TITLE: Construction of new ellipse
QUESTION [6 upvotes]: Using a pencil, the thread was pulled on the ellipse. Then the pencil started to rotate around the ellipse. How to prove that the new geometric figure which the pencil drew is also an ellipse (with the same foci as the first ellipse)?

REPLY [2 votes]: Partial answer
Given the two ellipses, we'll check if the thread around the inner ellipse is constant from any point on the outer ellipse.
We can describe the inner and outer ellipses as follows:
$$
\begin{align}
x &= a \cos(\alpha) && 0<a<e,\ 0<b, \label{1}\tag{1}\\
y &= b \sin(\alpha) && \alpha \in (-\pi, \pi],\ \beta \in (-\pi, \pi]\\
x' &= e \cos(\beta)\\
y' &= \sqrt{e^2-a^2+b^2}\ \sin(\beta)
\end{align}
$$
The $\sqrt{e^2-a^2+b^2}$ is to make the ellipses have the same foci.
First we want to know where (which two values of $\alpha$) the thread leaves the inner ellipse.
We do this by looking which tangent lines of the inner ellipse go through $(x', y')$.
The tangent lines can be found by adding a multiple of the derivative:
$$
\begin{align}
x' &= a \cos(\alpha) + t \cdot (-a\sin(\alpha)) \qquad t \in \mathbb{R} \\
y' &= b \sin(\alpha) + t \cdot (b \cos(\alpha))
\end{align}
$$
Let $c = \frac{x'}{a}$ and $d = \frac{y'}{b}$, we have to solve:
$$
\begin{align}
c &= \cos(\alpha) - t \sin(\alpha)\\
d &= \sin(\alpha) + t \cos(\alpha)
\end{align}
$$
This seems pretty hard, but using Mathematica we get:
$$
\begin{align}
 m &= d \pm \sqrt{c^2+d^2-1} \label{2}\tag{2} \\
 \alpha &= 2 \arctan \left( \displaystyle \frac{m}{c+1} \right) \\
 t &= \frac{1-c^2-m^2}{2 m}
\end{align}
$$
The formula for $\alpha$ is undefined for $c = -1$, but that's no problem, because
we'll only check the right part ($\beta \in [-\pi/2, \pi/2]$) of the outer ellipse,
which is sufficient because of symmetry. This also prevents us to run into a potential
problem with the discontinuity of $\alpha$ and $\beta$ at $\pi$.
We can plug $\alpha$ back in $\eqref{1}$ to find the points on the ellipse:
$$
\begin{align}
  x &= a \cos \left(2 \arctan \left( \frac{m}{c+1} \right) \right) = a \frac{-d m+c+1}{d m+c^2+c} \\
  y &= b \sin \left(2 \arctan \left( \frac{m}{c+1} \right) \right) = b \frac{m(c+1)}{d m+c^2+c}
\end{align}
$$
Now we can make a formula for the length of the thread $L$. In $L_3$ the $\alpha_1$ refers to the lower of
the two values of $\alpha$ (choosing $-$ in $\eqref{2}$), and $\alpha_2$ to the higher value (choosing $+$ in $\eqref{2}$).
The integrand is the Pythagorean theorem applied to the derivative of $\eqref{1}$.
$$
\begin{align}
 L_1 &= \sqrt{(x' - x_1)^2 + (y' - y_1)^2} \\
 L_2 &= \sqrt{(x' - x_2)^2 + (y' - y_2)^2} \\
 L_3 &= \int_{\alpha_2}^{\alpha_1+2\pi} \sqrt{a^2 \sin(\alpha)^2 + b^2 \cos(\alpha)^2} \ \mathrm{d}\alpha \\
 L &= L_1 + L_2 + L_3
\end{align}
$$
The only thing left to do is proving $L$ is constant. We can do this by proving that its derivative to $\beta$ is equal to 0.
When writing $L$ as function of $a$, $b$, $e$ and $\beta$, the formula gets pretty big. So you'll want to use a computer for this.
We can get rid of the integral in $L_3$ because:
$$
\begin{align}
 \frac{\mathrm{d}L_3}{\mathrm{d}\beta} &= \frac{\mathrm{d}\alpha_1}{\mathrm{d}\beta} \sqrt{a^2 \sin(\alpha_1)^2 + b^2 \cos(\alpha_1)^2} -
  \frac{\mathrm{d}\alpha_2}{\mathrm{d}\beta} \sqrt{a^2 \sin(\alpha_2)^2 + b^2 \cos(\alpha_2)^2}
\end{align}
$$
With help of Mathematica we see that $\frac{\mathrm{d}L}{\mathrm{d}\beta}$ is equal to: (Mathematica code)
1/2 ((2 (Sqrt[-a^2 + b^2 + e^2] sinβ + (b^2 (a + e Sqrt[1 - sinβ^2]) (a Sqrt[-a^2 + b^2 + e^2] sinβ - Sqrt[(-a^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))/(-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))) (Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] - (b^2 e sinβ (a Sqrt[-a^2 + b^2 + e^2] sinβ - Sqrt[(-a^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))/(-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])) + (b^2 (a + e Sqrt[1 - sinβ^2]) Sqrt[(-1 + sinβ^2)/((a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2)))] (a^4 sinβ + b^2 e^2 sinβ - a^2 (b^2 + e^2) sinβ + a Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))/(-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])) - (a^2 b^2 (a + e Sqrt[1 - sinβ^2]) (-Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[(-a^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]/ a) (-b^2 e^2 Sqrt[-a^2 + b^2 + e^2] (1 - sinβ^2)^( 3/2) + Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] (a^2 b^2 + (2 a^4 + b^2 e^2 - 2 a^2 (b^2 + e^2)) sinβ^2) - b sinβ (a b^2 e + 2 (a^2 - b^2) (a^2 - e^2) Sqrt[ 1 - sinβ^2]) Sqrt[-1 + ((-a^2 + b^2 + e^2) sinβ^2)/b^2 - (e^2 (-1 + sinβ^2))/ a^2]))/(Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))] (b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) - a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[-(a^2 - e^2) (-a^2 + b^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))^2)) + (2 (a + e Sqrt[1 - sinβ^2]) (a^4 sinβ^2 + b^2 e^2 (-1 + sinβ^2) + a sinβ Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)] + a^2 (-e^2 sinβ^2 - b^2 (-1 + sinβ^2))) (-e sinβ + (a^3 (a b^2 - a (-a^2 + b^2 + e^2) sinβ^2 + b^2 e Sqrt[1 - sinβ^2] + sinβ Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]) (-b^2 e^2 Sqrt[-a^2 + b^2 + e^2] (1 - sinβ^2)^(3/2) + Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] (a^2 b^2 + (2 a^4 + b^2 e^2 - 2 a^2 (b^2 + e^2)) sinβ^2) - ( sinβ (a b^2 e + 2 (a^2 - b^2) (a^2 - e^2) Sqrt[ 1 - sinβ^2]) Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])/ a))/(Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))] (b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) - a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[-(a^2 - e^2) (-a^2 + b^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))^2) - (a^2 b (-b^2 e^2 Sqrt[-a^2 + b^2 + e^2] (1 - sinβ^2)^(3/2) + Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] (a^2 b^2 + (2 a^4 + b^2 e^2 - 2 a^2 (b^2 + e^2)) sinβ^2) + sinβ (b^2 e + 2 a (-a^2 + b^2 + e^2) Sqrt[ 1 - sinβ^2]) Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))/(Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))] (-b^3 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a b sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[-(a^2 - e^2) (-a^2 + b^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])))))/(-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])))/(\[Sqrt](((a + e Sqrt[1 - sinβ^2])^2 (a^4 sinβ^2 + b^2 e^2 (-1 + sinβ^2) + a sinβ Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)] + a^2 (-e^2 sinβ^2 - b^2 (-1 + sinβ^2)))^2)/(b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) - a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))^2 + (Sqrt[-a^2 + b^2 + e^2] sinβ + (b^2 (a + e Sqrt[1 - sinβ^2]) (a Sqrt[-a^2 + b^2 + e^2] sinβ - Sqrt[(-a^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))/(-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])))^2)) + (-((2 (a + e Sqrt[1 - sinβ^2]) (a^4 sinβ^2 + b^2 e^2 (-1 + sinβ^2) - a sinβ Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)] + a^2 (-e^2 sinβ^2 - b^2 (-1 + sinβ^2))) (-e sinβ - (a^3 (-(( e sinβ)/a) - ( Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] (Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[(-a^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]/a))/b^2 - ( sinβ Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] (b Sqrt[-a^2 + b^2 + e^2] + ( b (-a^2 + b^2) (a^2 - e^2) sinβ)/( a Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])))/b^3))/(a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2) + ( a sinβ (-a^3 sinβ + a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))/ b^2) - (a^3 (a b^2 - a (-a^2 + b^2 + e^2) sinβ^2 + b^2 e Sqrt[1 - sinβ^2] - sinβ Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]) (-b^2 e^2 Sqrt[-a^2 + b^2 + e^2] (1 - sinβ^2)^(3/2) + Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] (a^2 b^2 + (2 a^4 + b^2 e^2 - 2 a^2 (b^2 + e^2)) sinβ^2) + ( sinβ (a b^2 e + 2 (a^2 - b^2) (a^2 - e^2) Sqrt[ 1 - sinβ^2]) Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])/ a))/(Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))] (b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (-a^3 sinβ + a e^2 sinβ + Sqrt[-(a^2 - e^2) (-a^2 + b^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))^2)))/(b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (-a^3 sinβ + a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))) + 2 (Sqrt[-a^2 + b^2 + e^2] sinβ + (b^2 (a + e Sqrt[1 - sinβ^2]) (a Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[-(a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))/(-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ - Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))) (Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] - (b^2 (a + e Sqrt[1 - sinβ^2]) Sqrt[(-1 + sinβ^2)/((a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2)))] (-a^4 sinβ - b^2 e^2 sinβ + a^2 (b^2 + e^2) sinβ + a Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))/(b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (-a^3 sinβ + a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])) - (b^2 e sinβ (a Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[-(a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))/(-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ - Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])) - (a^2 b^2 (a + e Sqrt[1 - sinβ^2]) (Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[(-a^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]/ a) (-b^2 e^2 Sqrt[-a^2 + b^2 + e^2] (1 - sinβ^2)^( 3/2) + Sqrt[(-a^2 + b^2 + e^2) (1 - sinβ^2)] (a^2 b^2 + (2 a^4 + b^2 e^2 - 2 a^2 (b^2 + e^2)) sinβ^2) + b sinβ (a b^2 e + 2 (a^2 - b^2) (a^2 - e^2) Sqrt[ 1 - sinβ^2]) Sqrt[-1 + ((-a^2 + b^2 + e^2) sinβ^2)/b^2 - (e^2 (-1 + sinβ^2))/ a^2]))/(Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))] (b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (-a^3 sinβ + a e^2 sinβ + Sqrt[-(a^2 - e^2) (-a^2 + b^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))^2)))/(\[Sqrt](((a + e Sqrt[1 - sinβ^2])^2 (a^4 sinβ^2 + b^2 e^2 (-1 + sinβ^2) - a sinβ Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)] + a^2 (-e^2 sinβ^2 - b^2 (-1 + sinβ^2)))^2)/(b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (-a^3 sinβ + a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]))^2 + (Sqrt[-a^2 + b^2 + e^2] sinβ + (b^2 (a + e Sqrt[1 - sinβ^2]) (a Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[-(a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))]))/(-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ - Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])))^2)) - (2 a b (a^3 e sinβ + a^4 sinβ Sqrt[1 - sinβ^2] - a^2 (b^2 + e^2) sinβ Sqrt[1 - sinβ^2] + a (-e^3 sinβ + Sqrt[-(a^2 - e^2) (a^2 - b^2 - e^2) (-1 + sinβ^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]) + e (b^2 e sinβ Sqrt[1 - sinβ^2] + Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])) Sqrt[-((( 4 a^2 (-a Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[(-a^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])^2)/( b^2 (a + e Sqrt[1 - sinβ^2])^2) + b^2 (-1 + (-a Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[-(a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])^2/( b^2 (a + e Sqrt[1 - sinβ^2])^2))^2)/((a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))))])/((-b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (a^3 sinβ - a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])) Abs[ 1 + (-a Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[-(a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])^2/( b^2 (a + e Sqrt[1 - sinβ^2])^2)]) - (2 a b (-a^3 e sinβ - a^4 sinβ Sqrt[1 - sinβ^2] + a^2 (b^2 + e^2) sinβ Sqrt[1 - sinβ^2] + a (e^3 sinβ + Sqrt[-(a^2 - e^2) (a^2 - b^2 - e^2) (-1 + sinβ^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)]) + e (-b^2 e sinβ Sqrt[1 - sinβ^2] + Sqrt[(-a^2 + e^2) (-a^2 + b^2 + e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])) Sqrt[-((( 4 (a^2 Sqrt[-a^2 + b^2 + e^2] sinβ + a Sqrt[(-a^2 + e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])^2)/( b^2 (a + e Sqrt[1 - sinβ^2])^2) + b^2 (-1 + (a Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[-(a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])^2/( b^2 (a + e Sqrt[1 - sinβ^2])^2))^2)/((a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))))])/((b^2 (a^2 sinβ^2 + a e Sqrt[1 - sinβ^2] - e^2 (-1 + sinβ^2)) + a sinβ (-a^3 sinβ + a e^2 sinβ + Sqrt[(a^2 - e^2) (a^2 - b^2 - e^2) (b^2 + a^2 sinβ^2 - b^2 sinβ^2)])) Abs[ 1 + (a Sqrt[-a^2 + b^2 + e^2] sinβ + Sqrt[-(a^2 - e^2) (a^2 sinβ^2 - b^2 (-1 + sinβ^2))])^2/( b^2 (a + e Sqrt[1 - sinβ^2])^2)]))

Inserting arbitrary values for $a$, $b$, $e$ and $sin\beta$ and calculating it numerically seems to always yield 0.
It is left as an exercise for the reader to prove this is true for all $0<a<e, 0<b, -1 \le sin\beta \le 1$.