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In the previous post of this mini-series
Properties of ellipses by matrix coefficients – I – Two defining matrices
I have discussed how the coefficients of two matrices which can be used to define a centered, rotated ellipse can be used to calculate geometrical properties of the ellipse:
The lengths σ1, σ2 of the ellipse’s principal axes and the rotation angle by which the major axis is rotated against the x-axis of the Euclidean coordinate system [ECS] we work with.
But there are other properties which are interesting, too. A centered, rotated ellipse has two points with extremal values in their y-coordinates. Can we express the coordinates – or equivalently the components of respective position vectors – in terms of the basic matrix coefficients?
The answer is, of course, yes. This post provides a derivation of respective formulas.
Matrix equation for an ellipse
In the last post we have shown that a centered ellipse is defined by a quadratic form, i.e. by a polynomial equation with quadratic terms in the components xe and ye of position vectors for points of the ellipse:
\alpha\,x_e^2 \, + \, \beta \, x_e y_e \, + \, \gamma \, y_e^2 \:=\: \delta
\]
The quadratic polynomial can be formulated as a matrix operation applied to position vectors ve of points on an ellipse. With the the quadratic and symmetric matrix Aq
\begin{pmatrix} \alpha & \beta / 2 \\ \beta / 2 & \gamma \end{pmatrix}
\]
we can rewrite the polynomial equation for the centered ellipse as
\pmb{v}_e^T \circ \pmb{\operatorname{A}}_q \circ \pmb{v}_e^T \:=\: \delta \,=\, \sigma_1^2\, \sigma_2^2, \quad \operatorname{with}\: \pmb{v_e} \,=\, \begin{pmatrix} x_e \\ y_e \end{pmatrix}.
\]
We could have artificially included the δ-term in a (3×3)-matrix, but this formal aspect will not help much to solve the equations coming below.
We also know already:
\operatorname{det}\left( \pmb{\operatorname{A}}_q \right) \:=\: \left(\alpha\gamma \,-\, {1\over 4}\beta^2\right) \, \ge \, 0
\]
Aq is an invertible matrix (as was to be expected).
Points of the ellipse with extremal ye-values
Our first step to get the xe and ye coordinates for extremal points is to evaluate the quadratic form with respect to ye. We define:
a_h \,& =\, {\alpha \over \gamma} \\
b_h \,& =\, {1 \over 2 } {\beta \over \gamma} \\
d_h \,& =\, {\delta \over \gamma}
\end{align}
\]
We reorder terms of the equation and add a supplemental quadratic term on both sides:
y_e^2 \, +\, b_h\,x_e\,y_e \, +\, b_h^2 \, x_e^2 \:=\: d_h \,-\, a_h \, x_e^2 \,+\, b_h^2 \, x_e^2
\]
We evaluate the complete quadratic term ob the left side to get
y_e \:=\: – b_h \, x_e \, \pm \, \left[\,d_h \,-\, \left( a_h \,-\, b_h^2 \right)\, x_e^2 \, \right]^{1/2}
\]
Let us first focus on the positive of the two alternative terms:
y_e \:=\: – b_h \, x_e \,+\, \left[\,d_h \,-\, \left( a_h \,-\, b_h^2 \right)\, x_e^2 \, \right]^{1/2}
\]
We get an extremal ye by evaluating the derivative with respect to xe
{ \partial \, y_e \over \partial \, x_e } \:=\: 0
\]
This means
– b_h \, – \, { \left(a_h \,-\, b_h^2 \right)\,x_e \over \left[\,d_h \,-\, \left( a_h \,-\, b_h^2 \right)\, x_e^2 \, \right]^{1/2} } \:=\: 0
\]
Getting rid of the denominator gives
– b_h \, \left[\,d_h \,-\, \left( a_h \,-\, b_h^2 \right)\, x_e^2 \, \right]^{1/2} \, – \, \left(a_h \,-\, b_h^2 \right)\,x_e \:=\: 0
\]
and
x_e \:=\: { – b_h \, \left[\,d_h \,-\, \left( a_h \,-\, b_h^2 \right)\, x_e^2 \, \right]^{1/2} \over \left(a_h \,-\, b_h^2 \right) }
\]
Squaring both sides and reordering leads to
x_e^2 \:=\: { – b_h^2 \, d_h \over a_h \, \left(a_h \,-\, b_h^2 \right) }.
\]
Using the old coefficients provides the final result for xe:
x_e \:=\: \pm \, {1 \over 2} \, \beta \, \left[ { \delta \over \alpha \,\left( \alpha \, \gamma \,-\, {1 \over 4} \beta^2 \right) } \right]^{1/2}
\]
We now follow the alternative solution for ye (see above). After a calculation of the ye-values from the derived xe values, we get the components for the two position vectors to the points with extremal y-values on the ellipse:
x_{e1}^{max} \:&=\: – \, {1 \over 2} \, \beta \, \left[ { \delta \over \alpha \,\left( \alpha \, \gamma \,-\, {1 \over 4} \beta^2 \right) } \right]^{1/2} \\
y_{e1}^{max} \:&=\: – \, {1 \over 2} \, {\beta \over \gamma} \, x_{e1}^{max} \,+\,
\left[ {\delta \over \gamma} \,-\, \left( {\alpha \over \gamma} \,-\, {1 \over 4} {\beta^2 \over \gamma^2} \right) \, x_{e1}^{max} \right]^{1/2}
\end{align}
\]
and
x_{e2}^{max} \:&=\: + \, {1 \over 2} \, \beta \, \left[ { \delta \over \alpha \,\left( \alpha \, \gamma \,-\, {1 \over 4} \beta^2 \right) } \right]^{1/2} \\
y_{e2}^{max} \:&=\: + \, {1 \over 2} \, {\beta \over \gamma} \, x_{e2}^{max} \,-\,
\left[ {\delta \over \gamma} \,-\, \left( {\alpha \over \gamma} \,-\, {1 \over 4} {\beta^2 \over \gamma^2} \right) \, x_{e2}^{max} \right]^{1/2}
\end{align}
\]
Coefficients of the matrix Ae
In my previous post I have discussed yet another matrix AE which also can be used to define an ellipse. This matrix summarizes two affine transformations of a centered unit circle: AE = Rφ ○ Dσ1, σ2. Dσ1, σ2.
You find the relations between the coefficients (a, b, c, d) of matrix AE and the coefficients (α, β, γ) of matrix Aq and δ in my previous post. This will allow you to calculate the vectors to the extremal points of an ellipse in terms of the coefficients (a, b, c, d).
Conclusion
In this post we have again used the coefficients of a matrix which defines an ellipse via a quadratic form to get information about a geometrical property.
We can now calculatethe components of the position vectors to the two points of an ellipse with extremal y-values as functions of the matrix coefficients.
In the next post
Properties of ellipses by matrix coefficients – III – coordinates of points with extremal radii
I will show you how to calculate the components of the end points of the principal axes of the ellipse with the help of our matrix for a quadratic form. I will also use our theoretical results for plots of some ellipses’ axes and of their extremal points. We will also compare theoretical predictions with numerically evaluated values.