Characterization of envelopes of moving rotational cones

Let $\Phi$ be a complete nondevelopable real analytic oriented surface in $\mathbb{R}^3$, and let $C_t$ be a real analytic family of oriented cones with opening angle $\theta$. The directions of the cones are unit vectors on the unit sphere; for a vertex $V_t$, write $V_t^i$ for its dual plane in isotropic space, and let the Gaussian image and inverse stereographic projection be understood as in the stated geometric construction.

Envelope characterization. The surface $\Phi$ contains the envelope of the family $C_t$ if and only if one of the following conditions holds:

• $\Phi$ is a sphere of radius $r$, the axes of $C_t$ are directed towards its center, and the vertices of $C_t$ are at distance $r/\sin\theta$ from the center;
• $\Phi$ is a parabolic cyclide, and $C_t$ is $\dots$;
• the Gaussian image of the parabolic-points set of $\Phi$ coincides with the $\pm(\pi/2-\theta)$-offset of the directions-set of $C_t$ in the unit sphere, and, for each $t$, the inverse stereographic projection of the top view of $\Phi^i\cap V_t^i$ is the $(\pi/2-\theta)$-offset of the direction of $C_t$ in the unit sphere.

The second alternative is incomplete in the source, and the status of this characterization cannot be determined from the supplied text.