Finding Parameters of Arbitrarily Rotated and Translated Hyperbola from Implicit Coordinate Equation

1. As given in Derivation of Implicit Coordinate Equation for Arbitrarily Rotated and Translated Hyperbolas, the Implicit Coordinate Equation for Arbitrarily Rotated and Translated Hyperbolas is given by
   
   \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F=0\)   ...(1)
   
   where
   
   \(A=b^2\cos^2\theta - a^2\sin^2\theta\)
   
   \(B=(b^2 + a^2)\sin2\theta\)
   
   \(C=b^2 \sin^2\theta - a^2\cos^2\theta\)
   
   \(D=-(2Ax_c+By_c)\)
   
   \(E=-(2Cy_c+Bx_c)\)
   
   \(F=A{x_c}^2+C{y_c}^2+Bx_cy_c - a^2b^2\)
   
   \(\theta\) = Counter Clockwize Rotation Angle from Positive \(X\) Axis
   
   \((x_c,y_c)\) = Coordinates of the Center of Hyperbola
   
   \(a\) = Length of Semi-Transverse Axis of Hyperbola
   
   \(b\) = Length of Semi-Conjugate Axis of Hyperbola
2. Following are the Steps to Find the Parameters of an Arbitrarily Rotated and Translated Hyperbola as given by equation (1) above
   1. Determine whether the equation (1) actually represents an Hyperbola by Calculating the Determinant of \(E\) Matrix and the \(e\) Matrix. Equation (1) represents an Hyperbola only if the Determinant of \(E\) Matrix is Non-Zero and the Determinant of \(e\) Matrix is \(< 0\).
   2. If the equation (1) actually represents an Hyperbola, Find the Coordinates of Center of the Hyperbola \((x_c,y_c)\).
   3. Translate the Equation of Hyperbola as given in equation (1) by \((-x_c,-y_c)\) so that the Center of the Hyperbola comes to lie at the Origin. This will get rid of the \(x\) and the \(y\) terms of the equation of the Hyperbola as given below
      
      \(A {(x+x_c)}^2 + B(x+x_c) (y+y_c) + C{(y+y_c)}^2 + D(x+x_c) + E(y+y_c) + F=0\)
      
      \(\Rightarrow A (x^2+{x_c}^2 + 2x_cx) + B(xy+y_cx+x_cy + x_cy_c) + C(y^2+{y_c}^2+2y_cy) + Dx+ Dx_c + Ey + Ey_c + F=0\)
      
      \(\Rightarrow Ax^2 + Bxy + Cy^2 + Dx + 2Ax_cx +By_cx + Ey + 2Cy_cy + Bx_cy + A{x_c}^2 + Bx_cy_c +C{y_c}^2 + Dx_c + Ey_c + F=0\)   ...(3)
      
      Putting the value of \(D\), \(E\) and \(F\) as given for equation (1) above in equation (3) above we get
      
      \( Ax^2 + Bxy + Cy^2 -(2Ax_c + By_c)x + 2Ax_cx +By_cx -(2Cy_c+Bx_c)y + 2Cy_cy + Bx_cy + A{x_c}^2 + Bx_cy_c +C{y_c}^2 -(2Ax_c + By_c)x_c -(2Cy_c+Bx_c)y_c + A{x_c}^2+C{y_c}^2+Bx_cy_c - a^2b^2=0\)
      
      \(\Rightarrow Ax^2 + Bxy + Cy^2 -2Ax_cx - By_cx + 2Ax_cx +By_cx -2Cy_cy - Bx_cy + 2Cy_cy + Bx_cy + 2A{x_c}^2 + 2Bx_cy_c + 2C{y_c}^2 -2A{x_c}^2 - 2Bx_cy_c -2C{y_c}^2 - a^2b^2=0\)
      
      \(\Rightarrow Ax^2 + Bxy + Cy^2 - a^2b^2=0\)   ...(4)
      
      The equation (4) above gives the Equation of the Rotated Hyperbola having Center at the Origin.
   4. The Angle of Rotation \(\theta\) is determined as follows
      
      Dividing equation (4) above by \(a^2b^2\) we get
      
      \(\frac{A}{a^2b^2}x^2 + \frac{B}{a^2b^2}xy + \frac{C}{a^2b^2}y^2 - 1=0\)   (Normalized Equation of the Rotated Hyperbola having Center at the Origin)...(5)
      
      Setting \(\frac{A}{a^2b^2} = \mathbf{A_1}\), \(\frac{B}{a^2b^2} = \mathbf{B_1}\) and \(\frac{C}{a^2b^2} = \mathbf{C_1}\) in equation (12) above we get
      
      \(A_1x^2 + B_1xy + C_1y^2 - 1=0\)   ...(6)
      
      Using the values of \(A\), \(B\) and \(C\) from equation (1) above calculate \(A_1-C_1\) as
      
      \(A_1-C_1={\Large \frac{b^2\cos^2\theta - a^2\sin^2\theta}{a^2b^2}} - {\Large \frac{b^2 \sin^2\theta - a^2\cos^2\theta}{a^2b^2}} \)
      
      \(\Rightarrow A_1-C_1={\Large \frac{b^2\cos^2\theta - a^2\sin^2\theta - b^2 \sin^2\theta + a^2\cos^2\theta}{a^2b^2}} \)
      
      \(\Rightarrow A_1-C_1={\Large \frac{b^2(\cos^2\theta-\sin^2\theta) + a^2( \cos^2\theta - \sin^2\theta)}{a^2b^2}} \)
      
      \(\Rightarrow A_1-C_1={\Large \frac{(a^2+b^2)(\cos^2\theta-\sin^2\theta)}{a^2b^2}}\)
      
      \(\Rightarrow A_1-C_1={\Large \frac{(a^2+b^2)\cos 2\theta}{a^2b^2}}\)
      
      \(\Rightarrow A_1-C_1={\Large \frac{(a^2+b^2)\cos 2\theta}{a^2b^2}}\)   ...(7)
      
      Now, Counter Clockwize Rotation Angle \(\theta\) from Positive \(X\) Axis can be found as
      
      \(2\theta=m\_arctan2(A_1-C_1,B_1)=m\_arctan2(\frac{(a^2 + b^2)\cos 2\theta}{a^2b^2},\frac{(a^2 + b^2)\sin2\theta}{a^2b^2})= m\_arctan2(\cos 2\theta,\sin2\theta)\)
      
      \(\Rightarrow \theta={\Large \frac{m\_arctan2(\cos 2\theta,\sin2\theta)}{2}}\)   ...(8)
      
      The \(m\_arctan2\) is the Modified Arctangent Function, which calculates the angle \(2\theta\) such that \(0 \leq 2\theta < 2\pi\), so that the Angle \(\theta\) obtained above is such that \(0 \leq \theta < \pi\).
   5. Rotate the Equation of Hyperbola as given in equation (4) by Angle \(\theta\) Clockwize. This will get rid of the \(xy\) term of the equation of Hyperbola and convert it into an Axis Aligned \(X\)-Transverse Hyperbola as given below
      
      \(A{(x\cos\theta - y\sin\theta)}^2 + B(x\cos\theta - y\sin\theta)(y\cos\theta + x\sin\theta) + C{(y\cos\theta + x\sin\theta)}^2 - a^2b^2=0\)
      
      \(\Rightarrow A(x^2 {\cos}^2\theta + y^2{\sin}^2\theta - 2xy\sin\theta\cos \theta) + B(xy {\cos}^2\theta + x^2\sin \theta\cos \theta - xy{\sin}^2\theta -y^2\sin \theta\cos \theta) + C(y^2 {\cos}^2\theta + x^2{\sin}^2\theta + 2xy\sin\theta\cos \theta) - a^2b^2=0\)
      
      \(\Rightarrow Ax^2 {\cos}^2\theta + Ay^2{\sin}^2\theta - 2Axy\sin\theta\cos \theta + Bxy {\cos}^2\theta + Bx^2\sin \theta\cos \theta - Bxy{\sin}^2\theta -By^2\sin \theta\cos \theta + Cy^2 {\cos}^2\theta + Cx^2{\sin}^2\theta+ 2Cxy\sin\theta\cos \theta - a^2b^2=0\)
      
      \(\Rightarrow (A{\cos}^2\theta + B\sin\theta\cos\theta + C{\sin}^2\theta)x^2 + (A{\sin}^2\theta -B\sin\theta\cos\theta+ C{\cos}^2\theta)y^2 + (- 2A\sin\theta\cos\theta + B{\cos}^2\theta - B{\sin}^2\theta + 2C\sin\theta\cos\theta)xy - a^2b^2=0\)
      
      \(\Rightarrow (A{\cos}^2\theta + B\sin\theta\cos\theta + C{\sin}^2\theta)x^2 + (A{\sin}^2\theta -B\sin\theta\cos\theta+ C{\cos}^2\theta)y^2 + (- A\sin2\theta + B\cos2\theta + C\sin2\theta)xy - a^2b^2=0\)
      
      \(\Rightarrow (A{\cos}^2\theta + B\sin\theta\cos\theta + C{\sin}^2\theta)x^2 + (A{\sin}^2\theta -B\sin\theta\cos\theta+ C{\cos}^2\theta)y^2 + ((C- A)\sin2\theta + B\cos2\theta)xy - a^2b^2=0\)   ...(9)
      
      Putting the value of \(A\), \(B\) and \(C\) from equation (1) above in equation (9) above, we get
      
      \(\Rightarrow (A{\cos}^2\theta + B\sin\theta\cos\theta + C{\sin}^2\theta)x^2 + (A{\sin}^2\theta -B\sin\theta\cos\theta+ C{\cos}^2\theta)y^2 + (((b^2 \sin^2\theta - a^2\cos^2\theta)- (b^2\cos^2\theta - a^2\sin^2\theta))\sin2\theta + ((b^2 + a^2)\sin2\theta)\cos2\theta)xy - a^2b^2=0\)
      
      \(\Rightarrow (A{\cos}^2\theta + B\sin\theta\cos\theta + C{\sin}^2\theta)x^2 + (A{\sin}^2\theta -B\sin\theta\cos\theta+ C{\cos}^2\theta)y^2 + ((b^2 \sin^2\theta - a^2\cos^2\theta - b^2\cos^2\theta + a^2\sin^2\theta)\sin2\theta + (b^2 + a^2)\sin2\theta\cos2\theta)xy - a^2b^2=0\)
      
      \(\Rightarrow (A{\cos}^2\theta + B\sin\theta\cos\theta + C{\sin}^2\theta)x^2 + (A{\sin}^2\theta -B\sin\theta\cos\theta+ C{\cos}^2\theta)y^2 + ( (-a^2 (\cos^2\theta-\sin^2\theta) -b^2(\cos^2\theta-\sin^2\theta))\sin2\theta + (b^2 + a^2)\sin2\theta\cos2\theta)xy - a^2b^2=0\)
      
      \(\Rightarrow (A{\cos}^2\theta + B\sin\theta\cos\theta + C{\sin}^2\theta)x^2 + (A{\sin}^2\theta -B\sin\theta\cos\theta+ C{\cos}^2\theta)y^2 + ((-a^2-b^2)\sin2\theta\cos2\theta + (b^2 + a^2)\sin2\theta\cos2\theta)xy - a^2b^2=0\)
      
      \(\Rightarrow (A{\cos}^2\theta + B\sin\theta\cos\theta + C{\sin}^2\theta)x^2 + (A{\sin}^2\theta -B\sin\theta\cos\theta+ C{\cos}^2\theta)y^2 - a^2b^2=0\)
      
      \(\Rightarrow ((b^2\cos^2\theta - a^2\sin^2\theta){\cos}^2\theta + ((b^2 + a^2)\sin2\theta)\sin\theta\cos\theta + (b^2 \sin^2\theta - a^2\cos^2\theta){\sin}^2\theta)x^2 + ((b^2\cos^2\theta - a^2\sin^2\theta){\sin}^2\theta -((b^2 + a^2)\sin2\theta)\sin\theta\cos\theta+ (b^2 \sin^2\theta - a^2\cos^2\theta){\cos}^2\theta)y^2 - a^2b^2=0\)
      
      \(\Rightarrow (b^2\cos^4\theta - a^2\sin^2\theta{\cos}^2\theta + ((b^2 + a^2)2\sin\theta\cos\theta)\sin\theta\cos\theta + b^2 \sin^4\theta - a^2\cos^2\theta{\sin}^2\theta)x^2 + (b^2\cos^2\theta{\sin}^2\theta - a^2{\sin}^4\theta -((b^2 + a^2)2\sin\theta\cos\theta)\sin\theta\cos\theta+ b^2 \sin^2\theta{\cos}^2\theta - a^2\cos^4\theta)y^2 - a^2b^2=0\)
      
      \(\Rightarrow (b^2\cos^4\theta + b^2 \sin^4\theta + 2b^2\sin^2\theta\cos^2\theta + 2a^2\sin^2\theta{\cos}^2\theta - 2a^2\sin^2\theta{\cos}^2\theta)x^2 + (-a^2{\sin}^4\theta - a^2\cos^4\theta - 2a^2\sin^2\theta\cos^2\theta + 2b^2 \sin^2\theta{\cos}^2\theta-2b^2 \sin^2\theta{\cos}^2\theta )y^2 - a^2b^2=0\)
      
      \(\Rightarrow (b^2(\cos^4\theta + \sin^4\theta + 2\sin^2\theta\cos^2\theta))x^2 + (-a^2({\sin}^4\theta + \cos^4\theta + 2\sin^2\theta\cos^2\theta))y^2 - a^2b^2=0\)
      
      \(\Rightarrow (b^2{(\cos^2\theta + \sin^2\theta)}^2 )x^2 - (a^2{({\sin}^2\theta + \cos^2\theta)}^2)y^2 - a^2b^2=0\)
      
      \(\Rightarrow b^2x^2 - a^2y^2 - a^2b^2=0\)
      
      \(\Rightarrow b^2x^2 - a^2y^2=a^2b^2\)   ...(10)
      
      Dividing equation (10) above by \(a^2b^2\) on both sides we get
      
      \(\frac{x^2}{a^2} - \frac{y^2}{b^2}=1\)   ...(11)
      
      The equation (11) above gives the Standard Coordinate Equation of \(X\)-Transverse Hyperbola corresponding to the Hyperbola as given in equation (1).
      
      In order to obtain a Standard Coordinate Equation of \(Y\)-Transverse Hyperbola, the equation (4) must be Rotated by Angle \(\phi=(\theta + 90^\circ) \mod 180^\circ \) Clockwize.
      
      Once the Standard Coordinate Equation of \(X\)-Transverse Hyperbola is found, it's Parameters can be calculated as given below.
      1. Length of Semi-Transverse Axis: \(a\)
      2. Length of Transverse Axis: \(L=2a\)
      3. Length of Semi-Conjugate Axis: \(b\)
      4. Length of Conjugate Axis: \(l=2b\)
      5. Length for Latus Rectum : \({LR}_d=\frac{l^2}{L}=\frac{2b^2}{a}\)
      6. Distance Between 2 Foci: \(F=\sqrt{L^2+l^2}=2c=2\sqrt{a^2+b^2}\)
      7. Distance of Directrices from the Center : \(D=\frac{L^2}{2F}=\frac{a^2}{c}\)
      8. Eccentricity \(e\) : \(\frac{F}{L}=\frac{c}{a}\)
      9. Angular Orientation : \(\theta={\Large \frac{m\_arctan2(A_1-C_1,B_1)}{2}}\)
      10. Coordinates of the Center: \((x_c,y_c)\)=\((\frac{BE-2CD}{4AC-B^2},\frac{BD-2AE}{4AC-B^2})\)
      11. Coordinates of the 2 Vertices : \((x_{v1},y_{v1})=(x_c + a\cos\theta,y_c + a\sin\theta),\hspace{.3cm}(x_{v2},y_{v2})=(x_c - a\cos\theta,y_c- a\sin\theta)\)
      12. Coordinates of the 2 Co-Vertices : \((x_{cv1},y_{cv1})=(x_c - b\sin\theta,y_c + b\cos\theta),\hspace{.3cm}(x_{cv2},y_{cv2})=(x_c + b\sin\theta,y_c- b\cos\theta)\)
      13. Coordinates of the 2 Foci : \((x_{f1},y_{f1})=(x_c + c\cos\theta,y_c + c\sin\theta),\hspace{.3cm}(x_{f2},y_{f2})=(x_c - c\cos\theta,y_c- c\sin\theta)\)
      14. Coordinates of Points of Intersection of Directrix-1 and Transverse Axis: \((x_{d1},y_{d1})=(x_c+D\cos\theta,y_c+D\sin\theta)\)
      15. Coordinates of Points of Intersection of Directrix-2 and Transverse Axis: \((x_{d2},y_{d2})=(x_c-D\cos\theta,y_c-D\sin\theta)\)
      16. Coordinates of Points of Intersection of Latus Rectum-1 and Hyperbola: \((x_{f1}-\frac{{LR}_d}{2}\sin\theta,y_{f1}+\frac{{LR}_d}{2}\cos\theta)\),\((x_{f1}+\frac{{LR}_d}{2}\sin\theta,y_{f1}-\frac{{LR}_d}{2}\cos\theta)\)
      17. Coordinates of Points of Intersection of Latus Rectum-2 and Hyperbola: \((x_{f2}-\frac{{LR}_d}{2}\sin\theta,y_{f2}+\frac{{LR}_d}{2}\cos\theta)\),\((x_{f2}+\frac{{LR}_d}{2}\sin\theta,y_{f2}-\frac{{LR}_d}{2}\cos\theta)\)
      18. Equation of Transverse Axis : \(x\sin\theta - y \cos\theta -x_c\sin\theta + y_c\cos\theta=0\)
      19. Equation of Conjugate Axis : \(x\cos\theta + y \sin\theta -x_c\cos\theta - y_c\sin\theta=0\)
      20. Equation of 2 Directrices : \(x\cos\theta + y \sin\theta -x_{d1}\cos\theta - y_{d1}\sin\theta=0,\hspace{.3cm}x\cos\theta + y \sin\theta -x_{d2}\cos\theta - y_{d2}\sin\theta=0\)
      21. Equation of 2 Latus Recta : \(x\cos\theta + y \sin\theta -x_{f1}\cos\theta - y_{f1}\sin\theta=0,\hspace{.3cm}x\cos\theta + y \sin\theta -x_{f2}\cos\theta - y_{f2}\sin\theta=0\)
      22. Equation of 2 Asymptotic Lines : \((b\cos\theta - a\sin\theta) x + (a\cos\theta + b\sin\theta) y - (b\cos\theta - a\sin\theta)x_c - (a\cos\theta + b\sin\theta)y_c = 0\),   \((b\cos\theta + a\sin\theta) x - (a\cos\theta - b\sin\theta) y - (b\cos\theta + a\sin\theta)x_c + (a\cos\theta - b\sin\theta)y_c = 0\)
      23. Equation of Conjugate Hyperbola : \(-Ax^2 -Bxy -Cy^2 -Dx -Ey + F + Dx_c + Ey_c=0\)