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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 σ_{1}, σ_{2} 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 *x*_{e} and *y*_{e} 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 *v*_{e} of points on an ellipse. With the the quadratic and symmetric matrix **A**_{q}

\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

\]

**A**_{q} is an invertible matrix (as was to be expected).

# Points of the ellipse with extremal y_{e}-values

Our first step to get the *x*_{e} and *y*_{e} coordinates for extremal points is to evaluate the quadratic form with respect to *y*_{e}. 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 *y*_{e} by evaluating the derivative with respect to *x*_{e}

{ \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 *x*_{e}:

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 *y*_{e} (see above). After a calculation of the *y*_{e}-values from the derived *x*_{e} 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 A_{e}

In my previous post I have discussed yet another matrix **A**_{E} which also can be used to define an ellipse. This matrix summarizes two affine transformations of a centered unit circle: **A**_{E} = **R**_{φ} ○ **D**_{σ1, σ2}. **D**_{σ1, σ2}.

You find the relations between the coefficients (a, b, c, d) of matrix **A**_{E} and the coefficients (α, β, γ) of matrix **A**_{q} 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.