求椭圆的长短轴,或者化成标准方程.XYABC五个参数.A*(x-X)^2 + 2*B*(x-X)*(y-Y) + C*(

这个涉及两步,第一步是平移,第二步试旋转.
1. 平移
将椭圆向左平移X, 向下平移Y得到: Ax² + 2Bxy + Cy² = 1 (i)




2. 旋转
设将坐标系逆时针旋转θ, 新坐标系为x'Oy'
x' = xcosθ - ysinθ
y' = xsinθ + ycosθ
可解得:
x = x'cosθ + y'sinθ
y = -x'sinθ + y'cosθ
代入(i):
A(x'cosθ + y'sinθ)² + 2B(x'cosθ + y'sinθ)(-x'sinθ + y'cosθ) + C(-x'sinθ + y'cosθ)² = 1
(Acos²θ - 2Bsinθcosθ + Csin²θ)x'² + (Asin²θ + 2Bsinθcosθ + Ccos²θ)y'²
+ (2Asinθcosθ + 2Bcos²θ - 2Bsin²θ - 2Csinθcosθ)x'y' = 1
化成标准方程, 须2Asinθcosθ + 2Bcos²θ - 2Bsin²θ - 2Csinθcosθ = 0
(A - C)sin(2θ) - 2B(cos²θ - sin²θ) = 0
(A- C)sin(2θ) = 2Bcos(2θ)
tan(2θ) = 2B/(A - C)
由此可求出θ, 以及x'², y'²的系数


图中是用A = 3, B = 3, C = 4作例子.