Axis area induced by a spherical cap along other axis

In 3 dimensions, assume an axis that is transformed through rotations into all points on a spherical cap (with angle $\theta$ between the axis and a vector starting from the origin and pointing towards a point on the cap boundary). During these transformations an area is covered by the other 2 axis. What is the form of this area? My guess is that this area is a strip of length $2\theta$ over the great circle defined by the 2 axes initial configuration.

Edit: My exact question can be stated as follows: I want to find the possible locations of axis $x,y$ occurred by tilting the $z$-axis along the spherical cap shown in the figure. These transformations include all rotations along the $z$-axis (covering respectively great circles in the $xy$-plane). This amounts to finding all vectors which are normal to vectors starting from the origin and pointing towards a point on the spherical cap surface.

• Can you please describe this more clearly? Maybe give us some hints as to what you mean by the first sentence? Possibly some formulas to explain it? How does a straight line become "spherical" by rotations? And how you form an angle between a line and a point? – Paul Sinclair Apr 19 '17 at 23:21
• @PaulSinclair You are right I didn't state my question properly. I added a figure for clarification and edited the question. – RTJ Apr 20 '17 at 7:55
• Much better. I'd offer a solution, but Andrew D. Hwang has beat me to it. – Paul Sinclair Apr 20 '17 at 17:20

$\newcommand{\Basis}{\mathbf{e}}$If $\Basis = (a, b, c)$ is a unit vector making an angle less than $\theta$ with the positive $z$-axis, i.e., satisfying $c > \cos\theta$, the set of unit vectors orthogonal to $\Basis$ is a great circle $C_{\Basis}$, the intersection of the plane $ax + by + cz = 0$ with the unit sphere $x^{2} + y^{2} + z^{2} = 1$. The $z$-component of a point on $C_{\Basis}$ is easily shown to satisfy $|z| \leq \sin\theta$ (look in the plane containing $\Basis$ and the $z$-axis), and sweeping $\Basis$ by rotation around the $z$-axis causes $C_{\Basis}$ to sweep out the band of points whose latitude $\phi$ satisfies $|\phi| \leq \theta$.