Saga University 2015: Locus in Space

Problem

In a coordinate space with the origin O, there are points A(1, 0, 0), B(2, 0, 0), and C(1, 0, 1). Let P be a point on the circumference of a circle with radius 1 centered at A on the xy-plane, and let ∠BAP=θ.
Find the locus of the intersection Q of the line CP and the yz-plane as θ varies in the range of $${\dfrac{\pi}{2} < \theta < \dfrac{3}{2}\pi}$$.
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Drawing the figure and performing the calculations should not be very difficult. The coordinates of point P can be expressed as P$${(\cos \theta +1,\ \sin \theta, 0)}$$. If we let R$${(x,\ y, \ z)}$$ be a point on the line CP, then from $${ \overrightarrow{\rm{OR}}=\overrightarrow{\rm{OC}}+t\overrightarrow{\rm{CP}} }$$, we can express $${x,\ y, \ z}$$ in terms of $${t}$$ and $${ \theta }$$. Since Q is a point on the yz-plane, if we set $${x=0}$$, we can express $${t}$$ in terms of $${ \theta }$$, resulting in Q$${ \left( 0, \ - \tan \theta, \ 1+\dfrac{1}{\cos \theta}\right) }$$. This is a parametric representation of a hyperbola.

I have created something to view the situation on the web.

The initial screen looks like this.
On the right, there is a slider and a "Show Locus" button. If the button is not displayed on the screen, please reload the page.

Let's change the value of θ with the slider. You can see the movement of Q, and it looks like a parabola. Let's display the locus using the "Show Locus" button. You will then see that it is not a parabola. The calculation result was a hyperbola.
The grid represents the yz-plane, but it might be difficult to understand the depth as it is, so let's rotate the figure using the circular slider on the left. This will help you grasp the sense of depth.

※ The figures are created using Cinderella (CindyJS).