In The Sirens of Titan, a major plot point surrounds the astronaut Winston Niles Rumfoord who purposefully steered his spaceship into a "chronosynclastic infundibulum" and consequently was turned into a type of spiral spread throughout the solar system. This allows him to exist at multiple points at the same time. In the book, it is mentioned that he simultaneously exists on four celestial bodies in our solar system: Mercury, Earth, Mars, and Titan. However, he only exists on these planets/moons when his spiral intercepts them.
We can make a simplistic model of the solar system, and thus the post-chronosynclastic-infundibulum Winston Niles Rumfoord with the following. In all cases we are assuming that every heavenly body has a perfectly circular orbit and is degenerated to a singular point. Assume that all planets begin aligned on the positive x-axis.
- Mercury orbits the Sun making one full revolution every $87.969$ days at a radius of $0.387$ AU. Winston's spiral intercepts Mercury's orbit every $14$ days.
- Earth orbits the Sun making one full revolution every $365.256$ days at a radius of $1.000$ AU. Winston's spiral intercepts Earth's orbit every $59$ days.
- Mars orbits the Sun making one full revolution every $686.980$ days at a radius of $1.524$ AU. Winston's spiral intercepts Mars' orbit every $111$ days.
- Saturn orbits the Sun making one full revolution every $10759.220$ days at a radius of $9.583$ AU. Furthermore, Titan orbits Saturn making one full revolution every $15.945$ days at a radius of $0.00816$ AU from Saturn. Part of Winston's spiral matches up one-to-one with Titan's orbit.
I will now here break Winston's spiral into two discrete curves, $C_1$ and $C_2$. $C_1$ is the part of Winston's spiral that aligns with the orbit of Titan relative to the Sun. This is easy enough to figure out given the above parameters. In particular, $C_1$ is parameterized by
$$x(t)=9.583\cos{\biggl(\frac{2\pi t}{10759.220}\biggr)}+0.00816\cos{\bigg(\frac{2\pi t}{15.945}\bigg)},\ y(t)=9.583\sin{\biggl(\frac{2\pi t}{10759.220}\biggr)}+0.00816\sin{\bigg(\frac{2\pi t}{15.945}\bigg)}$$
Meanwhile, we know that for $k=0,1,2...$, we know that $C_2$ passes through the following points:
$$\bigg(0.378\cos{\bigg(\frac{28k\pi}{87.969}\bigg)},\ 0.378\sin{\bigg(\frac{28k\pi}{87.969}\bigg)}\bigg)$$ $$\bigg(\cos{\bigg(\frac{108k\pi}{365.256}\bigg)},\ \sin{\bigg(\frac{108k\pi}{365.256}\bigg)}\bigg)$$ $$\bigg(1.524\cos{\bigg(\frac{222k\pi}{686.980}\bigg)},\ 1.524\sin{\bigg(\frac{222k\pi}{686.980}\bigg)}\bigg)$$
Given that we have three points that we want joined by a smooth curve, I think a natural instinct would be to join them with a parabola, but this feels inappropriate given the context. My next thought was a Bezier curve, but again, this doesn't feel right. Instead, I was hoping for something more akin to a spirographic curve, but I can't figure out how to start with this.
For what it's worth, I've made a little plaything here that shows the interceptions on Mercury, Earth, and Mars, respectively. I will say that the graph is a bit deceptive because it shows the first interceptions, then all the second interceptions, and then the third, and so on, when this may not accurately reflect the path taken. For example, here it may look like the pattern is always Mercury, Earth, and Mars in sequence, while really the pattern would start
Mercury, Mercury, Mercury, Mercury, Earth, Mercury, Mercury, Mercury, Earth, Mars, Mercury...
But, then again, Vonnegut never seemed too concerned about linear time, so that may be a non-issue here too ;)
I guess you could extend this to "is there a bespoke curve that passes through a sequence of given points that lay on a series of concentric circles", but I don't know if that would be easier or harder than the question as it is posed. Thank you for your time!