So this problem is massively difficult to describe in words, making it very difficult to do any kind of search for; if it's a duplicate, please point me in the right direction.
Below is a gif of what's going on:
As the gif hopefully shows, the shape of the ground plane changes significantly as the camera moves.
I discovered this behavior when I noticed some odd rendering behavior in Freestyle, where some of the lines were disappearing for no readily apparent reason, which I eventually realized was the ground plane being in unexpected places in the render (I was only rendering the lines, not colors).
It should be noted that this ground plane is huge; it's around 2km x 1km in size. I don't know if that has any kind of affect, but if it does, I could probably cut it into smaller pieces.
The purpose of this is to be a large map that I can render from a variety of angles for illustration purposes, so I can't have the ground position/shape being so mutable. What's going on?