对于N=3,m=5,n=3起点为(m*cos(theta),n*sin(theta))
设cos(theta)=(1-s^2)/(1+s^2),sin(theta)=(2*s)/(1+s^2)
存在内接等边三角形的条件为：
6.185895742>a>5.995560490
得到的f(a,s)=0方程中，s都是偶次的，完全可以由s^2={1-cos\theta}/{1+cos\theta}代回去得到方程g(a,\theta)=0, 从而在极坐标系中画出图像，观察周期性
按照楼上的建议：
我们得到N=3 时
起点(m*cos(t),n*sin(t))
-2048*m^4*n^4*(108*m^8*cos(2*t)-81*m^8-27*m^8*cos(4*t)+168*m^6*n^2*cos(2*t)-228*m^6*n^2+60*m^6*n^2*cos(4*t)+
490*m^4*n^4-66*m^4*n^4*cos(4*t)-228*m^2*n^6+60*m^2*n^6*cos(4*t)-168*m^2*n^6*cos(2*t)-27*n^8*cos(4*t)-108*n^8*cos(2*t)-81*n^8)*a^2+
1536*m^2*n^2*(m-n)*(m+n)*(-27*m^8+27*m^8*cos(2*t)-186*m^6*n^2+36*m^6*n^2*cos(2*t)+130*n^4*cos(2*t)*m^4+36*m^2*n^6*cos(2*t)+
186*m^2*n^6+27*n^8*cos(2*t)+27*n^8)*a^4+576*(m-n)^2*(m+n)^2*(n^2+3*m^2)^2*(3*n^2+m^2)^2*a^6+24576*m^6*n^6*(m^6*cos(6*t)-
6*m^6*cos(4*t)+15*m^6*cos(2*t)-10*m^6+6*m^4*n^2*cos(4*t)-6*m^4*n^2-3*m^4*n^2*cos(6*t)+3*m^4*n^2*cos(2*t)+3*m^2*n^4*cos(6*t)-
6*m^2*n^4-3*m^2*n^4*cos(2*t)+6*m^2*n^4*cos(4*t)-n^6*cos(6*t)-6*n^6*cos(4*t)-15*n^6*cos(2*t)-10*n^6)=0
取m=5,n=3得到
a^6-(539325/8281)*a^4+(749475/16562)*a^4*cos(2*t)-(18933750/8281)*a^2*cos(2*t)+(212810625/132496)*a^2+
(1383750/8281)*a^2*cos(4*t)+(386015625/16562)*cos(2*t)-394453125/18928+(3375000/8281)*cos(6*t)-(43031250/8281)*cos(4*t)=0
对于N=5的内接于椭圆的N等边形问题似乎很难分析
与所有满足条件的五边形相切的椭圆与原椭圆似乎并不共焦点，也不相似。。。。
根据15#的类似求解方案：
对于内接一般的N等边形可以得到
cos(t_1)+cos(t_2)+...+cos(t_n)=0 (1)
sin(t_1)+sin(t_2)+...+sin(t_n)=0 (2)
很巧的是：t_k=t+(2*pi*(k-1))/n,k=1..n (3)均满足(1)(2).
若正确的话可以得到：
2*pi=arccos(1-1/2*(cos(t_1)^2/a^2+sin(t_1)^2/b^2)*L^2)+arccos(1-1/2*(cos(t_2)^2/a^2+sin(t_2)^2/b^2)*L^2)+arccos(1-1/2*(cos(t_3)^2/a^2+sin(t_3)^2/b^2)*L^2)+...+arccos(1-1/2*(cos(t_n)^2/a^2+sin(t_n)^2/b^2)*L^2) (3)
theta_k-theta_(k-1)=arccos(1-1/2*(cos(t_k)^2/a^2+sin(t_k)^2/b^2)*L^2) k=2...n+1,注 theta_(n+1)=theta_1
theta_1=arctan(b/a*ctan(t))-1/2*arccos(1-1/2*(cos(t)^2/a^2+sin(t)^2/b^2)*L^2)
N等边形的顶点坐标为(a*cos(theta_i),b*sin(theta_i)),i=1..n
可惜对于N=5, 以上的结果并不正确，即得到的是非等边形
不知对于(1),(2) 是否还能找到合适的解，即满足题意且t_k是特殊的一组解
突然发现楼上的问题与http://bbs.emath.ac.cn/forum.php?mod=viewthread&tid=4267&extra=page%3D6&page=3
中的两个问题有点关系：
对于双椭圆（内接和外切)的五边形ABCDE(五边依次为a,b,c,d,e),存在的条件？
内接椭圆为x^2/m^2+y^2/n^2=1
(m*cos(t)-m*cos(t+(2*pi)/5))^2+(n*sin(t)-n*sin(t+(2*pi)/5))^2=a^2
(m*cos(t+(2*pi)/5)-m*cos(t+(4*pi)/5))^2+(n*sin(t+(2*pi)/5)-n*sin(t+(4*pi)/5))^2=b^2
(m*cos(t+(4*pi)/5)-m*cos(t+(6*pi)/5))^2+(n*sin(t+(4*pi)/5)-n*sin(t+(6*pi)/5))^2=c^2
(m*cos(t+(6*pi)/5)-m*cos(t+(8*pi)/5))^2+(n*sin(t+(6*pi)/5)-n*sin(t+(8*pi)/5))^2=d^2
(m*cos(t+(8*pi)/5)-m*cos(t+2*pi))^2+(n*sin(t+(8*pi)/5)-n*sin(t+2*pi))^2=e^2
化简为：
c^2*m^2+c^2*n^2+(5*m^4)/2-(5*n^4)/2-(sqrt(5)+1)/2*m^2*b^2-m^2*a^2+(sqrt(5)+1)/2*n^2*b^2+n^2*a^2=0
7.184447626*d^2*m^2*n^2-11.62468045*m^2*b^2*n^2-11.62468045*m^2*a^2*n^2-3.592223813*d^2*m^4-
3.592223813*d^2*n^4+5.550291030*m^4*n^2+5.550291030*m^2*n^4+5.812340223*m^4*b^2+5.812340224*m^4*a^2+
5.812340223*n^4*b^2+5.812340224*n^4*a^2-5.550291030*m^6-5.550291030*n^6=0
7.184447624*m^2*b^2*n^2+11.62468045*m^2*a^2*n^2+7.184447626*e^2*m^2*n^2-8.980559532*m^4*n^2-8.980559533*m^2*n^4-3.592223812*m^4*b^2-5.812340224*m^4*a^2-3.592223812*n^4*b^2-5.812340224*n^4*a^2-3.592223813*e^2*m^4-3.592223813*e^2*n^4+8.980559532*m^6+8.980559532*n^6=0
有谁能给出简化的最终结果？？
(a^2-c^2)/(d^2-e^2)=(sqrt(5)-1)/2
我们可以仿上面的方法得到下面不等式
设t_k 为正实数，k=1..n, t_k<=t_(k+1),t_(n+1)=t_1则有
|sum_(i=1)^n sin(t_(k+1)-t_k)|<=n*sin((2*pi)/n)
仅当t_(k+1)=t_k+(2*pi)/n时取等号
在网上收集到有关椭圆仿射的等价结果：
http://www.docin.com/p-543662416.html
即在什么条件下，椭圆内接及其外切N边形与圆内接及其外切正N边形仿射等价？
定理A:椭圆及其内接N边形与圆及其内接正N边形仿射等价的充要条件：
椭圆内接N边形的相邻两点(x_1,y_1),(x_2,y_2)满足(b^2*x_1*x_2+a^2*y_1*y_2)/(a^2*b^2)=cos({2*pi}/n)
定理B:椭圆及其外切N边形与圆及其内接正N边形仿射等价的充要条件：
在定理A成立的条件下，其椭圆外形N边形的切点是内接N边形各边的中点。
我今天公布一下近几天计算的结果：
根据 http://bbs.emath.ac.cn/thread-5252-2-3.html14#的计算及mathe提供的计算方案有
y/{{b*cos(v)}/sin(u)}+x/{{a*cos(v)}/cos(u)}=1 （1）
4*sin(v)^2*(a^2*sin(u)^2+b^2*cos(u)^2)=L^2 （2）
对上面的消元v得到：
4*sin(u)^4*y^2*a^4+4*sin(u)^2*y^2*a^2*b^2*cos(u)^2+8*sin(u)^3*y*a^3*cos(u)*x*b+8*sin(u)*y*a*cos(u)^3*x*b^3+4*cos(u)^2*x^2*b^2*a^2*sin(u)^2+4*cos(u)^4*x^2*b^4+L^2*b^2*a^2-4*b^2*a^4*sin(u)^2-4*b^4*a^2*cos(u)^2＝0（3）
然后对（3）求导并令 sin(u) = {2*t}/(t^2+1), cos(u) = (-t^2+1)/(t^2+1)代入消元t得到
L^8*a^10*b^2*y^4+2*L^8*a^8*b^4*x^2*y^2-2*L^8*a^8*b^4*y^4+L^8*a^6*b^6*x^4-4*L^8*a^6*b^6*x^2*y^2+L^8*a^6*b^6*y^4-2*L^8*a^4*b^8*x^4+2*L^8*a^4*b^8*x^2*y^2+L^8*a^2*b^10*x^4-2*L^6*a^12*b^4*y^2-
4*L^6*a^12*b^2*y^4+4*L^6*a^12*y^6+2*L^6*a^10*b^6*x^2+6*L^6*a^10*b^6*y^2-12*L^6*a^10*b^4*x^2*y^2+10*L^6*a^10*b^2*x^2*y^4-8*L^6*a^10*b^2*y^6-6*L^6*a^8*b^8*x^2-6*L^6*a^8*b^8*y^2-8*L^6*a^8*b^6*x^4+
12*L^6*a^8*b^6*x^2*y^2+12*L^6*a^8*b^6*y^4+8*L^6*a^8*b^4*x^4*y^2-14*L^6*a^8*b^4*x^2*y^4+2*L^6*a^8*b^4*y^6+6*L^6*a^6*b^10*x^2+2*L^6*a^6*b^10*y^2+12*L^6*a^6*b^8*x^4+12*L^6*a^6*b^8*x^2*y^2-
8*L^6*a^6*b^8*y^4+2*L^6*a^6*b^6*x^6-4*L^6*a^6*b^6*x^4*y^2-4*L^6*a^6*b^6*x^2*y^4+2*L^6*a^6*b^6*y^6-2*L^6*a^4*b^12*x^2-12*L^6*a^4*b^10*x^2*y^2+2*L^6*a^4*b^8*x^6-14*L^6*a^4*b^8*x^4*y^2+
8*L^6*a^4*b^8*x^2*y^4-4*L^6*a^2*b^12*x^4-8*L^6*a^2*b^10*x^6+10*L^6*a^2*b^10*x^4*y^2+4*L^6*b^12*x^6+L^4*a^14*b^6+8*L^4*a^14*b^4*y^2-8*L^4*a^14*b^2*y^4-4*L^4*a^12*b^8-12*L^4*a^12*b^6*x^2-
12*L^4*a^12*b^6*y^2+22*L^4*a^12*b^4*x^2*y^2+56*L^4*a^12*b^4*y^4-20*L^4*a^12*b^2*x^2*y^4-32*L^4*a^12*b^2*y^6+6*L^4*a^10*b^10+28*L^4*a^10*b^8*x^2-12*L^4*a^10*b^8*y^2+22*L^4*a^10*b^6*x^4+
4*L^4*a^10*b^6*x^2*y^2-66*L^4*a^10*b^6*y^4-32*L^4*a^10*b^4*x^4*y^2-44*L^4*a^10*b^4*x^2*y^4+72*L^4*a^10*b^4*y^6+L^4*a^10*b^2*x^4*y^4+20*L^4*a^10*b^2*x^2*y^6-8*L^4*a^10*b^2*y^8-4*L^4*a^8*b^12-12*L^4*a^8*b^10*x^2+28*L^4*a^8*b^10*y^2-4*L^4*a^8*b^8*x^4-52*L^4*a^8*b^8*x^2*y^2-4*L^4*a^8*b^8*y^4-12*L^4*a^8*b^6*x^6-40*L^4*a^8*b^6*x^4*y^2+136*L^4*a^8*b^6*x^2*y^4-28*L^4*a^8*b^6*y^6+
2*L^4*a^8*b^4*x^6*y^2+48*L^4*a^8*b^4*x^4*y^4-54*L^4*a^8*b^4*x^2*y^6+8*L^4*a^8*b^4*y^8+L^4*a^6*b^14-12*L^4*a^6*b^12*x^2-12*L^4*a^6*b^12*y^2-66*L^4*a^6*b^10*x^4+4*L^4*a^6*b^10*x^2*y^2+22*L^4*a^6*b^10*y^4-28*L^4*a^6*b^8*x^6+136*L^4*a^6*b^8*x^4*y^2-40*L^4*a^6*b^8*x^2*y^4-12*L^4*a^6*b^8*y^6+
L^4*a^6*b^6*x^8+36*L^4*a^6*b^6*x^6*y^2-92*L^4*a^6*b^6*x^4*y^4+36*L^4*a^6*b^6*x^2*y^6+L^4*a^6*b^6*y^8+8*L^4*a^4*b^14*x^2+56*L^4*a^4*b^12*x^4+22*L^4*a^4*b^12*x^2*y^2+72*L^4*a^4*b^10*x^6-
44*L^4*a^4*b^10*x^4*y^2-32*L^4*a^4*b^10*x^2*y^4+8*L^4*a^4*b^8*x^8-54*L^4*a^4*b^8*x^6*y^2+48*L^4*a^4*b^8*x^4*y^4+2*L^4*a^4*b^8*x^2*y^6-8*L^4*a^2*b^14*x^4-32*L^4*a^2*b^12*x^6-20*L^4*a^2*b^12*x^4*y^2-
8*L^4*a^2*b^10*x^8+20*L^4*a^2*b^10*x^6*y^2+L^4*a^2*b^10*x^4*y^4-4*L^2*a^16*b^6+4*L^2*a^16*b^4*y^2+12*L^2*a^14*b^8+16*L^2*a^14*b^6*x^2-64*L^2*a^14*b^6*y^2-12*L^2*a^14*b^4*x^2*y^2+
48*L^2*a^14*b^4*y^4-8*L^2*a^12*b^10+4*L^2*a^12*b^8*x^2+152*L^2*a^12*b^8*y^2-24*L^2*a^12*b^6*x^4-12*L^2*a^12*b^6*x^2*y^2-232*L^2*a^12*b^6*y^4+12*L^2*a^12*b^4*x^4*y^2+52*L^2*a^12*b^4*x^2*y^4+
88*L^2*a^12*b^4*y^6-8*L^2*a^10*b^12-112*L^2*a^10*b^10*x^2-112*L^2*a^10*b^10*y^2-88*L^2*a^10*b^8*x^4+24*L^2*a^10*b^8*x^2*y^2+296*L^2*a^10*b^8*y^4+16*L^2*a^10*b^6*x^6+224*L^2*a^10*b^6*x^4*y^2+
80*L^2*a^10*b^6*x^2*y^4-224*L^2*a^10*b^6*y^6-4*L^2*a^10*b^4*x^6*y^2-104*L^2*a^10*b^4*x^4*y^4-52*L^2*a^10*b^4*x^2*y^6+48*L^2*a^10*b^4*y^8+12*L^2*a^8*b^14+152*L^2*a^8*b^12*x^2+4*L^2*a^8*b^12*y^2+
296*L^2*a^8*b^10*x^4+24*L^2*a^8*b^10*x^2*y^2-88*L^2*a^8*b^10*y^4+120*L^2*a^8*b^8*x^6-368*L^2*a^8*b^8*x^4*y^2-368*L^2*a^8*b^8*x^2*y^4+120*L^2*a^8*b^8*y^6-4*L^2*a^8*b^6*x^8-156*L^2*a^8*b^6*x^6*y^2+
80*L^2*a^8*b^6*x^4*y^4+180*L^2*a^8*b^6*x^2*y^6-52*L^2*a^8*b^6*y^8+4*L^2*a^8*b^4*x^6*y^4+12*L^2*a^8*b^4*x^4*y^6+12*L^2*a^8*b^4*x^2*y^8+4*L^2*a^8*b^4*y^10-4*L^2*a^6*b^16-64*L^2*a^6*b^14*x^2+16*L^2*a^6*b^14*y^2-232*L^2*a^6*b^12*x^4-12*L^2*a^6*b^12*x^2*y^2-24*L^2*a^6*b^12*y^4-224*L^2*a^6*b^10*x^6+80*L^2*a^6*b^10*x^4*y^2+224*L^2*a^6*b^10*x^2*y^4+16*L^2*a^6*b^10*y^6-
52*L^2*a^6*b^8*x^8+180*L^2*a^6*b^8*x^6*y^2+80*L^2*a^6*b^8*x^4*y^4-156*L^2*a^6*b^8*x^2*y^6-4*L^2*a^6*b^8*y^8+8*L^2*a^6*b^6*x^8*y^2+24*L^2*a^6*b^6*x^6*y^4+24*L^2*a^6*b^6*x^4*y^6+8*L^2*a^6*b^6*x^2*y^8+
4*L^2*a^4*b^16*x^2+48*L^2*a^4*b^14*x^4-12*L^2*a^4*b^14*x^2*y^2+88*L^2*a^4*b^12*x^6+52*L^2*a^4*b^12*x^4*y^2+12*L^2*a^4*b^12*x^2*y^4+48*L^2*a^4*b^10*x^8-52*L^2*a^4*b^10*x^6*y^2-104*L^2*a^4*b^10*x^4*y^4-
4*L^2*a^4*b^10*x^2*y^6+4*L^2*a^4*b^8*x^10+12*L^2*a^4*b^8*x^8*y^2+12*L^2*a^4*b^8*x^6*y^4+4*L^2*a^4*b^8*x^4*y^6+16*a^16*b^8-16*a^16*b^6*y^2-64*a^14*b^10-80*a^14*b^8*x^2+128*a^14*b^8*y^2+
64*a^14*b^6*x^2*y^2-64*a^14*b^6*y^4+96*a^12*b^12+256*a^12*b^10*x^2-288*a^12*b^10*y^2+160*a^12*b^8*x^4-320*a^12*b^8*x^2*y^2+288*a^12*b^8*y^4-96*a^12*b^6*x^4*y^2+64*a^12*b^6*x^2*y^4-96*a^12*b^6*y^6-
64*a^10*b^14-288*a^10*b^12*x^2+256*a^10*b^12*y^2-384*a^10*b^10*x^4+512*a^10*b^10*x^2*y^2-384*a^10*b^10*y^4-160*a^10*b^8*x^6+192*a^10*b^8*x^4*y^2-160*a^10*b^8*x^2*y^4+256*a^10*b^8*y^6+
64*a^10*b^6*x^6*y^2+64*a^10*b^6*x^4*y^4-64*a^10*b^6*x^2*y^6-64*a^10*b^6*y^8+16*a^8*b^16+128*a^8*b^14*x^2-80*a^8*b^14*y^2+288*a^8*b^12*x^4-320*a^8*b^12*x^2*y^2+160*a^8*b^12*y^4+
256*a^8*b^10*x^6-160*a^8*b^10*x^4*y^2+192*a^8*b^10*x^2*y^4-160*a^8*b^10*y^6+80*a^8*b^8*x^8+64*a^8*b^8*x^6*y^2-32*a^8*b^8*x^4*y^4+64*a^8*b^8*x^2*y^6+80*a^8*b^8*y^8-16*a^8*b^6*x^8*y^2-
64*a^8*b^6*x^6*y^4-96*a^8*b^6*x^4*y^6-64*a^8*b^6*x^2*y^8-16*a^8*b^6*y^10-16*a^6*b^16*x^2-64*a^6*b^14*x^4+64*a^6*b^14*x^2*y^2-96*a^6*b^12*x^6+64*a^6*b^12*x^4*y^2-
96*a^6*b^12*x^2*y^4-64*a^6*b^10*x^8-64*a^6*b^10*x^6*y^2+64*a^6*b^10*x^4*y^4+64*a^6*b^10*x^2*y^6-16*a^6*b^8*x^10-64*a^6*b^8*x^8*y^2-96*a^6*b^8*x^6*y^4-64*a^6*b^8*x^4*y^6-16*a^6*b^8*x^2*y^8＝0 (5)
对楼上的（5）代入
a=5,b=3得到
然后分别令L＝3,4,5,6,7,8作图得到