中秋画月饼

wayne

mathe 发表于 2020-10-5 08:16
可以在我上面推导的表达式中取$\alpha=\beta\gt \frac{\pi}4$,得到函数在$\alpha=1.1267645149718208888226 ...

我是直接设的三角形的三个角$\alpha,\beta,\gamma$,表达式结构相似,应该是相同的. 目标是求下面的最大值.\(k = \pi  \left(\cot ^4\left(\frac{\alpha +\pi }{4}\right)+\cot ^4\left(\frac{\beta +\pi }{4}\right)+1\right) \tan \left(\frac{\alpha }{2}\right) \tan \left(\frac{\beta }{2}\right) \tan \left(\frac{\gamma }{2}\right),0<\alpha <\frac{\pi }{2},0<\beta <\frac{\pi }{2}\)
取得最大值是 $k_{max}=\pi y = 0.76223544633708454009202606640962263986525340540534..$ ,对应的条件是 $\alpha=\beta =8 arctan( x) =0.88806362364615146081730681981125665012752212979001 $,
其中$x$满足 $3-40 x+142 x^2+744 x^3-10681 x^4+16608 x^5+23336 x^6-86112 x^7+11446 x^8+160208 x^9-45420 x^10-160208 x^11+11446 x^12+86112 x^13+23336 x^14-16608 x^15-10681 x^16-744 x^17+142 x^18+40 x^19+3 x^20=0$
$y$满足$2300257521-78919447741 y^2+880150583171 y^4-3377753432143 y^6-1290437695232 y^8+45538633728 y^10= 0$

.

王守恩

dlpg070 发表于 2020-10-5 14:25
利用你的例子三角形验算,并且画图
显示
mathe的例子的三角形和我的例11 是相似三角形,结果除尺寸大小 ...

NMaximize[{((1^2 + 2 r^2) \[Pi])/(x y),
  x^2 == (4 r)/(1 - r)^2 == (x^2 + y^2)/(y - 1)^2,
y > 0, x > 0, r > 0}, {y, x, r}]
{0.762236, {y -> 2.58499, x -> 2.1021, r -> 0.399009}}

NMaximize中的N可以去掉吗？

dlpg070

mathe 发表于 2020-10-5 08:16
可以在我上面推导的表达式中取$\alpha=\beta\gt \frac{\pi}4$,得到函数在$\alpha=1.1267645149718208888226 ...

用我的计算方法验算mathe的例子,我的结果与mathe的 40位完全相同
a=4.2041950602464528636635007683629823534000        b=3.3318208        c=3.3318208        A=78.235405度
             k=0.76223544633708454009202606640962263987
mathe    k=0.76223544633708454009202606640962263987

waynekmax=Pi*y = 0.76223544633708454009202606640962263986525340540534..
我的结果 53位于之完全相同
a=4.2041950602464528636635007683629823534000000000000000       
b=3.3318207698458268712795327719486244618000000000000000       
c=3.3318207698458268712795327719486244618000000000000000       
A=78.23540485196237318771526383779000071014172988756026       
A=1.3654654062974903168280297436569895838416127796608872       
k=0.76223544633708454009202606640962263986525340540534
看来最大值计算问题已解决
余下问题是如何证明:
1 最大园为内切圆方案是最佳的
2 最大值存在于 等腰三角形中
显然成立,但我不会证明

王守恩

dlpg070 发表于 2020-10-5 14:25
利用你的例子三角形验算,并且画图
显示
mathe的例子的三角形和我的例11 是相似三角形,结果除尺寸大小 ...

谢谢dlpg070！利用11#的图。
1，记\(BD=x，AD=y，AB=\sqrt{x^2+y^2}\)，三角形\(ABC\)面积\(=xy\)
2，记大圆半径\(=1\)，小圆半径\(=r，BO=\sqrt{x^2+1}\)
3，已知\(\frac{\sqrt{x^2+1}}{1}=\frac{1+r}{1-r}，\Rightarrow x^2=\frac{4r}{(1-r)^2}\)
4，已知\(\frac{x}{\sqrt{x^2+y^2}}=\frac{1}{y-1}，\Rightarrow x^2=\frac{x^2+y^2}{(y-1)^2}\)
5，求\(\frac{(1+2r^2)\pi}{xy}\)的最大值

dlpg070

王守恩 发表于 2020-10-6 09:56
谢谢dlpg070！利用11#的图。
1，记\(BD=x，AD=y，AB=\sqrt{x^2+y^2}\)，三角形\(ABC\)面积\(=xy\)
2， ...

公式与mathe基本相同,但没有转变为角度形式,计算正确,简单,漂亮
(x,y) :顶点A的坐标
r     :小园的半径
大园半径给定为1
我做了些代码的改进,使精度达到和wayne的50位完全相同
--------

1. (*test 王守恩 *)
   
2. Clear["Global`*"];n0=50
   
3. nsol=NMaximize[{((1^2+2 r^2) \[Pi])/(x y),x^2==(4 r)/(1-r)^2==(x^2+y^2)/(y-1)^2,y>0,x>0,r>0},{y,x,r},AccuracyGoal->50,WorkingPrecision->66,Method->"NelderMead"]
   
4. k=N[nsol[[1]],n0];
   
5. y=y/.nsol[[2,1]];
   
6. x=x/.nsol[[2,2]];
   
7. r=r/.nsol[[2,3]];
   
8. a=N[2 x,n0];
   
9. b=N[Sqrt[x^2+y^2],n0]
   
10. c=b;
    
11. 
    
12. Print["x=",x,"\ny=",y,"\nr=",r,"\na=",a,"\nb=",b,"\nc=",c,"\nk=",k]
    
13. Print["=== end ==="]
    

复制代码

--------

计算结果:
x=2.10209753012322643183175038418149117687216126504439631071825884082
y=2.58499818495601067591169418057115581507112785117745105076875157459
r=0.399008596479768339006532766592411131000001697400887502955663381212
a=4.2041950602464528636635007683629823537443225300888
b=3.3318207698458268712795327719486244618669647956151
c=3.3318207698458268712795327719486244618669647956151
k=0.76223544633708454009202606640962263986525340540534 王守恩
k=0.76223544633708454009202606640962263986525340540534 wayne
                                                       全相同

王守恩

dlpg070 发表于 2020-10-5 14:25
利用你的例子三角形验算,并且画图
显示
mathe的例子的三角形和我的例11 是相似三角形,结果除尺寸大小 ...

谢谢dlpg070！利用11#的图。这样也可以。

记\(∠ABC=2a\)，小圆半径\(=r\)，大圆半径\(=1\)

NMaximize[{\(\D\frac{(1 + 2 r^2)\ \pi}{\cot^2(a) \tan(2 a)}， \sin(a)=\frac{1 - r}{1 + r}\), a > 0, r > 0}, {a, r},]

NMaximize[{\(\D\frac{((1 + \sin(a))^2 + 2 (1 - \sin(a))^2)\pi}{(1 +\sin(a))^2 \tan(2 a)\cot(a)^2}\),a > 0}, {a}]

dlpg070

王守恩 发表于 2020-10-7 08:57
谢谢dlpg070！利用11#的图。这样也可以。

记\(∠ABC=2a\)，小圆半径\(=r\)，大圆半径\(=1\)

以王守恩公式为基础化简,向Wayne学习,试着提高计算精度得如下公式和代码
-----------
代码:

1. (*dlpangong070单变量角度公式,容易实现高精度
   
2. 角ABC= A，小圆半径 r，大圆半径 1*)
   
3. Clear["Global`*"];
   
4. n0=100
   
5. nsol=NMaximize[{((1+2 (1-Sin[A/2])^2/(1+Sin[A/2])^2) \[Pi])/(Cot[A/2]^2 Tan[A]),\[Pi]/2>A>0(*,(1-Sin[A/2])/(1+Sin[A/2])>0*)},{A},AccuracyGoal->200,WorkingPrecision->300(*,Method->"NelderMead"*)];
   
6. k=nsol[[1]];
   
7. A=N[A/.nsol[[2,1]],n0];
   
8. Print["单变量角度公式,100位精度,300位内部工作精度,与Wayne公布的200位完全相同:\nABC=",(A)*180/Pi,"度\nk =",k]
   
9. 
   
10. Print["=== end ==="]
    
11. 
    
12. 
    
13. 
    

复制代码

------------
结果:
单变量角度公式,150位精度,300位内部工作精度,给出300位结果,与Wayne公布的约200位完全相同:
ABC=50.88229757401881340614236808110499964204966804392550842499507014347107679199837489947260353385153468
k =0.76223544633708454009202606640962263986525340540533889578262658943321662118415496116357923653885418532917002994643317926679466390226142462875257937425579051176419131724267383654279704698608268408382666297789470479797119151751767306133419167949787323115902932452287595417297851402897864631749201128824329170029946433179266794663902261424628752579374255790511764191317242673836542797046986082684083826662977894704797971191517517673061334191679497873231159029324522875954172978514028978646317492011288243 dlpg070 300位
k =0.762235446337084540092026066409622639865253405405338895782626589433216621184154961163579236538854185329170029946433179266794663902261424628752579374255790511764191317242673836542797046986082... wayne 200位

dlpg070

dlpg070 发表于 2020-10-7 12:56
以王守恩公式为基础化简,向Wayne学习,试着提高计算精度得如下公式和代码
-----------
代码:

单变量角度公式,300位精度,600位内部工作精度计算结果:
ABC=50.88229757401881340614236808110499964204966804392550842499507014347107679199837489947260353385153468度
k =0.762235446337084540092026066409622639865253405405338895782626589433216621184154961163579236538854185329170029946433179266794663902261424628752579374255790511764191317242673836542797046986082684083826662977894704797971191517517673061334191679497873231159029324522875954172978514028978646317492011288242869670124336854288271520949610356358042493152292541674963352074157706153943014489257302300900248701879914222656654430737723712387527965844919646281094368870981175194600041222839905754421013623909134952273586110890866156213173965521777222392666524654609572851916020733252466462455782838999984734585514

wayne

嗯? 你想跟我比? 那好,反正是开心茶馆,那我再给出小数点后100000000位(一亿位哈),求超越, :

1. 0.7622354463370845400920260664096226398652534054053388957826265894332166211841549611635792365388541853291700299464331792667946639022614246287525793742557905117641913172426738365427970469860826840838266629778947047979711915175176730613341916794978732311590293245228759541729785140289786463174920112882428696701243368542882715209496103563580424931522925416749633520741577061539430144892573023009002487018799142226566544307377237123875279658449196462810943688709811751946000412228399057544210136239091349522735861108908661562131739655217772223926665246546095728519160207332524664624557828389999847345855141627434421353266652774253960434921479778954407657477678280858275074623380508558573386073535735022950452278339566631113135083154676372174295876514626476470439413480303198566968617957038897958981494086971679613637657315458139234435866462367415262386505402055396907055636434053727546405336759319735045155672511567380821598749550793617808252587890392037718637439288438821930178672476818326529428419202636
   
2. [此处省掉中间最无聊的99998000个数字]
   
3. 0258435688720118463303852633386720970982705135884310234348367547806974864950128341089836156441898182397216247290024318822536369575611957266956828066077684094406532148408875750176446859750636674815485336704365392769131678590057636783527488071530274061288344716742323555915915315151527061509777040395662428944037435515663651654148663742973728703646649699976151855093045446145522913825220022772794769403607676753148345991639851858073210426976379864872558008585704389852256853811145509578048515157248057180201958131409832586557737475328743461718654068424040627042556979851788733457679904926722152864633719375417139355195934667773942424731131055254171241233704166047680178087406531836273794128405054745491651466550041384390838998405690836550099571210200576746503703679501319431755122190248943109739729271256964728679754967232136693907831212228744144859546638807782553303799843761761606218397671793624146901739917433749683276183666934360958158134979396469095079602777452445890909321462774499298588167196671

复制代码

wayne

1. N[\[Pi] Root[2300257521 - 78919447741 #^2 + 880150583171 #^4 - 3377753432143 #^6 - 1290437695232 #^8 + 45538633728 #^10& , 3, 0], 1000]

复制代码