Derivation of Implicit Coordinate Equation for Arbitrarily Rotated and Translated Ellipses

1. As given in Derivation of Standard and Implicit Coordinate Equation for Axis Aligned Ellipses, the Standard Coordinate Equation for Ellipses having their Major Axes Parallel to the \(X\)-Axis and Center at Origin, Length of Semi-Major Axis \(a\) and Length of Semi-Minor Axis \(b\) is given as
   
   \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = \pm\hspace{1mm}1\)   ...(1)
   
   where the Right Hand Side of the equation (1) above is \(1\) for Real Ellipse and \(-1\) for Imaginary Ellipse
   
   The Equation of an Arbitrary Rotated Ellipse with its Center Translated to a given Point can be obtained using following steps in order
   1. Rotate the Standard Coordinate Equation of Ellipse as given in equation (1) by the Arbitrary Angle \(\theta\).
   2. Translate the Equation of Rotated Ellipse to the given Point.
2. The following derives the Equation of Rotated Ellipse from Standard Coordinate Equation of Ellipse given in equation (1)
   
   Multiplying equation (1) by \(a^2b^2\)on Both Sides we get
   
   \(b^2x^2+ a^2y^2 = \pm\hspace{1mm}a^2b^2 \)
   
   \(\Rightarrow b^2x^2+ a^2y^2 \pm\hspace{1mm}a^2b^2 = 0\)   ...(2)
   
   Rotating the equation (2) above Counter Clockwise with angle \(\theta\) with respect to Positive Direction of \(X\) Axis we get
   
   \(b^2{(x \cos\theta + y \sin\theta)}^2+ a^2{(y \cos\theta - x \sin\theta)}^2 \pm\hspace{1mm}a^2b^2 = 0 \)
   
   \(\Rightarrow b^2(x^2 \cos^2\theta + y^2 \sin^2\theta + 2xy\sin\theta\cos\theta )+ a^2(y^2\cos^2\theta + x^2\sin^2\theta - 2xy\sin\theta\cos\theta ) \pm\hspace{1mm}a^2b^2 = 0 \)
   
   \(\Rightarrow b^2x^2\cos^2\theta + b^2y^2 \sin^2\theta + 2b^2xy\sin\theta\cos\theta + a^2y^2\cos^2\theta + a^2x^2\sin^2\theta - 2a^2xy\sin\theta\cos\theta \pm\hspace{1mm}a^2b^2 = 0 \)
   
   \(\Rightarrow (b^2\cos^2\theta + a^2\sin^2\theta)x^2 + (2b^2\sin\theta\cos\theta - 2a^2\sin\theta\cos\theta)xy + (b^2 \sin^2\theta + a^2\cos^2\theta)y^2 \pm\hspace{1mm}a^2b^2 = 0 \)
   
   \(\Rightarrow (b^2\cos^2\theta + a^2\sin^2\theta)x^2 + (2\sin\theta\cos\theta(b^2 - a^2))xy + (b^2 \sin^2\theta + a^2\cos^2\theta)y^2 \pm\hspace{1mm}a^2b^2 = 0\)
   
   \(\Rightarrow (b^2\cos^2\theta + a^2\sin^2\theta)x^2 + ((b^2 - a^2)\sin2\theta)xy + (b^2 \sin^2\theta + a^2\cos^2\theta)y^2 \pm\hspace{1mm}a^2b^2 = 0\)   ...(3)
   
   The equation (3) above gives the Equation of the Rotated Ellipse having Center at Origin
   
   Setting \(b^2\cos^2\theta + a^2\sin^2\theta=\mathbf{A}\), \((b^2 - a^2)\sin2\theta=\mathbf{B}\) and \(b^2 \sin^2\theta + a^2\cos^2\theta=\mathbf{C}\) in equation (3) above we get
   
   \(Ax^2 + Bxy + Cy^2 \pm\hspace{1mm}a^2b^2 = 0\)   ...(4)
   
   Now, Translating the Equation of the Rotated Ellipse as given in equation (4) above so that it's Center is at \((x_c,y_c)\) we get
   
   \(A{(x-x_c)}^2 + B(x-x_c)(y-y_c) + C{(y-y_c)}^2 \pm\hspace{1mm}a^2b^2 = 0\)
   
   \(\Rightarrow A(x^2+{x_c}^2 - 2x_cx) + B(xy-x_cy-y_cx+x_cy_c) + C(y^2+{y_c}^2-2y_cy) \pm\hspace{1mm}a^2b^2 = 0\)
   
   \(\Rightarrow Ax^2+A{x_c}^2 - 2Ax_cx + Bxy-Bx_cy-By_cx+Bx_cy_c + Cy^2+C{y_c}^2-2Cy_cy \pm\hspace{1mm}a^2b^2 = 0\)
   
   \(\Rightarrow Ax^2+ Bxy + Cy^2 - (2Ax_c+By_c)x -(2Cy_c+Bx_c)y +A{x_c}^2+C{y_c}^2+Bx_cy_c \pm\hspace{1mm}a^2b^2=0\)   ...(5)
   
   Setting \(- (2Ax_c+By_c)=\mathbf{D}\), \(-(2Cy_c+Bx_c)=\mathbf{E}\) and \(A{x_c}^2+C{y_c}^2+Bx_cy_c \pm\hspace{1mm}a^2b^2=\mathbf{F}\) in equation (5) above we get
   
   \(Ax^2 + Bxy + Cy^2 +Dx +Ey +F=0\)   ...(6)
   
   The equation (5) and (6) above give the Implicit Coordinate Equation of the Ellipse Rotated Counter Clockwise by Angle \(\theta\) with respect to Positive Direction of \(X\) Axis and having Center at \((x_c,y_c)\) where
   
   \(A = b^2\cos^2\theta + a^2\sin^2\theta\)
   
   \(B = (b^2 - a^2)\sin2\theta\)
   
   \(C = b^2 \sin^2\theta + a^2\cos^2\theta\)
   
   \(D = -(2Ax_c+By_c) = -(2(b^2\cos^2\theta + a^2\sin^2\theta) x_c + ((b^2 - a^2)\sin2\theta) y_c)\)
   
   \(E = -(2Cy_c+Bx_c) = -(2(b^2\sin^2\theta + a^2\cos^2\theta) y_c + ((b^2 - a^2)\sin2\theta) x_c)\)
   
   \(F = A{x_c}^2+C{y_c}^2+Bx_cy_c -a^2b^2 = (b^2\cos^2\theta + a^2\sin^2\theta) {x_c}^2 + (b^2 \sin^2\theta + a^2\cos^2\theta) {y_c}^2 + ((b^2 - a^2)\sin2\theta) x_cy_c \pm a^2b^2\)