Finding Parameters of Arbitrarily Rotated Parabola from Implicit Coordinate Equation

As given in Derivation and Properties of Implicit Coordinate Equation for Axis Aligned and Arbitrarily Rotated Parabolas, the Implicit Coordinate Equation for Arbitrarily Rotated Parabola is given by the General Quadratic Equation in 2 Variables as follows

\(Ax^2 + Bxy + Cy^2 + Dx + Ey + F =0\) ...(1)

It can be determined whether the equation (1) actually represents an Parabola by Calculating the Determinant of \(E\) Matrix and the \(e\) Matrix. Equation (1) represents a Parabola only if the Determinant of \(E\) Matrix is Non-Zero and the Determinant of \(e\) Matrix is \(0\).

If the equation (1) represents a Parabola then as given in Derivation and Properties of Implicit Coordinate Equation for Axis Aligned and Arbitrarily Rotated Parabolas, for a Parabola having it's Focus at a point \((x_f,y_f)\) and having Directrix given by the equation \(A_Dx + B_Dy + C_D=0\) the equation (1) becomes

\({B_D}^2x^2 - 2A_DB_Dxy + {A_D}^2y^2 - 2(Px_f + A_DC_D)x - 2(Py_f + B_DC_D)y + P{x_f}^2 + P{y_f}^2 - {C_D}^2 =0\) ...(2)

where \(\mathbf{A}={B_D}^2\), \(\mathbf{B}=-2A_DB_D\), \(\mathbf{C}={A_D}^2\), \(\mathbf{D}=-2(Px_f + A_DC_D)\), \(\mathbf{E}=-2(Py_f + B_DC_D)\), \(\mathbf{F}=P{x_f}^2 + P{y_f}^2 - {C_D}^2\) and \(\mathbf{P}={A_D}^2 + {B_D}^2\)
The following steps can be used to find the Focus and Equation of Directrix of the Parabola represented by equation (2)
1. If the Co-efficient of \(x^2\) or Co-efficient of \(y^2\) in the Equation of Parabola is less than 0 then Multiply the Equation by -1. All the calculations must be done (including calculating the value of \(P={A_D}^2+{B_D}^2\)) must be done after this step.
2. As can be seen from equation (2), the Square Root of the Co-efficient of \(x^2\) in the Equation of Parabola gives the Co-efficient of \(y\) (i.e value of \(B_D\)) of the Equation of the Directrix and the Square Root of the Co-efficient of \(y^2\) in the Equation of Parabola gives the Co-efficient of \(x\) (i.e value of \(A_D\)) of the Equation of the Directrix. Also if the coefficient of \(xy\) in the Equation of Parabola is Negative then either both Co-efficients of \(x\) and \(y\) in the Equation of the Directrix are Positive or both Co-efficients are Negative. If the Co-efficient of \(xy\) in the Equation of Parabola is Positive then one of the Co-efficients in the Equation of the Directrix is Negative and one of them is Positive.
3. The \(x\) coordinate of the Focus of the Parabola \(x_f\) can be found out using the value of constant \(D\) from equation (2) in the following manner
  
  \(D=-2(Px_f+A_DC_D)\)
  
  \(\Rightarrow x_f=\frac{-D-2A_DC_D}{2P}\) ...(3)
4. Similarly, the \(y\) coordinate of the Focus of the Parabola \(y_f\) can be found out using the value of constant \(E\) from equation (2) in the following manner
  
  \(E=-2(Py_f+B_DC_D)\)
  
  \(\Rightarrow y_f=\frac{-E-2B_DC_D}{2P}\) ...(4)
5. Using the Coordinates of Focus from equations (3) and (4) and the value of constant \(F\) from equation (2), the constant \(C\) of the Equation of Directrix can be calculated as follows
  
  \(F=P{x_f}^2 + P{y_f}^2 - {C_D}^2\) ...(5)
  
  Putting the value of \(x_f\) and \(y_f\) from equations (3) and (4) in equation (5) we get
  
  \(F=P{(\frac{-D-2A_DC_D}{2P})}^2 + P{(\frac{-E-2B_DC_D}{2P})}^2 - {C_D}^2\)
  
  \(\Rightarrow F=P{(\frac{{D}^2+4{A_D}^2{C_D}^2 +4DA_DC_D}{4P^2})} + P{(\frac{{E}^2+4{B_D}^2{C_D}^2 + 4EB_DC_D}{4P^2})} - {C_D}^2\)
  
  \(\Rightarrow F={(\frac{{D}^2+4{A_D}^2{C_D}^2 +4DA_DC_D}{4P})} + {(\frac{{E}^2+4{B_D}^2{C_D}^2 + 4EB_DC_D}{4P})} - {C_D}^2\)
  
  \(\Rightarrow 4PF={D}^2 + {E}^2 + 4DA_DC_D + 4EB_DC_D + 4{A_D}^2{C_D}^2 + 4{B_D}^2{C_D}^2 - 4P{C_D}^2\)
  
  \(\Rightarrow 4PF={D}^2 + {E}^2 + 4DA_DC_D + 4EB_DC_D +4{A_D}^2{C_D}^2 + 4{B_D}^2{C_D}^2 - 4{A_D}^2{C_D}^2 - 4{B_D}^2{C_D}^2\)
  
  \(\Rightarrow 4PF - ({D}^2 + {E}^2 ) = 4DA_DC_D + 4EB_DC_D\)
  
  \(\Rightarrow C_D(4DA_D + 4EB_D) = 4PF - ({D}^2 + {E}^2)\)
  
  \(\Rightarrow C_D = \frac{4PF - ({D}^2 + {E}^2)}{4DA_D + 4EB_D}\) ...(6)
  
  The equation (6) given above gives the Constant \(C_D\) of the Equation of Directrix. Please note that the value of the constant \(C_D\) adjusts automatically based on the value and sign taken for Co-efficients \(A_D\) and \(B_D\).
6. Now the value of \(x_f\) and \(y_f\) of the Focus can be found by putting the value of \(A_D\), \(B_D\) and \(C_D\) in equations (3) and (4).
The Signed Focal Length \(f\) of the Parabola can be calculated as Half the Signed Distance between the Focus and the Directrix as follows

\(f=\frac{A_Dx_f + B_Dy_f + C_D}{2\sqrt{{A_D}^2 + {B_D}^2}}\)
The Length of Latus Rectum of Parabola is 4 times the Unsigned Focal Length \(f\) of the Parabola or 2 times Unsigned the Distance between the Focus and the Directrix and can be calculated as follows

\(4|f|=2(|\frac{A_Dx_f + B_Dy_f + C_D}{\sqrt{{A_D}^2 + {B_D}^2}}|)\)
The Coordinates of the Vertex \((x_v,y_v)\) of Parabola can be calculated as follows

\(\begin{bmatrix}x_v\\y_v\end{bmatrix}=\begin{bmatrix}x_f\\y_f\end{bmatrix}- f \begin{bmatrix}\frac{A_D}{\sqrt{{A_D}^2 + {B_D}^2}}\\\frac{B_D}{\sqrt{{A_D}^2 + {B_D}^2}}\end{bmatrix}\)
Since the Direction of Normal to Base of the Parabola is Same as the Direction of Normal to Directrix of Parabola, the Equation of Base of Parabola can be calculated as follows

\(A_Dx + B_Dy -A_Dx_v -B_Dy_v=0\)
Since the Direction of Normal to Latus Rectum of the Parabola is Same as the Direction of Normal to Directrix of Parabola, the Equation of Latus Rectum of Parabola can be calculated as follows

\(A_Dx + B_Dy -A_Dx_f -B_Dy_f=0\)
Since the Direction of Normal to Axis of the Parabola is Perpendicular to the Direction of Normal to Directrix of Parabola, the Equation of Axis of Parabola can be calculated as follows

\(-B_Dx + A_Dy +B_Dx_f -A_Dy_f=0\)

OR

\(-B_Dx + A_Dy +B_Dx_v -A_Dy_v=0\)
The Coordinates of the Points of Intersection of the Latus Rectum and Parabola \((x_1,y_1)\) and \((x_2,y_2)\) can be calculated as follows

\(\begin{bmatrix}x_1\\y_1\end{bmatrix}=\begin{bmatrix}x_f\\y_f\end{bmatrix}- 2f \begin{bmatrix}\frac{-B_D}{\sqrt{{A_D}^2 + {B_D}^2}}\\\frac{A_D}{\sqrt{{A_D}^2 + {B_D}^2}}\end{bmatrix}\)

\(\begin{bmatrix}x_2\\y_2\end{bmatrix}=\begin{bmatrix}x_f\\y_f\end{bmatrix}+ 2f \begin{bmatrix}\frac{-B_D}{\sqrt{{A_D}^2 + {B_D}^2}}\\\frac{A_D}{\sqrt{{A_D}^2 + {B_D}^2}}\end{bmatrix}\)

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