[平面幾何][Matlab] 平面橢圓參數與一般式之間的轉換

 

橢圓的一般式為：\[A{x^2} + Bxy + C{y^2} + Dx + Ey + F = 0\]

橢圓的參數為：長半軸 $a$  短半軸 $b$  橢圓中心 $(x_{0},y_{0})$  傾角為 $\theta$ (定義逆時針為正，長軸與x正方向的夾角)

 1.由橢圓參數轉化為一般式：

 

推導過程： 

橢圓 $C[3*3]$ ,中心在原點，長軸與x軸重合，經過旋轉矩陣  ${R} =f({\theta})$ , 平移矩陣 ${T}$$ =$$g$$(x_{0},y_{0})$ ,

后得到

$C^{'}=T^{T}*R^{T}*C*R*T$

 

ps:關於旋轉矩陣R和平移矩陣T的定義看上篇博文 直角坐標系下點/曲線平移與旋轉的矩陣計算

 

即 $H({\theta},x_{0},y_{0},a,b)=A{x^2} + Bxy + C{y^2} + Dx + Ey + F $

對應相等可以得到：

 

①  $A = \frac{{{{\cos }^2}\theta }}{{{a^2}}} + \frac{{{{\sin }^2}\theta }}{{{b^2}}}$

 

② $B = 2 \cdot \sin \theta  \cdot \cos \theta  \cdot (\frac{1}{{{a^2}}} - \frac{1}{{{b^2}}})$

 

③ $C = \frac{{{{\cos }^2}\theta }}{{{b^2}}} + \frac{{{{\sin }^2}\theta }}{{{a^2}}}$

 

④ $D =  - 2 \cdot [{x_0} \cdot (\frac{{{{\cos }^2}\theta }}{{{a^2}}} + \frac{{{{\sin }^2}\theta }}{{{b^2}}}) + {y_0} \cdot \sin \theta  \cdot \cos \theta  \cdot (\frac{1}{{{a^2}}} - \frac{1}{{{b^2}}})]$

 

⑤ $E =  - 2 \cdot [{x_0} \cdot \sin \theta  \cdot \cos \theta  \cdot (\frac{1}{{{a^2}}} - \frac{1}{{{b^2}}}) + {y_0} \cdot (\frac{{{{\cos }^2}\theta }}{{{b^2}}} + \frac{{{{\sin }^2}\theta }}{{{a^2}}})]$

 

⑥ $F = \frac{{{{({x_0} \cdot \cos \theta  + {y_0} \cdot \sin \theta )}^2}}}{{{a^2}}} + \frac{{{{({x_0} \cdot \sin \theta  - {y_0} \cdot \cos \theta )}^2}}}{{{b^2}}} - 1$

matlab推導過程

+驗證

 

clc
syms a b theta x0 y0
% 公式推導
C  = [1/a.^2  0       0;
        0       1/b.^2 0;
        0   	0       -1;];
Rot = [cos(theta) sin(theta) 0;
        -sin(theta) cos(theta) 0;
        0       0       1;];
T = [1 0 -x0;
     0 1 -y0;
     0 0 1;];
C1 =  T'*Rot'*C*Rot*T;

%公式驗證
as = a*a;
bs = b*b;
coss = cos(theta).^2;
sins = sin(theta).^2;
cs = sin(theta)*cos(theta);

A =coss/as+sins/bs;
B =2*cs*(1/as-1/bs);
C = coss/bs +sins/as;
D = -(2*A*x0 +B*y0);
E = -(B*x0 +2*A*y0);
F = (x0*cos(theta)+y0*sin(theta)).^2/as+(x0*sin(theta)-y0*cos(theta)).^2/bs-1

a = 3;
b = 2;
x0 = 1;
y0 = 0.5;
theta = 0.1;
A = eval(A)
B = eval(B)
C = eval(C)
D = eval(D)
E = eval(E)
F = eval(F)

syms x y
f1 = ezplot( A*x^2+ C*y^2 +F + B*x*y + D*x +E*y,[-2,6],[-2,6]);
set(f1,'Color','r','LineWidth',1.5)
xlim([-2,6])
ylim([-2,6])
axis equal
grid on

 

 

 

 

 

 

 

 

2.由一般式得到橢圓參數式：

橢圓的一般式為：\[A{x^2} + Bxy + C{y^2} + Dx + Ey + F = 0\]

由①②③式可以得到：

　長半軸： $a^{2}=\frac{2}{A+C-\sqrt{B^{2}+(A-C)^{2}}}$

　短半軸： $b^{2}=\frac{2}{A+C+\sqrt{B^{2}+(A-C)^{2}}}$

　傾角：　 ${\theta} = arcsin({ sign(-B) \sqrt{\frac{(Aa^{2}-Cb^{2})a^{2}b^{2}}{a^{4}-b^{4}}}})$

　偏移：alpha = cos(theta).^2/a^2+sin(theta).^2/b^2;
　　　　beta = sin(theta)*cos(theta)*(1/a^2-1/b^2);
　　　　gama = cos(theta).^2/b^2+sin(theta).^2/a^2;
　　　　y0 = (E/2 - beta*D/(2*alpha))/(beta^2/alpha - gama)
　　　　x0 = (-D/2 - beta*y0)/alpha

 

%接上面程序運行
aa = 2/(A+C-sqrt(B^2+(A-C).^2))
bb = 2/(A+C+sqrt(B^2+(A-C).^2))
if(bb > aa)
    temp = aa;
    aa = bb;
    bb = temp;
end
theta
theta2 = asin(sign(-B)*sqrt((A*aa-C*bb)*aa*bb/(aa*aa-bb*bb)))
a = sqrt(aa)
b = sqrt(bb)
alpha = cos(theta).^2/a^2+sin(theta).^2/b^2;
beta = sin(theta)*cos(theta)*(1/a^2-1/b^2);
gama = cos(theta).^2/b^2+sin(theta).^2/a^2;
y0 = (E/2 - beta*D/(2*alpha))/(beta^2/alpha - gama)
x0 = (-D/2 - beta*y0)/alpha