Visibility of Brilli's Orbit

We are asked to find out when Brilli disappears from view and when she reappears. This happens at the far points when the line of vision is tangent to the sphere, with a distance of 10 from the origin. The line through $\mathbf{v}=(100,100,50)$ and $\mathbf{w}=(18\cos t,18\sin t,0)$ has a distance of $d=\frac{||\mathbf{v} \times \mathbf{w}||}{||\mathbf{v}-\mathbf{w}||}$ from the origin. Setting this distance equal to 10 gives $t\approx 3.3547$ and $t\approx 4.4993$ as the time interval when Brilli disappears from view. Thus about $1-\frac{4.4993-3.3547}{2\pi}\approx \boxed{81.8\text{\%}}$ of the orbit are visible.

Maybe one with a cube next?