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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. (From a desire for brevity as well as 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, the curves 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 \(y\) 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 \(x\) for \(y\) 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 of the monic form
\begin{equation}
x^{2}+Bxy+\frac{B^{2}}{4A}y^{2}+Dx+Ey+F=0
\label{eq:skew-parabola}
\end{equation}
This is identical to the form \(ax^{2}+bxy+\frac{b^{2}}{4}y^{2}+dx+ey+f=0\) divided through by \(a\). We can also 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 (shown here in rough form), we obtain the following results.
Coefficients: Standard Form
First, we let
\begin{equation*}
B=\frac{-U+\sqrt{U^{2}-4TV}}{2T}
\end{equation*}
Each one of the roots will provide the coefficients for one of the two possible parabolas. Here,
\begin{multline*}
T=\frac{\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)y_{k}^{2}}{4\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}\\
+\frac{\left(y_{h}-y_{i}\right)\left(y_{h}-y_{j}\right)\left(y_{i}-y_{j}\right)x_{k}+\left(\left(y_{i}^{2}-y_{j}^{2}\right)x_{h}-\left(y_{h}^{2}-y_{j}^{2}\right)x_{i}+\left(y_{h}^{2}-y_{i}^{2}\right)x_{j}\right)y_{k}}{4\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}\\
+\frac{-\left(\left(y_{i}-y_{j}\right)x_{h}y_{i}y_{j}-\left(y_{h}-y_{j}\right)x_{i}y_{h}y_{j}+\left(y_{h}-y_{i}\right)x_{j}y_{h}y_{i}\right)}{4\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}
\end{multline*}
\begin{multline*}
U=\frac{\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)x_{k}y_{k}}{\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}\\
+\frac{\left(\left(x_{h}-x_{i}\right)y_{h}y_{i}-\left(x_{h}-x_{j}\right)y_{h}y_{j}\left(x_{i}-x_{j}\right)y_{i}y_{j}\right)x_{k}}{\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}\\
+\frac{-\left(x_{h}x_{i}\left(y_{h}-y_{i}\right)-x_{h}x_{j}\left(y_{h}-y_{j}\right)+x_{i}x_{j}\left(y_{i}-y_{j}\right)\right)y_{k}}{\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}\\
+\frac{\left(\left(y_{h}-y_{i}\right)x_{h}x_{i}y_{j}-\left(y_{h}-y_{j}\right)x_{h}x_{j}y_{i}+\left(y_{i}-y_{j}\right)x_{i}x_{j}y_{h}\right)}{\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}
\end{multline*}
\begin{multline*}
V=\frac{\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)x_{k}^{2}}{\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}\\
+\frac{-\left(\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}\right)x_{k}}{\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}\\
+\frac{-\left(x_{h}-x_{i}\right)\left(x_{h}-x_{j}\right)\left(x_{i}-x_{j}\right)y_{k}}{\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}\\
+\frac{\left(\left(x_{h}-x_{i}\right)x_{h}x_{i}y_{j}-\left(x_{h}-x_{j}\right)x_{h}x_{j}y_{i}+\left(x_{i}-x_{j}\right)x_{i}x_{j}y_{h}\right)}{\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}
\end{multline*}
We then define the other coefficients in terms of \(B\):
\begin{multline*}
D=\frac{B^{2}\left(y_{h}-y_{i}\right)\left(y_{h}-y_{j}\right)\left(y_{i}-y_{j}\right)}{4\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}\\
+\frac{4B\left(\left(x_{h}-x_{i}\right)y_{h}y_{i}-\left(x_{h}-x_{j}\right)y_{h}y_{j}+\left(x_{i}-x_{j}\right)y_{i}y_{j}\right)}{4\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}\\
+\frac{-4\left(\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}\right)}{4\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}
\end{multline*}
\begin{multline*}
E=\frac{B^{2}\left(\left(y_{i}^{2}-y_{j}^{2}\right)x_{h}-\left(y_{h}^{2}-y_{j}^{2}\right)x_{i}+\left(y_{h}^{2}-y_{i}^{2}\right)x_{j}\right)}{4\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}\\
+\frac{-4B\left(x_{h}x_{i}\left(y_{h}-y_{i}\right)-x_{h}x_{j}\left(y_{h}-y_{j}\right)+x_{i}x_{j}\left(y_{i}-y_{j}\right)\right)}{4\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}\\
+\frac{-4\left(x_{h}-x_{i}\right)\left(x_{h}-x_{j}\right)\left(x_{i}-x_{j}\right)}{4\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}
\end{multline*}
\begin{multline*}
F=\frac{-B^{2}\left(\left(y_{i}-y_{j}\right)x_{h}y_{i}y_{j}-\left(y_{h}-y_{j}\right)x_{i}y_{h}y_{j}+\left(y_{h}-y_{i}\right)x_{j}y_{h}y_{i}\right)}{4\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}\\
+\frac{4B\left(\left(y_{h}-y_{i}\right)x_{h}x_{i}y_{j}-\left(y_{h}-y_{j}\right)x_{h}x_{j}y_{i}+\left(y_{i}-y_{j}\right)x_{i}x_{j}y_{h}\right)}{4\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}\\
+\frac{4\left(\left(x_{h}-x_{i}\right)x_{h}x_{i}y_{j}-\left(x_{h}-x_{j}\right)x_{h}x_{j}y_{i}+\left(x_{i}-x_{j}\right)x_{i}x_{j}y_{h}\right)}{4\left(\left(x_{i}-x_{j}\right)y_{h}-\left(x_{h}-x_{j}\right)y_{i}+\left(x_{h}-x_{i}\right)y_{j}\right)}
\end{multline*}
Coefficients: Slope Intercept Form
We may also write the parabola in terms of the slope of the directrix \(M\), the intercept of the directrix \(N\), and the focus \(\left(x_{f},y_{f}\right)\):
\begin{equation*}
\left(x-x_{f}\right)^{2}+\left(y-y_{f}\right)^{2}=\frac{\left(Mx-y+N\right)^{2}}{M^{2}+1}
\end{equation*}
Expanding this equation gives
\begin{multline*}
x^{2}+2Mxy+M^{2}y^{2}-2\left(M^{2}x_{f}+MN+x_{f}\right)x\\
-2\left(M^{2}y_{f}-N+y_{f}\right)y+\left(\left(M^{2}+1\right)x_{f}^{2}+\left(M^{2}+1\right)y_{f}^{2}-N^{2}\right)=0
\end{multline*}
By comparing the coefficients, we see
\begin{equation*}
M=\frac{B}{2}
\end{equation*}
\begin{equation*}
D=-2\left(M^{2}x_{f}+MN+x_{f}\right)
\end{equation*}
\begin{equation*}
E=-2\left(M^{2}y_{f}-N+y_{f}\right)
\end{equation*}
\begin{equation*}
F=\left(M^{2}+1\right)x_{f}^{2}+\left(M^{2}+1\right)y_{f}^{2}-N^{2}
\end{equation*}
Directrix & Axis
The directrix is given by
\begin{equation*}
y=Mx+N
\end{equation*}
And from the above we derive
\begin{equation*}
N=\frac{F\left(B^{2}+4\right)-\left(D^{2}+E^{2}\right)}{2\left(BD-2E\right)}
\end{equation*}
The axis runs through the focus (shown below) and is given by
\begin{equation*}
y=-\frac{1}{M}\left(x-x_{f}\right)+y_{f}
\end{equation*}
The axis intersects the directrix at
\begin{equation*}
x_{p}=\frac{x_{f}+M\left(y_{f}-N\right)}{M^{2}+1}
\end{equation*}
\begin{equation*}
y_{p}=\frac{Mx_{f}+M^{2}y_{f}+N}{M^{2}+1}
\end{equation*}
or
\begin{equation*}
x_{p}=-2\left(\frac{\left(3B^{2}+4\right)D^{2}-F\left(B^{2}+4\right)^{2}+E\left(BD-E\right)\left(B^{2}-4\right)}{\left(B^{2}+4\right)^{2}\left(BD-2E\right)}\right)
\end{equation*}
\begin{equation*}
y_{p}=-2\left(\frac{\left(3B^{2}+4\right)D^{2}-F\left(B^{2}+4\right)^{2}+E\left(BD-E\right)\left(B^{2}-4\right)}{\left(B^{2}+4\right)^{2}\left(BD-2E\right)}\right)
\end{equation*}
Focus & Vertex
The focus is given by
\begin{equation*}
x_{f}=-\frac{BF\left(B^{2}+4\right)+B\left(D^{2}-E^{2}\right)-4DE}{\left(B^{2}+4\right)\left(BD-2E\right)}
\end{equation*}
\begin{equation*}
y_{f}=2\left(\frac{F\left(B^{2}+4\right)-\left(D^{2}-E^{2}\right)-BDE}{\left(B^{2}+4\right)\left(BD-2E\right)}\right)
\end{equation*}
The vertex is simpler to write in terms of the focus, as
\begin{equation*}
x_{v}=\frac{\left(M^{2}+2\right)x_{f}+M\left(y_{f}-N\right)}{2\left(M^{2}+1\right)}
\end{equation*}
\begin{equation*}
y_{v}=\frac{\left(2M^{2}+1\right)y_{f}+Mx_{f}+N}{2\left(M^{2}+1\right)}
\end{equation*}
Or, we may express it as
\begin{equation*}
x_{v}=-\frac{4D\left(BD-4E\right)-BE^{2}\left(B^{2}+8\right)+BF\left(B^{2}+4\right)^{2}}{\left(B^{2}+4\right)^{2}\left(BD-2E\right)}
\end{equation*}
\begin{equation*}
y_{v}=-2\left(\frac{2D^{2}\left(B^{2}+2\right)+B^{2}E\left(BD-E\right)-F\left(B^{2}+4\right)^{2}}{\left(B^{2}+4\right)^{2}\left(BD-2E\right)}\right)
\end{equation*}
The distance from the focus to the vertex (and from the vertex to the point of intersection between the directrix and the axis) is given by
\begin{equation*}
G=\sqrt{\left(x_{f}-x_{v}\right)^{2}+\left(y_{f}-y_{v}\right)^{2}}
\end{equation*}
or
\begin{equation*}
G=\frac{-1}{\sqrt{B^{2}+4}}\left(\frac{BD-2E}{B^{2}+4}\right)
\end{equation*}
which is one quarter the length of the latus rectum.
Derivative
Finally, the derivative can be expressed either as
\begin{equation*}
y=-\frac{2}{B}\pm\frac{2E-BD}{B\sqrt{B^{2}x^{2}+2BEx+E^{2}-B^{2}x^{2}-B^{2}\left(Dx+F\right)}}
\end{equation*}
or
\begin{multline*}
y=-\frac{1}{M}\pm\\
\frac{\left(M^{2}+1\right)\left(Mx_{f1}-y_{f1}+N\right)}{M\sqrt{\left(M^{2}+1\right)\left(2M\left(Mx_{f1}-y_{f1}+N\right)x-M^{2}x_{f1}^{2}+\left(y_{f1}-2N\right)y_{f1}+N^{2}\right)}}
\end{multline*}