We can descend from these three-dimensional vistas to the more familiar two-dimensional one by asking what happens when we intersect this cone with some plane $P$.
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If the plane is parallel to the horizontal plane, there's certainly no mystery-the intersection is just a circle. Otherwise, the plane $P$ intersects the horizontal plane in a straight line. We can make things a lot simpler for ourselves if we rotate everything around the vertical axis so that this intersection line points straight out from the plane of the paper, while the first axis is in the usual position that we are familiar with. The plane $P$ is thus viewed "straight on," so that all we see (Figure 4) is its intersection $L$ with the plane of the first and third axes; from this view-point the cone itself simply appears as two straight lines.
I've kind of had a little trouble thinking about what it means when we "rotate everything around the vertical axis." Does this mean that we rotate Figure 3 about the first axis/$x$-axis?