Elliptical Cross Sections

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Elliptic Conic Section

If the conic is an ellipse, 0<e<1. This implies

$e=\frac{\sin \beta}{\sin \alpha}=\frac{\sin\ (25^{\circ})}{\sin (90^{\circ}-\theta)}$

$\frac{sin (25^{\circ})}{cos (\theta)}=e$

since e must be less than 1, this sets the limit of theta at less than 65 degrees. Since the limit of $\theta=0$, this implies the minimum eccentricity will be $e\approx .4291$

This implies that the shape made on the the plane of the sector is an ellipse for angles

$0\lt \theta\lt 65^{\circ}$