The Perseid meteor shower is caused by debris left behind Comet Swift-Tutte. My understanding is that the meteor shower occurs once per year because that's when the Earth passes near the orbital path of the comet.
My question is why this shower only occurs once per year. I'm imagining that the Earth has an elliptical orbit around the sun, as does the comet. If the comet's orbit intersects the Earth's orbit at some point, since they're both ellipses, I would have thought that there would be (at least) two intersection points with the Earth's orbit, which would mean that we'd see this meteor shower twice or four times in the year as the Earth passes through these intersection points.
I'm assuming there's something wrong with my reasoning - can anyone clarify this for me?