r1 = 1 として，以下の連立方程式を立て，nlsolve() で数値解を求める。

include("julia-source.txt");

using SymPy

@syms r0::positive, a::positive, r1::positive,
     r2::positive, x2::positive, y2::positive,
     r3::positive, x3::positive, y3::positive,
     r4::positive, x5::positive, y5::positive;

r1 = 1
eq1 = x2^2 + (y2 - r0)^2 - (r0 - r2)^2
eq2 = x3^2 + (y3 - r0)^2 - (r0 - r3)^2
eq3 = (x2 - x3)^2 + (y2 - y3)^2 - (r2 + r3)^2
eq4 = 2(r0 - r4)^2 - (r0 +r4)^2
eq5 = r1^2*(4r0^2 + a^2) - a^2*(2r0 - r1)^2
eq6 = 2r0*(y2 - r0) - a*x2
eq7 = (r0 - 2r2)*(2r0 - r1) - r1*r0
eq8 = distance(0, 2r0, a, 0, r0 - r4, r4) - r4^2
eq9 = distance(0, 2r0, a, 0, x3, y3) - r3^2;
eq10 = x5^2 + (y5 - r0)^2 - (r0 -r3)^2
eq11 = distance(0, 2r0, a, 0, x5, y5) - r3^2;

# solve([eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8, eq9])

using NLsolve

function nls(func, params...; ini = [0.0])
   if typeof(ini) <: Number
       r = nlsolve((vout, vin) -> vout[1] = func(vin[1], params..., [ini]), ftol=1e-14)
       v = r.zero[1]
   else
       r = nlsolve((vout, vin)->vout .= func(vin, params...), ini, ftol=1e-14)
       v = r.zero
   end
   return v, r.f_converged
end;

function H(u)
   (r0, a, r2, x2, y2, r3, x3, y3, r4, x5, y5) = u
   return [
       x2^2 + (-r0 + y2)^2 - (r0 - r2)^2,  # eq1
       x3^2 + (-r0 + y3)^2 - (r0 - r3)^2,  # eq2
       -(r2 + r3)^2 + (x2 - x3)^2 + (y2 - y3)^2,  # eq3
       2*(r0 - r4)^2 - (r0 + r4)^2,  # eq4
       -a^2*(2*r0 - 1)^2 + a^2 + 4*r0^2,  # eq5
       -a*x2 + 2*r0*(-r0 + y2),  # eq6
       -r0 + (r0 - 2*r2)*(2*r0 - 1),  # eq7
       -r4^2 + (-2*r0*(a^2 - a*r0 + a*r4 + 2*r0*r4)/(a^2 + 4*r0^2) + r4)^2 + (-a*(a*r0 - a*r4 + 4*r0^2 - 2*r0*r4)/(a^2 + 4*r0^2) + r0 - r4)^2,  # eq8
       -r3^2 + (-a*(a*x3 + 4*r0^2 - 2*r0*y3)/(a^2 + 4*r0^2) + x3)^2 + (-2*r0*(a^2 - a*x3 + 2*r0*y3)/(a^2 + 4*r0^2) + y3)^2,  # eq9
       x5^2 + (-r0 + y5)^2 - (r0 - r3)^2,  # eq10
       -r3^2 + (-a*(a*x5 + 4*r0^2 - 2*r0*y5)/(a^2 + 4*r0^2) + x5)^2 + (-2*r0*(a^2 - a*x5 + 2*r0*y5)/(a^2 + 4*r0^2) + y5)^2,  # eq11
   ]
end;

iniv = [big"2.1", 1.41, 0.67, 1.26, 2.44, 0.44, 1.41, 1.34, 0.34, 0.68, 3.4]
res = nls(H, ini=iniv);
println(res);

  (BigFloat[2.000000000000000000000000000320970390259289336928171646885307055660835194112333, 1.414213562373095048801688724010844988825749298548807658294693530490981696954068, 0.6666666666666666666666666668515396947153689160851700548263921689551120494197228, 1.257078722109417821157056643922483726053671707035922941395627750409082770279752, 2.444444444444444444444444444692495891824273215136865120549810095106586884337016, 0.4444444444444444444444444445433658437202953697363922783119495000794485144891305, 1.410452971059059754400848547640885881999130012137408290274003768454414456979157, 1.343969891811035677909661427194763924578403051134536294523138065736756827990007, 0.3431457505076198047932451032161729608070508078480639448376432554356357666557888, 0.6846782324566366141942458588087396030640151534840055426317378671228636978868817, 3.396770848929705062831079314009928699095283423608233263143025331890301809943354], true)

   r0 = 2;  a = 1.41421;  r2 = 0.666667;  x2 = 1.25708;  y2 = 2.44444;  r3 = 0.444444;  x3 = 1.41045;  y3 = 1.34397;  r4 = 0.343146;  x5 = 0.684678;  y5 = 3.39677

中円の直径が1のとき，甲円の直径は 0.4444(4/9) である。

using Plots

function draw(more)
    pyplot(size=(500, 500), grid=false, aspectratio=1, label="", fontfamily="IPAMincho")
   r1 = 1
   (r0, a, r2, x2, y2, r3, x3, y3, r4, x5, y5) = res[1]
   @printf("r0 = %g;  a = %g;  r2 = %g;  x2 = %g;  y2 = %g;  r3 = %g;  x3 = %g;  y3 = %g;  r4 = %g;  x5 = %g;  y5 = %g\n", r0, a, r2, x2, y2, r3, x3, y3, r4, x5, y5)
   #@printf("甲円の直径が %g 寸のとき，乙円の直径は %g 寸\n", 2r1, 2r2)
   plot([r0, r0, -r0, -r0, r0], [0, 2r0, 2r0, 0, 0], color=:black, lw=0.5)
   circle(0, r0, r0)
   circle(0, r1, r1, :brown)
   circle(x2, y2, r2, :blue)
   circle(x3, y3, r3, :magenta)
   circle(x5, y5, r3, :magenta)
   circle(r0 - r4, r4, r4, :green)
   plot!([a, 0, -a], [0, 2r0, 0], color=:black, lw=0.5)
   if more
       point(0, r0, " r0", :red, :left, :bottom)
       point(0, r1, " r1", :brown)
       point(x2, y2, "(x2,y2)")
       point(x3, y3, "(x3,y3)", :magenta)
       point(x5, y5, "(x5,y5)", :magenta)
       point(r0 - r4, r4, "(r0-r4,r4)")
       point(a, 0, "a ", :black, :right, :bottom)
       point(r0, 0, " r0", :black, :left, :bottom)
       point(0, 2r0, " 2r0", :black, :left, :bottom)
       delta = (fontheight = (ylims()[2]- ylims()[1]) / 500 * 10 * 2) /3  # size[2] * fontsize * 2
       hline!([0], color=:black, lw=0.5)
       vline!([0], color=:black, lw=0.5)
   else
       plot!(showaxis=false)
   end
end;