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| Two parabolas through four points, plotted using Desmos.com |
Conic sections are fascinating, and the internet is flush with articles describing them. I make my own small contribution here, providing implicit formulas for generating circles from three points and parabolas from either three or four points. The explanations are below--though either from brevity or laziness, I have not provided the full (and painful) derivation of the four-point parabola. Click the links to go to the specific section:
- Fitting a Vertical Parabola to Three Points
- Fitting a Circle to Three Points
- Fitting Skewed Parabolas to Four Points
More engaging are these scenarios I set up on the Desmos.com Graphing Calculator, where you can adjust the points and move around the circle and parabolas; recall that in the four-point case, they may become imaginary and disappear.
- A Circle and a Parabola from Three Points (on Desmos.com)
- Two Parabolas from Four Points (on Desmos.com)
Fitting a Vertical Parabola to Three Points
Summary
Any three points \(\left( x_{h},y_{h} \right)\), \(\left( x_{i},y_{i} \right),\) and \(\left( x_{j},y_{j} \right)\) can be fitted to a standard vertical parabola using the following formula (Equation \ref{eq:quadratic-three}):
\begin{multline*}
g\left(x\right)=\left( \frac{M_{hi}-M_{hj}}{x_{{i}}-x_{{j}}} \right) x^{2} +
\left( \frac{M_{hj}\left(x_{h}+x_{i} \right)-M_{hi}\left(x_{h}+x_{j} \right)}{x_{i}-x_{j}} \right) x \\
+ \left( \frac{x_{h}\left(M_{hi}x_{j}-M_{hj}x_{i} \right)}{x_{i}-x_{j}}+y_{h} \right)
\end{multline*}
Where (Equations \ref{eq:m_hi} and \ref{eq:m_hj}),
\begin{equation*}
M_{hi}=\frac{y_{h}-y_{i}}{x_{h}-x_{i}}~~~~~~
M_{hj}=\frac{y_{h}-y_{j}}{x_{h}-x_{j}}
\end{equation*}
For the vertex form of the equation (and coordinates of the vertex and the focus), see Equation \ref{eq:vertex-parabola}; for an expanded version of the equation that does not use \(M_{hi}\) and \(M_{hj}\), see Equation \ref{eq:parabola-expanded}.
Derivation
Any three points in the plane \(\left( x_{h},y_{h} \right)\), \(\left( x_{i},y_{i} \right),\) \(\left( x_{j},y_{j} \right)\) can be connected to form a standard parabola with a vertical axis, of the form
\begin{equation}
g\left(x\right)=ax^{2}+bx+c
\label{eq:quadratic-basic}
\end{equation}
To begin, we note that since all three points lie on the same parabola, their \emph{y} coordinates can be rewritten:
\begin{equation}
y_{h}= g\left(x_{h} \right)= ax_{h}^{2}+ bx_{h}+ c
\label{eq:x_h}
\end{equation}
\begin{equation}
y_{i}= g\left(x_{i} \right)= ax_{i}^{2}+ bx_{i}+ c
\label{eq:x_i}
\end{equation}
\begin{equation}
y_{j}= g\left(x_{j} \right)= ax_{j}^{2}+ bx_{j}+ c
\label{eq:x_j}
\end{equation}
Equations \ref{eq:x_h}, \ref{eq:x_i}, and \ref{eq:x_j} will allow us to solve for \(a\), \(b\), and \(c\) in four steps.
Step 1: Partly Solve for b
By subtracting Equation \ref{eq:x_j} from Equation \ref{eq:x_h}, we obtain
\begin{equation*}
y_{h} - y_{j} = \left( ax_{h}^{2} + bx_{h} + c \right) - \left( ax_{j}^{2} + bx_{j} + c \right)
\end{equation*}
\begin{equation*}
y_{h} - y_{j} = ax_{h}^{2} + bx_{h} + c - ax_{j}^{2} - bx_{j} - c
\end{equation*}
\begin{equation*}
y_{h} - y_{j} = a\left( x_{h}^{2} - x_{j}^{2} \right) + b\left( x_{h} - x_{j} \right)
\end{equation*}
\begin{equation*}
\frac{y_{h} - y_{j}}{x_{h} - x_{j}} = a\left( x_{h} + x_{j} \right) + b
\end{equation*}
\begin{equation*}
b = \frac{y_{h} - y_{j}}{x_{h} - x_{j}} - a\left( x_{h} + x_{j} \right)
\end{equation*}
If we define
\begin{equation}
M_{hj}=\frac{y_{h}-y_{j}}{x_{h}-x_{j}}
\label{eq:m_hj}
\end{equation}
Then
\begin{equation}
b = M_{hj} - a\left( x_{h} + x_{j} \right)
\label{eq:b-interim}
\end{equation}
Step 2: Fully Solve for a
Using Equation \ref{eq:b-interim}, we substitute for \(b\) into Equations \ref{eq:x_h} and \ref{eq:x_i}:
\begin{equation*}
y_{h} = ax_{h}^{2} + \left( M_{hj} - a\left( x_{h} + x_{j} \right) \right)x_{h} + c
\end{equation*}
\begin{equation*}
y_{i} = ax_{i}^{2} + \left( M_{hj} - a\left( x_{h} + x_{j} \right) \right)x_{i} + c
\end{equation*}
We may now fully solve for \(a\):
\begin{equation*}
y_{h} - y_{i} = \left( ax_{h}^{2} + \left( M_{hj} - a\left( x_{h} + x_{j} \right) \right)x_{h} + c \right) - \left( ax_{i}^{2} + \left( M_{hj} - a\left( x_{h} + x_{j} \right) \right) + c \right)
\end{equation*}
\begin{equation*}
y_{h} - y_{i} = ax_{h}^{2} + \left( M_{hj} - a\left( x_{h} + x_{j} \right) \right)x_{h} - ax_{i}^{2} - \left( M_{hj} - a\left( x_{h} + x_{j} \right) \right)x_{i}
\end{equation*}
\begin{equation*}
y_{h} - y_{i} = ax_{h}^{2} + \left( M_{hj}x_{h} - ax_{h}^{2} - ax_{h}x_{j} \right) - ax_{i}^{2} - \left( M_{hj}x_{i} - ax_{h}x_{i} - ax_{i}x_{j} \right)
\end{equation*}
\begin{equation*}
y_{h} - y_{i} = M_{hj}x_{h} - ax_{h}x_{j} - ax_{i}^{2} - M_{hj}x_{i} + ax_{h}x_{i} + ax_{i}x_{j}
\end{equation*}
\begin{equation*}
y_{h} - y_{i} = M_{hj}x_{h} - M_{hj}x_{i} + ax_{h}x_{i} - ax_{i}^{2} - ax_{h}x_{j} + ax_{i}x_{j}
\end{equation*}
\begin{equation*}
y_{h} - y_{i} = M_{hj}\left( x_{h} - x_{i} \right) + ax_{i}\left( x_{h} - x_{i} \right) - ax_{j}\left( x_{h} - x_{i} \right)
\end{equation*}
\begin{equation*}
y_{h} - y_{i} - M_{hj}\left( x_{h} - x_{i} \right) = a\left( x_{i} - x_{j} \right)\left( x_{h} - x_{i} \right)
\end{equation*}
\begin{equation*}
a = \frac{1}{x_{i} - x_{j}}\left( \frac{y_{h} - y_{i}}{x_{h} - x_{i}} - M_{hj} \right)
\end{equation*}
If we define
\begin{equation}
M_{hi}=\frac{y_{h}-y_{i}}{x_{h}-x_{i}}
\label{eq:m_hi}
\end{equation}
Then
\begin{equation}
a=\frac{M_{hi}-M_{hj}}{x_{i}-x_{j}}
\label{eq:a-full}
\end{equation}
Step 3: Fully Solve for b
Using Equation \ref{eq:a-full}, we substitute for \(a\) into Equation \ref{eq:b-interim}, to obtain \(b\):
\begin{equation*}
b = M_{hj} - \left( \frac{M_{hi} - M_{hj}}{x_{i} - x_{j}} \right)\left( x_{h} + x_{j} \right)
\end{equation*}
\begin{equation*}
b = \frac{M_{hj}\left( x_{i} - x_{j} \right) - M_{hi}\left( x_{h} + x_{j} \right) + M_{hj}\left( x_{h} + x_{i} \right)}{x_{i} - x_{j}}
\end{equation*}
\begin{equation}
b=\frac{M_{hj}\left(x_{h}+x_{i}\right)-M_{hi}\left(x_{h}+x_{j} \right)}{x_{i}-x_{j}}
\label{eq:b-full}
\end{equation}
Step 4: Fully Solve for c
Using Equation \ref{eq:b-interim}, we substitute for \(b\) into Equation \ref{eq:x_i}. This gives,
\begin{equation*}
y_{h} = ax_{h}^{2} + \left( M_{hj} - a\left( x_{h} + x_{j} \right) \right)x_{h} + c
\end{equation*}
\begin{equation*}
y_{h} = ax_{h}^{2} + M_{hj}x_{h} - ax_{h}\left( x_{h} + x_{j} \right) + c
\end{equation*}
\begin{equation*}
y_{h} = M_{hj}x_{h} - ax_{h}x_{j} + c
\end{equation*}
\begin{equation*}
c = y_{h} - M_{hj}x_{h} + ax_{h}x_{j}
\end{equation*}
\begin{equation*}
c = \left( ax_{j} - M_{hj} \right)x_{h} + y_{h}
\end{equation*}
Using Equation \ref{eq:a-full}, we substitute for \(a\), to obtain \(c\):
\begin{equation*}
c = \left( \left( \frac{M_{hi} - M_{hj}}{x_{i} - x_{j}} \right)x_{j} - M_{hj} \right)x_{h} + y_{h}
\end{equation*}
\begin{equation*}
c = \left( \frac{M_{hi}x_{j} - M_{hj}x_{j} - M_{hj}\left( x_{i} - x_{j} \right)}{x_{i} - x_{j}} \right)x_{h} + y_{h}
\end{equation*}
\begin{equation}
c =\frac{\left(M_{hi}x_{j}-M_{hj}x_{i} \right)x_{h}}{x_{i}-x_{j}}+y_{h}
\label{eq:c-full}
\end{equation}
The Quadratic Equation
Substituting these results (Equations \ref{eq:a-full}, \ref{eq:b-full}, and \ref{eq:c-full}) into Equation \ref{eq:quadratic-basic} gives the general form of the quadratic equation:
\begin{multline}
g\left(x\right)=\left( \frac{M_{hi}-M_{hj}}{x_{{i}}-x_{{j}}} \right) x^{2} +
\left( \frac{M_{hj}\left(x_{h}+x_{i} \right)-M_{hi}\left(x_{h}+x_{j} \right)}{x_{i}-x_{j}} \right) x \\
+ \left( \frac{x_{h}\left(M_{hi}x_{j}-M_{hj}x_{i} \right)}{x_{i}-x_{j}}+y_{h} \right)
\label{eq:quadratic-three}
\end{multline}
The derivative will be
\begin{equation*}
g'\left(x\right) = 2ax + b
\end{equation*}
\begin{equation}
g'\left(x\right)=2\left( \frac{M_{hi}-M_{hj}}{x_{{i}}-x_{{j}}} \right) x +
\frac{M_{hj}\left(x_{h}+x_{i} \right)-M_{hi}\left(x_{h}+x_{j} \right)}{x_{i}-x_{j}}
\label{eq:quadratic-derivative}
\end{equation}
Vertex Form of the Quadratic Equation
In vertex form, a parabola is described as
\begin{equation}
y = a(x - h)^{2} + k
\label{eq:vertex-parabola}
\end{equation}
With the vertex \((h,k)\) and the focus \(\left( h,\ k + \frac{1}{4a} \right)\). The directrix is given by
\begin{equation}
y = k - \frac{1}{4a}
\label{eq:directrix}
\end{equation}
Expanding Equation \ref{eq:vertex-parabola} gives
\begin{equation*}
y = ax^{2} - 2ahx + {ah}^{2} + k
\end{equation*}
Comparing the coefficients to Equation \ref{eq:quadratic-basic}, we now have
\begin{equation*}
b = - 2ah~~~~~~c = ah^{2} + k
\end{equation*}
and
\begin{equation*}
h = - \frac{b}{2a}~~~~~~k = c - ah^{2}
\end{equation*}
To derive \(h\), we substitute for \(a\) and \(b\), using Equations \ref{eq:a-full} and \ref{eq:b-full}:
\begin{equation*}
h = - \frac{\frac{M_{hj}\left( x_{h} + x_{i} \right) - M_{hi}\left( x_{h} + x_{j} \right)}{x_{i} - x_{j}}}{2\left( \frac{M_{hi} - M_{hj}}{x_{i} - x_{j}} \right)}
\end{equation*}
\begin{equation}
h = \frac{M_{hi}\left( x_{h} + x_{j} \right) - M_{hj}\left( x_{h} + x_{i} \right)}{2\left( M_{hi} - M_{hj} \right)}
\label{eq:h}
\end{equation}
We perform the same substitutions to derive \(k\), though this requires many steps:
\begin{equation*}
k = \frac{\left( M_{hi}x_{j} - M_{hj}x_{i} \right)x_{h}}{x_{i} - x_{j}} + y_{h} - \frac{\left( \frac{M_{hj}\left( x_{h} + x_{i} \right) - M_{hi}\left( x_{h} + x_{j} \right)}{x_{i} - x_{j}} \right)^{2}}{4\left( \frac{M_{hi} - M_{hj}}{x_{i} - x_{j}} \right)}
\end{equation*}
\begin{equation*}
k = \frac{\left( M_{hi}x_{j} - M_{hj}x_{i} \right)x_{h}}{x_{i} - x_{j}} - \frac{\left( M_{hj}\left( x_{h} + x_{i} \right) - M_{hi}\left( x_{h} + x_{j} \right) \right)^{2}}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} + y_{h}
\end{equation*}
\begin{multline*}
k = \frac{4\left( M_{hi} - M_{hj} \right)\left( M_{hi}x_{j} - M_{hj}x_{i} \right)x_{h}}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} \\
-\frac{M_{hj}^{2}\left( x_{h} + x_{i} \right)^{2} - 2M_{hi}M_{hj}\left( x_{h} + x_{i} \right)\left( x_{h} + x_{j} \right) + M_{hi}^{2}\left( x_{h} + x_{j} \right)^{2}}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} + y_{h}
\end{multline*}
\begin{multline*}
k = \frac{4\left( M_{hi}^{2}x_{j} - M_{hi}M_{hj}x_{i} - M_{hi}M_{hj}x_{j} + M_{hj}^{2}x_{i} \right)x_{h}}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} \\
- \frac{M_{hj}^{2}\left( x_{h} + x_{i} \right)^{2} + 2M_{hi}M_{hj}\left( x_{h} + x_{i} \right)\left( x_{h} + x_{j} \right)}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} \\
- \frac{M_{hi}^{2}\left( x_{h} + x_{j} \right)^{2}}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} + y_{h}
\end{multline*}
\begin{multline*}
k = \frac{4M_{hi}^{2}x_{h}x_{j} - M_{hi}^{2}\left( x_{h} + x_{j} \right)^{2} + 2M_{hi}M_{hj}\left( x_{h} + x_{i} \right)\left( x_{h} + x_{j} \right)}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} \\
- \frac{4M_{hi}M_{hj}x_{h}\left( x_{i} + x_{j} \right) + 4M_{hj}^{2}x_{h}x_{i} - M_{hj}^{2}\left( x_{h} + x_{i} \right)^{2}}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} + y_{h}
\end{multline*}
\begin{multline*}
k = \frac{M_{hi}^{2}\left( 4x_{h}x_{j} - \left( x_{h}^{2} + 2x_{h}x_{j} + x_{j}^{2} \right) \right)}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} \\
+\frac{2M_{hi}M_{hj}\left( \left( x_{h} + x_{i} \right)\left( x_{h} + x_{j} \right) - 2x_{h}\left( x_{i} + x_{j} \right) \right)}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} \\
+\frac{M_{hj}^{2}\left( 4x_{h}x_{i} - \left( x_{h}^{2} + 2x_{h}x_{i} + x_{i}^{2} \right) \right)}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} + y_{h}
\end{multline*}
\begin{multline*}
k = \frac{M_{hi}^{2}\left( - x_{h}^{2} + 2x_{h}x_{j} - x_{j}^{2} \right)}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} \\
+ \frac{2M_{hi}M_{hj}\left( x_{h}^{2} + x_{h}x_{j} + x_{h}x_{i} + x_{i}x_{j} - 2x_{h}x_{i} - 2x_{h}x_{j} \right)}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} \\
+\frac {M_{hj}^{2}\left( 4 - x_{h}^{2} + 2x_{h}x_{i} - x_{i}^{2} \right)}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} + y_{h}
\end{multline*}
\begin{equation*}
k = \frac{- M_{hi}^{2}\left( x_{h} - x_{j} \right)^{2} + 2M_{hi}M_{hj}\left( x_{h}^{2} - x_{h}x_{j} - x_{h}x_{i} + x_{i}x_{j} \right) - M_{hj}^{2}\left( x_{h} - x_{i} \right)^{2}}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} + y_{h}
\end{equation*}
\begin{equation*}
k = \frac{- M_{hi}^{2}\left( x_{h} - x_{j} \right)^{2} + 2M_{hi}M_{hj}\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right) - M_{hj}^{2}\left( x_{h} - x_{i} \right)^{2}}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} + y_{h}
\end{equation*}
\begin{equation}
k = - \frac{\left( M_{hj}\left( x_{h} - x_{i} \right) - M_{hi}\left( x_{h} - x_{j} \right) \right)^{2}}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} + y_{h}
\label{eq:k}
\end{equation}
Substituting the values of \(a\), \(h\), and \(k\) into Equation \ref{eq:a-full} will give the vertex form of the parabola:
\begin{multline}
g(x) = \left( \frac{M_{hi} - M_{hj}}{x_{i} - x_{j}} \right)\left( x - \frac{M_{hi}\left( x_{h} + x_{j} \right) - M_{hj}\left( x_{h} + x_{i} \right)}{2\left( M_{hi} - M_{hj} \right)} \right)^{2} \\
- \frac{\left( M_{hj}\left( x_{h} - x_{i} \right) - M_{hi}\left( x_{h} - x_{j} \right) \right)^{2}}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} + y_{h}
\label{eq:vertex-full}
\end{multline}
With the vertex \(\left(h,k\right)\) given by
\begin{equation*}
\left( \frac{M_{hi}\left( x_{h} + x_{j} \right) - M_{hj}\left( x_{h} + x_{i} \right)}{2\left( M_{hi} - M_{hj} \right)}, - \frac{\left( M_{hj}\left( x_{h} - x_{i} \right) - M_{hi}\left( x_{h} - x_{j} \right) \right)^{2}}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} + y_{h} \right)
\end{equation*}
And the focus \(\left(h,k + \frac{1}{4a}\right)\) given by
\begin{multline*}
\left( \frac{M_{hi}\left( x_{h} + x_{j} \right) - M_{hj}\left( x_{h} + x_{i} \right)}{2\left( M_{hi} - M_{hj} \right)},\right. \\
\left.\frac{{\left( x_{i} - x_{j} \right)^{2} - \left( M_{hj}\left( x_{h} - x_{i} \right) - M_{hi}\left( x_{h} - x_{j} \right) \right)}^{2}}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} + y_{h} \right)
\end{multline*}
The directrix will be given by
\begin{equation*}
y = k - \frac{1}{4a}
\end{equation*}
\begin{equation*}
y = - \frac{\left( M_{hj}\left( x_{h} - x_{i} \right) - M_{hi}\left( x_{h} - x_{j} \right) \right)^{2}}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} + y_{h} - \frac{1}{4}\left( \frac{x_{i} - x_{j}}{M_{hi} - M_{hj}} \right)
\end{equation*}
\begin{equation}
y = - \frac{{\left( x_{i} - x_{j} \right)^{2} + \left( M_{hj}\left( x_{h} - x_{i} \right) - M_{hi}\left( x_{h} - x_{j} \right) \right)}^{2}}{4\left( M_{hi} - M_{hj} \right)\left( x_{i} - x_{j} \right)} + y_{h}
\label{eq:directrix-expanded}
\end{equation}
Fully Expanding the Quadratic Equation
We may expand Equation \ref{eq:quadratic-three} into a form that can be computed directly from the three points \(\left( x_{h},y_{h} \right)\), \(\left( x_{i},y_{i} \right),\), and \(\left( x_{j},y_{j} \right)\); i.e., without using \(M_{hi}\) and \(M_{hj}\).
We must first simplify the coefficients. Beginning with \(a\) (Equation \ref{eq:a-full}), we substitute for \(M_{hi}\) and \(M_{hj}\) (Equations \ref{eq:m_hi} and \ref{eq:m_hj}):
\begin{equation*}
a = \left( \frac{1}{x_{i} - x_{j}} \right)\left( \frac{y_{h} - y_{i}}{x_{h} - x_{i}} - \frac{y_{h} - y_{j}}{x_{h} - x_{j}} \right)
\end{equation*}
\begin{equation*}
a = \frac{\left( y_{h} - y_{i} \right)\left( x_{h} - x_{j} \right) - \left( y_{h} - y_{j} \right)\left( x_{h} - x_{i} \right)}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}
\end{equation*}
\begin{equation*}
a = \frac{\left( x_{h} - x_{j} \right)y_{h} - \left( x_{h} - x_{j} \right)y_{i} - \left( x_{h} - x_{i} \right)y_{h} + \left( x_{h} - x_{i} \right)y_{j}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}
\end{equation*}
\begin{equation}
a = \frac{\left( x_{i} - x_{j} \right)y_{h} - \left( x_{h} - x_{j} \right)y_{i} + \left( x_{h} - x_{i} \right)y_{j}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}
\label{eq:a-expanded}
\end{equation}
For \(b\) (Equation \ref{eq:b-full}), again we substitute for \(M_{hi}\) and \(M_{hj}\):
\begin{equation*}
b = \left( \frac{1}{x_{i} - x_{j}} \right)\left( \frac{y_{h} - y_{j}}{x_{h} - x_{j}}\left( x_{h} + x_{i} \right) - \frac{y_{h} - y_{i}}{x_{h} - x_{i}}\left( x_{h} + x_{j} \right) \right)
\end{equation*}
\begin{equation*}
b = \frac{\left( x_{h} + x_{i} \right)\left( x_{h} - x_{i} \right)\left( y_{h} - y_{j} \right)}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)} - \frac{\left( x_{h} + x_{j} \right)\left( x_{h} - x_{j} \right)\left( y_{h} - y_{i} \right)}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}
\end{equation*}
\begin{equation*}
b = \frac{\left( x_{h}^{2} - x_{i}^{2} \right)\left( y_{h} - y_{j} \right) - \left( x_{h}^{2} - x_{j}^{2} \right)\left( y_{h} - y_{i} \right)}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}
\end{equation*}
\begin{equation*}
b = \frac{\left( x_{h}^{2} - x_{i}^{2} \right)y_{h} - \left( x_{h}^{2} - x_{i}^{2} \right)y_{j} - \left( x_{h}^{2} - x_{j}^{2} \right)y_{h} + \left( x_{h}^{2} - x_{j}^{2} \right)y_{i}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}
\end{equation*}
\begin{equation}
b = \frac{- \left( x_{i}^{2} - x_{j}^{2} \right)y_{h} + \left( x_{h}^{2} - x_{j}^{2} \right)y_{i} - \left( x_{h}^{2} - x_{i}^{2} \right)y_{j}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}
\label{eq:b-expanded}
\end{equation}
For \(c\) (Equation \ref{eq:c-full}), again we substitute for \(M_{hi}\) and \(M_{hj}\), though this requires many steps:
\begin{equation*}
c = \frac{\left( \left( \frac{y_{h} - y_{i}}{x_{h} - x_{i}} \right)x_{j} - \left( \frac{y_{h} - y_{j}}{x_{h} - x_{j}} \right)x_{i} \right)x_{h}}{x_{i} - x_{j}} + y_{h}
\end{equation*}
\begin{equation*}
c = \frac{\left( y_{h} - y_{i} \right)x_{h}x_{j}}{\left( x_{h} - x_{i} \right)\left( x_{i} - x_{j} \right)} - \frac{\left( y_{h} - y_{j} \right){x_{h}x}_{i}}{\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)} + y_{h}
\end{equation*}
\begin{multline*}
c = \frac{\left( y_{h} - y_{i} \right)\left( x_{h} - x_{j} \right)x_{h}x_{j}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)} - \frac{\left( y_{h} - y_{j} \right)\left( x_{h} - x_{i} \right)x_{h}x_{i}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)} \\
+ \frac{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)y_{h}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}
\end{multline*}
\begin{multline*}
c = \frac{\left( x_{h} - x_{j} \right)\left( x_{h}x_{j} \right)y_{h} - \left( x_{h} - x_{j} \right)\left( x_{h}x_{j} \right)y_{i}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)} \\
- \frac{\left( x_{h} - x_{j} \right)\left( x_{h}x_{j} \right)y_{h} - \left( x_{h} - x_{j} \right)\left( x_{h}x_{j} \right)y_{j}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)} \\
+ \frac{\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)y_{h}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}
\end{multline*}
\begin{multline*}
c = \frac{\left( \left( x_{h} - x_{j} \right)\left( x_{h}x_{j} \right) - \left( x_{h} - x_{i} \right)\left( x_{h}x_{i} \right) + \left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right) \right)y_{h}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)} \\
+ \frac{\left( x_{h} - x_{i} \right)\left( x_{h}x_{i} \right)y_{j} - \left( x_{h} - x_{j} \right)\left( x_{h}x_{j} \right)y_{i}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}
\end{multline*}
\begin{multline*}
c = \frac{\left( \left( x_{h} - x_{j} \right)\left( x_{h}x_{j} \right) - \left( x_{h} - x_{i} \right)\left( x_{h}x_{i} \right) + \left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right) \right)y_{h}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)} \\
+ \frac{\left( x_{h} - x_{i} \right)\left( x_{h}x_{i} \right)y_{j} - \left( x_{h} - x_{j} \right)\left( x_{h}x_{j} \right)y_{i}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}
\end{multline*}
\begin{multline*}
c = \frac{\left( x_{h}^{2}x_{j} - x_{h}x_{j}^{2} - x_{h}^{2}x_{i} + x_{h}x_{i}^{2} + \left( x_{h}^{2} - x_{h}x_{j} - x_{h}x_{i} + x_{i}x_{j} \right)\left( x_{i} - x_{j} \right) \right)y_{h}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)} \\
+ \frac{\left( x_{h} - x_{i} \right)\left( x_{h}x_{i} \right)y_{j} - \left( x_{h} - x_{j} \right)\left( x_{h}x_{j} \right)y_{i}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}
\end{multline*}
\begin{multline*}
c = \left(\frac{ x_{h}^{2}x_{j} - x_{h}x_{j}^{2} - x_{h}^{2}x_{i} + x_{h}x_{i}^{2} + x_{h}^{2}x_{i} - x_{h}x_{i}x_{j} - x_{h}x_{i}^{2}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}\right. \\
+\left.\frac{x_{i}^{2}x_{j} - x_{h}^{2}x_{j} + x_{h}x_{j}^{2} + x_{h}x_{i}x_{j} - x_{i}x_{j}^{2}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}\right)y_{h} \\
+ \frac{\left( x_{h} - x_{i} \right)\left( x_{h}x_{i} \right)y_{j} - \left( x_{h} - x_{j} \right)\left( x_{h}x_{j} \right)y_{i}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}
\end{multline*}
\begin{equation*}
c = \frac{\left( x_{i}^{2}x_{j} - x_{i}x_{j}^{2} \right)y_{h}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)} + \frac{\left( x_{h} - x_{i} \right)\left( x_{h}x_{i} \right)y_{j} - \left( x_{h} - x_{j} \right)\left( x_{h}x_{j} \right)y_{i}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}
\end{equation*}
\begin{equation*}
c = \frac{\left( x_{i} - x_{j} \right)\left( x_{i}x_{j} \right)y_{h} - \left( x_{h} - x_{j} \right)\left( x_{h}x_{j} \right)y_{i} + \left( x_{h} - x_{i} \right)\left( x_{h}x_{i} \right)y_{j}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}
\end{equation*}
\begin{equation}
c = \frac{\left( x_{i} - x_{j} \right)\left( x_{i}x_{j} \right)y_{h} - \left( x_{h} - x_{j} \right)\left( x_{h}x_{j} \right)y_{i} + \left( x_{h} - x_{i} \right)\left( x_{h}x_{i} \right)y_{j}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}
\label{eq:c-expanded}
\end{equation}
Substituting the full forms of \(a\), \(b\), and \(c\) (Equations \ref{eq:a-expanded}, \ref{eq:b-expanded}, and \ref{eq:c-expanded}) into Equation \ref{eq:quadratic-three} will give
\begin{multline}
g(x) = \left( \frac{\left( x_{i} - x_{j} \right)y_{h} - \left( x_{h} - x_{j} \right)y_{i} + \left( x_{h} - x_{i} \right)y_{j}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)} \right) x^{2} \\
+ \left( \frac{- \left( x_{i}^{2} - x_{j}^{2} \right)y_{h} + \left( x_{h}^{2} - x_{j}^{2} \right)y_{i} - \left( x_{h}^{2} - x_{i}^{2} \right)y_{j}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)} \right) x \\
+ \frac{\left( x_{i} - x_{j} \right)\left( x_{i}x_{j} \right)y_{h} - \left( x_{h} - x_{j} \right)\left( x_{h}x_{j} \right)y_{i} + \left( x_{h} - x_{i} \right)\left( x_{h}x_{i} \right)y_{j}}{\left( x_{h} - x_{i} \right)\left( x_{h} - x_{j} \right)\left( x_{i} - x_{j} \right)}
\label{eq:parabola-expanded}
\end{multline}
Fitting a Circle to Three Points
Summary
We can fit the three points
\(\left( x_{h},y_{h} \right),\left( x_{i},y_{i} \right),\left( x_{j},y_{j} \right)\)
to the circle
\begin{equation}
\left( x - x_{c} \right)^{2} + \left( y - y_{c} \right)^{2} = r^{2}
\label{eq:circle}
\end{equation}
with center \(\left(x_{c},y_{c}\right)\) and radius \(r\), where
\begin{equation*}
x_{c} = \frac{1}{2}\left( \frac{M_{12}}{M_{11}} \right)
\end{equation*}
\begin{equation*}
y_{c} = - \frac{1}{2}\left( \frac{M_{13}}{M_{11}} \right)
\end{equation*}
\begin{equation*}
r^{2} = x_{c}^{2} + y_{c}^{2} + \frac{M_{14}}{M_{11}}
\end{equation*}
or
\begin{equation*}
r^{2} = \frac{M_{12}^{2} + M_{13}^{2} + 4M_{11}M_{14}}{4M_{11}^{2}}
\end{equation*}
Where \(M_{11}\), \(M_{12}\), \(M_{13}\), and \(M_{14}\) are given by Equations \ref{eq:M11}, \ref{eq:M12}, \ref{eq:M13}, and \ref{eq:M14}).
Derivation
We can represent the circle in matrix form as
\begin{equation*}
\left| \begin{matrix}
x^{2} + y^{2} & x & y & 1 \\
x_{h}^{2} + y_{h}^{2} & x_{h} & y_{h} & 1 \\
x_{i}^{2} + y_{i}^{2} & x_{i} & y_{i} & 1 \\
x_{j}^{2} + y_{j}^{2} & x_{j} & y_{j} & 1 \\
\end{matrix} \right| = 0
\end{equation*}
Let \(M_{pq}\) be the determinant of the minor the element \((p,q)\). Then, the overall determinant of the circle can be written as
\begin{equation*}
M_{11}\left( x^{2} + y^{2} \right) - M_{12}x + M_{13}y - M_{14} = 0
\end{equation*}
\begin{equation}
x^{2} + y^{2} - \frac{M_{12}}{M_{11}}x + \frac{M_{14}}{M_{11}}y - \frac{M_{14}}{M_{11}} = 0
\label{eq:determinant}
\end{equation}
Note the equation of the circle (Equation \ref{eq:circle}) can be expanded to give
\begin{equation*}
\left( x - x_{c} \right)^{2} + \left( y - y_{c} \right)^{2} - r^{2} = 0
\end{equation*}
\begin{equation*}
x^{2} - 2x_{c}x + x_{c}^{2} + y^{2} - 2y_{c}y + y_{c}^{2} - r^{2} = 0
\end{equation*}
\begin{equation*}
x^{2} + y^{2} - 2x_{c}x - 2y_{c}y + x_{c}^{2} + y_{c}^{2} - r^{2} = 0
\end{equation*}
By comparing the coefficients with the determinant of the circle (Equations \ref{eq:circle} and \ref{eq:determinant}), we see that
\begin{equation*}
- \frac{M_{12}}{M_{11}} = - 2x_{c}
\end{equation*}
\begin{equation*}
\frac{M_{14}}{M_{11}} = - 2y_{c}
\end{equation*}
\begin{equation*}
x_{c}^{2} + y_{c}^{2} - r^{2} = - \frac{M_{14}}{M_{11}}
\end{equation*}
We then fill in the determinants using Equations \ref{eq:M11}, \ref{eq:M12}, \ref{eq:M13}, and \ref{eq:M14} below.
Determinant \(M_{11}\)
\begin{equation*}
M_{11} = \left| \begin{matrix}
x_{h} & y_{h} & 1 \\
x_{i} & y_{i} & 1 \\
x_{j} & y_{j} & 1 \\
\end{matrix} \right|
\end{equation*}
\begin{equation*}
M_{11} = x_{h}y_{i} + x_{j}y_{h} + x_{i}y_{j} - x_{j}y_{i} - x_{i}y_{h} - x_{h}y_{j}
\end{equation*}
\begin{equation*}
M_{11} = x_{h}y_{i} - x_{h}y_{j} + x_{i}y_{j} - x_{i}y_{h} + x_{j}y_{h} - x_{j}y_{i}
\end{equation*}
\begin{equation*}
M_{11} = x_{h}\left( y_{i} - y_{j} \right) - x_{i}\left( y_{h} - y_{j} \right) + x_{j}\left( y_{h} - y_{i} \right)
\end{equation*}
\begin{equation}
M_{11} = - \left( x_{i} - x_{j} \right)y_{h} + \left( x_{h} - x_{j} \right)y_{i} - \left( x_{h} - x_{i} \right)y_{j}
\label{eq:M11}
\end{equation}
Determinant \(M_{12}\)
\begin{equation*}
M_{12} = \left| \begin{matrix}
x_{h}^{2} + y_{h}^{2} & y_{h} & 1 \\
x_{i}^{2} + y_{i}^{2} & y_{i} & 1 \\
x_{j}^{2} + y_{j}^{2} & y_{j} & 1 \\
\end{matrix} \right|
\end{equation*}
\begin{multline*}
M_{12} = \left( x_{h}^{2} + y_{h}^{2} \right)y_{i} + \left( x_{j}^{2} + y_{j}^{2} \right)y_{h} + \left( x_{i}^{2} + y_{i}^{2} \right)y_{j} \\ - \left( x_{j}^{2} + y_{j}^{2} \right)y_{i}
- \left( x_{i}^{2} + y_{i}^{2} \right)y_{h} - \left( x_{h}^{2} + y_{h}^{2} \right)y_{j}
\end{multline*}
\begin{multline*}
M_{12} = \left( x_{j}^{2} + y_{j}^{2} \right)y_{h} - \left( x_{i}^{2} + y_{i}^{2} \right)y_{h} + \left( x_{h}^{2} + y_{h}^{2} \right)y_{i} \\ - \left( x_{j}^{2} + y_{j}^{2} \right)y_{i} + \left( x_{i}^{2} + y_{i}^{2} \right)y_{j} - \left( x_{h}^{2} + y_{h}^{2} \right)y_{j}
\end{multline*}
\begin{equation*}
M_{12} = \left( x_{j}^{2} + y_{j}^{2} - x_{i}^{2} - y_{i}^{2} \right)y_{h} + \left( x_{h}^{2} + y_{h}^{2} - x_{j}^{2} - y_{j}^{2} \right)y_{i} + \left( x_{i}^{2} + y_{i}^{2} - x_{h}^{2} - y_{h}^{2} \right)y_{j}
\end{equation*}
\begin{equation*}
M_{12} = \left( x_{j}^{2} - x_{i}^{2} + y_{j}^{2} - y_{i}^{2} \right)y_{h} + \left( x_{h}^{2} - x_{j}^{2} + y_{h}^{2} - y_{j}^{2} \right)y_{i} + \left( x_{i}^{2} - x_{h}^{2} + y_{i}^{2} - y_{h}^{2} \right)y_{j}
\end{equation*}
\begin{equation*}
M_{12} = - \left( x_{i}^{2} - x_{j}^{2} + y_{i}^{2} - y_{j}^{2} \right)y_{h} + \left( x_{h}^{2} - x_{j}^{2} + y_{h}^{2} - y_{j}^{2} \right)y_{i} - \left( x_{h}^{2} - x_{i}^{2} + y_{h}^{2} - y_{i}^{2} \right)y_{j}
\end{equation*}
\begin{multline*}
M_{12} = - \left( \left( x_{i} + x_{j} \right)\left( x_{i} - x_{j} \right) + \left( y_{i} + y_{j} \right)\left( y_{i} - y_{j} \right) \right)y_{h} \\
+ \left( \left( x_{h} + x_{j} \right)\left( x_{h} - x_{j} \right) + \left( y_{h} + y_{j} \right)\left( y_{h} - y_{j} \right) \right)y_{i} \\
- \left( \left( x_{h} + x_{i} \right)\left( x_{h} - x_{i} \right) + \left( y_{h} + y_{i} \right)\left( y_{h} - y_{i} \right) \right)y_{j}
\end{multline*}
\begin{multline}
M_{12} = - \left( x_{i}^{2} - x_{j}^{2} + y_{i}^{2} - \ y_{j}^{2} \right)y_{h} + \left( x_{h}^{2} - x_{j}^{2} + \ y_{h}^{2} - \ y_{j}^{2} \right)y_{i} \\
- \left( x_{h}^{2} - x_{i}^{2} + \ y_{h}^{2} - \ y_{i}^{2} \right)y_{j}
\label{eq:M12}
\end{multline}
Determinant \(M_{13}\)
\begin{equation*}
M_{13} = \left| \begin{matrix}
x_{h}^{2} + y_{h}^{2} & x_{h} & 1 \\
x_{i}^{2} + y_{i}^{2} & x_{i} & 1 \\
x_{j}^{2} + y_{j}^{2} & x_{j} & 1 \\
\end{matrix} \right|
\end{equation*}
Following the same steps as with \(M_{12}\) and substituting \emph{x} for \emph{y} in the middle column gives
\begin{multline}
M_{13} = - \left( x_{i}^{2} - x_{j}^{2} + y_{i}^{2} - \ y_{j}^{2} \right)x_{h} + \left( x_{h}^{2} - x_{j}^{2} + \ y_{h}^{2} - \ y_{j}^{2} \right)x_{i} \\
- \left( x_{h}^{2} - x_{i}^{2} + \ y_{h}^{2} - \ y_{i}^{2} \right)x_{j}
\label{eq:M13}
\end{multline}
Determinant \(M_{14}\)
\begin{equation*}
M_{14} = \left| \begin{matrix}
x_{h}^{2} + y_{h}^{2} & x_{h} & y_{h} \\
x_{i}^{2} + y_{i}^{2} & x_{i} & y_{i} \\
x_{j}^{2} + y_{j}^{2} & x_{j} & y_{j} \\
\end{matrix} \right|
\end{equation*}
\begin{multline*}
M_{14} = \left( x_{h}^{2} + y_{h}^{2} \right)x_{i}y_{j} + \left( x_{j}^{2} + y_{j}^{2} \right)x_{h}y_{i} + \left( x_{i}^{2} + y_{i}^{2} \right)x_{j}y_{h} \\
- \left( x_{j}^{2} + y_{j}^{2} \right)x_{i}y_{h} - \left( x_{i}^{2} + y_{i}^{2} \right)x_{h}y_{j} - \left( x_{h}^{2} + y_{h}^{2} \right)x_{j}y_{i}
\end{multline*}
\begin{multline*}
M_{14} = \left( x_{h}^{2} + y_{h}^{2} \right)x_{i}y_{j} - \left( x_{h}^{2} + y_{h}^{2} \right)x_{j}y_{i} + \left( x_{i}^{2} + y_{i}^{2} \right)x_{j}y_{h} \\
- \left( x_{i}^{2} + y_{i}^{2} \right)x_{h}y_{j} + \left( x_{j}^{2} + y_{j}^{2} \right)x_{h}y_{i} - \left( x_{j}^{2} + y_{j}^{2} \right)x_{i}y_{h}
\end{multline*}
\begin{multline}
M_{14} = \left( x_{h}^{2} + y_{h}^{2} \right)\left( x_{i}y_{j} - x_{j}y_{i} \right) - \left( x_{i}^{2} + y_{i}^{2} \right)\left( x_{h}y_{j} - x_{j}y_{h} \right) \\
+ \left( x_{j}^{2} + y_{j}^{2} \right)\left( x_{h}y_{i} - x_{i}y_{h} \right)
\label{eq:M14}
\end{multline}
Fitting Skewed Parabolas to Four Points
Overview in Standard Form
The four points \(\left( x_{h},y_{h} \right)\), \(\left( x_{i},y_{i} \right),\), \(\left( x_{j},y_{j} \right)\), and \(\left( x_{k},y_{k} \right)\) can be fitted to two skewed parabolas using the following formula
\begin{equation}
Ax^{2}+Bxy+\frac{B^{2}}{4A}y^{2}+Dx+Ey+F=0
\label{eq:skew-parabola}
\end{equation}
We see that Equation \ref{eq:quadratic-basic} is a special case, where \(a=A\), \(b=D\), \(c=F\), and \(B=E=0\)). (Note that it may not always be possible to draw one or both of the parabolas given by Equation \ref{eq:skew-parabola} in the real plane.) After a lengthy and tedious derivation, we obtain the following results.
Coefficients
First, we let
\begin{equation*}
A=\frac{B^{2}}{4}
\end{equation*}
And
\begin{equation*}
B=\frac{2U \pm 2\sqrt{U^{2}+TV}}{T}
\end{equation*}
Where
\begin{multline*}
T=\left(x_{1}-x_{3}\right)\left(x_{1}-x_{4}\right)\left(x_{3}-x_{4}\right)\left(y_{1}-y_{2}\right)\\
-\left(x_{1}-x_{2}\right)\left(x_{1}-x_{4}\right)\left(x_{2}-x_{4}\right)\left(y_{1}-y_{3}\right)\\
+\left(x_{1}-x_{2}\right)\left(x_{1}-x_{3}\right)\left(x_{2}-x_{3}\right)\left(y_{1}-y_{4}\right)
\end{multline*}
\begin{multline*}
U=\left(\left(x_{3}y_{3}-x_{4}y_{4}\right)x_{1}-\left(x_{1}y_{1}-x_{4}y_{4}\right)x_{3}+\left(x_{1}y_{1}-x_{3}y_{3}\right)x_{4}\right)\left(y_{1}-y_{2}\right)\\
-\left(\left(x_{2}y_{2}-x_{4}y_{4}\right)x_{1}-\left(x_{1}y_{1}-x_{4}y_{4}\right)x_{2}+\left(x_{1}y_{1}-x_{2}y_{2}\right)x_{4}\right)\left(y_{1}-y_{3}\right)\\
+\left(\left(x_{2}y_{2}-x_{3}y_{3}\right)x_{1}-\left(x_{1}y_{1}-x_{3}y_{3}\right)x_{2}+\left(x_{1}y_{1}-x_{2}y_{2}\right)x_{3}\right)\left(y_{1}-y_{4}\right)
\end{multline*}
\begin{multline*}
V=\left(x_{1}-x_{2}\right)\left(y_{1}-y_{3}\right)\left(y_{1}-y_{4}\right)\left(y_{3}-y_{4}\right)\\
-\left(x_{1}-x_{3}\right)\left(y_{1}-y_{2}\right)\left(y_{1}-y_{4}\right)\left(y_{2}-y_{4}\right)\\
+\left(x_{1}-x_{4}\right)\left(y_{1}-y_{2}\right)\left(y_{1}-y_{3}\right)\left(y_{2}-y_{3}\right)
\end{multline*}
We then define the other coefficients in terms of \(B\):
\begin{multline*}
D=-\frac{1}{{4\left(\left(x_{2}-x_{3}\right)y_{1}-\left(x_{1}-x_{3}\right)y_{2}+\left(x_{1}-x_{2}\right)y_{3}\right)}}\times\\
\left[B^{2}\left(\left(x_{2}+x_{3}\right)\left(x_{2}-x_{3}\right)y_{1}-\left(x_{1}+x_{3}\right)\left(x_{1}-x_{3}\right)y_{2}+\left(x_{1}+x_{2}\right)\left(x_{1}-x_{2}\right)y_{3}\right)\right.\\
-
4B\left(\left(x_{1}-x_{2}\right)y_{1}y_{2}-\left(x_{1}-x_{3}\right)y_{1}y_{3}+\left(x_{2}-x_{3}\right)y_{2}y_{3}\right)\\
-
\left.4\left(y_{1}-y_{2}\right)\left(y_{1}-y_{3}\right)\left(y_{2}-y_{3}\right)\right]
\end{multline*}
\begin{multline*}
E=
-\frac{1}{{4\left(\left(x_{2}-x_{3}\right)y_{1}-\left(x_{1}-x_{3}\right)y_{2}+\left(x_{1}-x_{2}\right)y_{3}\right)}}\times\\
\left[B^{2}\left(x_{1}-x_{2}\right)\left(x_{1}-x_{3}\right)\left(x_{2}-x_{3}\right)\right.\\
+
4B\left(\left(x_{2}-x_{3}\right)x_{1}y_{1}-\left(x_{1}-x_{3}\right)x_{2}y_{2}+\left(x_{1}-x_{2}\right)x_{3}y_{3}\right)\\
+
\left.4\left(\left(x_{2}-x_{3}\right)y_{1}^{2}-\left(x_{1}-x_{3}\right)y_{2}^{2}+\left(x_{1}-x_{2}\right)y_{3}^{2}\right)\right]
\end{multline*}
\begin{multline*}
F=\frac{1}{4\left(\left(x_{2}-x_{3}\right)y_{1}-\left(x_{1}-x_{3}\right)y_{2}+\left(x_{1}-x_{2}\right)y_{3}\right)}\times\\
\left[B^{2}\left(\left(x_{2}-x_{3}\right)x_{2}x_{3}y_{1}-\left(x_{1}-x_{3}\right)x_{1}x_{3}y_{2}+\left(x_{1}-x_{2}\right)x_{1}x_{2}y_{3}\right)\right.\\
-4B\left(\left(x_{2}-x_{3}\right)x_{1}y_{2}y_{3}-\left(x_{1}-x_{3}\right)x_{2}y_{1}y_{3}+\left(x_{1}-x_{2}\right)x_{3}y_{1}y_{2}\right)\\
-\left.4\left(\left(y_{2}-y_{3}\right)x_{1}y_{2}y_{3}-\left(y_{1}-y_{3}\right)x_{2}y_{1}y_{3}+\left(y_{1}-y_{2}\right)x_{3}y_{1}y_{2}\right)\right]
\end{multline*}
Focus & Vertex
The coordinates of the focus \(\left(x_{f},y_{f}\right)\) will be given by
\begin{equation*}
x_{f}=\frac{\left(D^{2}+4F\right)\left(B^{2}+4\right)-\left(DB+2E\right)^{2}}{2\left(B^{2}+4\right)\left(BE-2D\right)}
\end{equation*}
\begin{equation*}
y_{f}=\frac{\left(E^{2}+B^{2}F\right)\left(B^{2}+4\right)-\left(DB+2E\right)^{2}}{\left(B^{2}+4\right)\left(2\left(DB+2E\right)-E\left(B^{2}+4\right)\right)}
\end{equation*}
The coordinates of the vertex \(\left(x_{v},y_{v}\right)\) can be expressed in terms of those of the focus:
\begin{equation*}
x_{v}=x_{f}-2\frac{BE-2D}{\left(B^{2}+4\right)^{2}}
\end{equation*}
\begin{equation*}
y_{v}=y_{f}+\frac{B\left(BE-2D\right)}{\left(B^{2}+4\right)^{2}}
\end{equation*}
The distance from the focus to the vertex (and from the vertex to the point of intersection) is given by
\begin{equation}
\frac{\left(BE-2D\right)\sqrt{B^{2}+4}}{\left(B^{2}+4\right)^{2}}
\end{equation}
which is one quarter the length of the latus rectum.
Note that we may also find the coordinates of the point of intersection between the axis and the directrix \(\left(x_{p},y_{p}\right)\) by taking these distances again,
\begin{equation*}
x_{p}=x_{f}-4\frac{BE-2D}{\left(B^{2}+4\right)^{2}}
\end{equation*}
\begin{equation*}
y_{p}=y_{f}+2\frac{B\left(BE-2D\right)}{\left(B^{2}+4\right)^{2}}
\end{equation*}
Directrix & Axis
The slope of the directrix will be given by \(M=\frac{2}{B}\) and its intercept will be
\begin{equation*}
N=-\frac{\left(B^{2}+4\right)\left(4F-D^{2}\right)+\left(BD-2E\right)\left(BD+2E\right)}{4B\left(BE-2D\right)}
\end{equation*}
The axis will be given by
\begin{equation*}
y=-\frac{Bx}{2}-\frac{DB+2E}{4+B^{2}}
\end{equation*}
Slope-Intercept of the Directrix Form of the Parabola
We may also write the equation in terms of the focus and the slope and intercept of the directrix, instead of using coefficients (as in Equation \ref{eq:skew-parabola}:
\begin{equation*}
\left(x-x_{f}\right)^{2}+\left(y-y_{f}\right)^{2}=\frac{\left(Mx-y+N\right)^{2}}{M^{2}+1}
\label{eq:slope-intercept-parabola}
\end{equation*}
Substituting for \(M\) gives
\begin{equation*}
\left(x-x_{f}\right)^{2}+\left(y-y_{f}\right)^{2}=\frac{\left(2x-B\left(y+N\right)\right)^{2}}{B^{2}+4}
\end{equation*}