算額（その438）

算額（その438）

社丈右衛門正常 天明2年壬寅4月(1782)

藤田貞資(1789)：神壁算法巻上
http://www.wasan.jp/jinpeki/jinpekisanpo1.pdf

交差する 2 個の大円の中に，等円 3 個，甲円，乙円，丙円がそれぞれ 4 こずつ入っている。大円の直径が与えられたとき，大円の直径が与えられたとき，各円の直径を求めよ。

大円の半径と中心座標を 2r, (0, r), (0, -r)
等円の半径と中心座標を r, (0, 2r), (0, 0), (0, -2r)
甲円の半径と中心座標を r1, (x1, y1)
乙円の半径と中心座標を r2, (x2, y2)
丙円の半径と中心座標を r3, (x3, y3)
として以下の連立方程式の解を求める。

include("julia-source.txt");

using SymPy
@syms r0::positive, r::positive,
     r1::positive, x1::positive, y1::positive,
     r2::positive, x2::positive, y2::positive,
     r3::positive, x3::positive, y3::positive,
     r4::positive, x4::positive, y4::positive,
     r5::positive, x5::positive, y5::positive,
     r6::positive, x6::positive, y6::positive;

r = r0//2
eq1 = x1^2 + (y1 + r)^2 - (2r + r1)^2
eq2 = x1^2 + (y1 - r)^2 - (2r - r1)^2
eq3 = x1^2 + (2r - y1)^2 - (r1 + r)^2

eq4 = (x1 - x2)^2 + (y1 - y2)^2 - (r1 + r2)^2
eq5 = x2^2 + (r - y2)^2 - (2r - r2)^2
eq6 = x2^2 + (y2 + r)^2 - (2r + r2)^2

eq7 = x3^2 + (r - y3)^2 - (2r - r3)^2
eq8 = (x2 - x3)^2 + (y2 - y3)^2 - (r2 + r3)^2
eq9 = x3^2 + (y3 + r)^2 - (2r + r3)^2;

solve([eq1, eq2, eq3, eq4, eq5, eq6, eq7, eq8, eq9], (r1, x1, y1, r2, x2, y2, r3, x3, y3))

   2-element Vector{NTuple{9, Sym}}:
    (3*r0/10, 2*sqrt(3)*r0/5, 3*r0/5, 9*r0/82, 20*sqrt(3)*r0/41, 9*r0/41, 27*r0/730, 182*sqrt(3)*r0/365, 27*r0/365)
    (3*r0/10, 2*sqrt(3)*r0/5, 3*r0/5, 9*r0/82, 20*sqrt(3)*r0/41, 9*r0/41, 3*r0/10, 2*sqrt(3)*r0/5, 3*r0/5)

最初の組が適解である。
(3*r0/10, 2*sqrt(3)*r0/5, 3*r0/5, 9*r0/82, 20*sqrt(3)*r0/41, 9*r0/41, 27*r0/730, 182*sqrt(3)*r0/365, 27*r0/365)

r0 = 36; r = 18;  r1 = 10.8; x1 = 24.9415; y1 = 21.6; r2 = 3.95122; x2 = 30.4165; y2 = 7.90244; r3 = 1.33151; x3 = 31.0915; y3 = 2.66301

円の直径だけをいえば，甲円，乙円，丙円の直径は大円の直径のそれぞれ 3/10, 9/82, 27/730 倍である。

大円の直径 = 72 のとき，等円の直径 = 36;  甲円の直径 = 21.6;  乙円の直径 = 7.90244;  丙円の直径 = 2.66301

72 .* [3/10, 9/82, 27/730] |> println

   [21.599999999999998, 7.902439024390244, 2.663013698630137]

using Plots

function draw(more=false)
   pyplot(size=(500, 500), grid=false, aspectratio=1, label="", fontfamily="IPAMincho")
   r0 = 72/2
   r = r0/2
   (r1, x1, y1, r2, x2, y2, r3, x3, y3) = (3*r0/10, 2*sqrt(3)*r0/5, 3*r0/5, 9*r0/82, 20*sqrt(3)*r0/41, 9*r0/41, 27*r0/730, 182*sqrt(3)*r0/365, 27*r0/365)
   @printf("r0 = %g; r = %g;  r1 = %g; x1 = %g; y1 = %g; r2 = %g; x2 = %g; y2 = %g; r3 = %g; x3 = %g; y3 = %g\n",
           r0, r, r1, x1, y1, r2, x2, y2, r3, x3, y3)
   @printf("大円の直径 = %g;  等円の直径 = %g;  甲円の直径 = %g;  乙円の直径 = %g;  丙円の直径 = %g\n", 2r0, 2r, 2r1, 2r2, 2r3)
   plot()
   circle(0, r, r0, :gray)
   circle(0, -r, r0, :gray)
   circle(0, 2r, r, :blue)
   circle(0, -2r, r, :blue)
   circle(0, 0, r, :blue)
   circle4(x1, y1, r1)
   circle4(x2, y2, r2, :blue)
   circle4(x3, y3, r3, :green)
   if more
       delta = (fontheight = (ylims()[2]- ylims()[1]) / 500 * 10 * 2) /  3  # size[2] * fontsize * 2
       point(x1, y1, "  甲円:r1,(x1,y1)", :black, :left, :vcenter)
       point(x2, y2, "   乙円:r2,(x2,y2)", :black, :left, :vcenter)
       point(x3, y3, "  丙円:r3,(x3,y3)", :black, :left, :vcenter)
       point(0, r, " r")
       point(0, -r, " -r")
       point(0, 2r, " 2r")
       point(0, 3r, " 3r")
       point(0.08r, 2.4r, "等円", :blue, mark=false)
       point(0.8x1, 2.1r, "大円", :gray, mark=false)
       vline!([0], color=:black, lw=0.5)
       hline!([0], color=:black, lw=0.5)
   else
       plot!(showaxis=false)
   end
end;