Timeline for Determining the first future intersection possible between ships and a planet
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Nov 25, 2017 at 17:30 | vote | accept | Samson Close | ||
| Nov 24, 2017 at 17:13 | comment | added | Stormwind | @DMGregory will try during closest days. My thought is to describe the scenario a bit differently (maybe a bit clearer/simpler) but the end conslusion will be the same as you already mentioned; cannot solve analytically (or way too challenging for ordinary deadly). The most relevant parameter is actually not travel angle, but start time. There are a huge amount of start times when the rocket cannot ever hit the planet (ie. no result exists), but that hasn't really been discussed. The 2nd and 3rd most important are rocket speed and direction. So it will be a 3-dimensional search for optimum. | |
| Nov 24, 2017 at 13:07 | comment | added | DMGregory♦ | @Stormwind I'd be curious to see more of your isometric space suggestion, if you'd be willing to write up an answer. :) | |
| Nov 23, 2017 at 3:10 | comment | added | Samson Close | Thank you for the edit, that cleared things up. I'll give a shot at implementing this and see how it goes. | |
| Nov 23, 2017 at 2:22 | history | edited | DMGregory♦ | CC BY-SA 3.0 |
Adding detail and an improved upper bound
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| Nov 22, 2017 at 22:19 | comment | added | Samson Close | Thank you! This solution makes a lot of sense. However, could you please elaborate as to how you are solving for t_0? What angle difference are we trying to find, and why are we using atan2 on the ship's distance? | |
| Nov 22, 2017 at 21:23 | comment | added | Tim Holt | True. I suppose my point is that there are potentially multiple solutions to the problem if some conditions aren't set (like "ship moves straight line entire time"). | |
| Nov 22, 2017 at 18:50 | comment | added | DMGregory♦ | @TimHolt in any such case, we could instead aim a little "upstream" in the orbit and travel a little longer to intercept it earlier in its path, instead of twiddling our thumbs. Think of the ship's potential positions as a circular ripple expanding continuously over time, or a widening cone in spacetime. There is always a point where the spacetime helix traced by the planet intersects this cone for the first time, and if we steer to that point we'll intercept it at the earliest possible moment, with no waiting required. | |
| Nov 22, 2017 at 18:46 | comment | added | Tim Holt | I wonder if there are cases where the most effective way to get to the planet the fastest is to just sit still for a certain amount of time, and then travel to the location? Or to travel to a point where the planet will be, then stop and wait for the planet? | |
| Nov 22, 2017 at 11:34 | comment | added | DMGregory♦ | @Bálint no, the ship doesn't yet have an absolute direction. We need to choose its direction such that, moving at its speed, it will reach a point in the orbit at the same time the planet does (effectively finding an intersection between the ship's future cone and the planet's future helix). Guessing a direction then checking whether the planet is ahead of or behind us in the orbit when we get there could be used for another style of iterative search though. | |
| Nov 22, 2017 at 10:02 | comment | added | Stormwind | An acceptable start DM, but actually a bit too complex. One could move to isometric space, ie. making the parabola a straight line and omitting cos() completely (as sin and cos work hand in hand). But one always ends up with sin(mT) = nT, which is the annoying part. | |
| Nov 22, 2017 at 7:28 | comment | added | Bálint | Couldn't you find the intersection point of the ship's path and the planet's orbit circle? Then you could check if the time the ship needs to get to those positions is the same as the planet needs. I might've misunderstood something though. | |
| Nov 22, 2017 at 3:50 | history | answered | DMGregory♦ | CC BY-SA 3.0 |