2

The original problem is

One fire, one plane at an initial location, on straight sea line that the plane must reach before going to the fire, what is the minimal path?

The solution is easy, with similar triangles, now I'm wondering what it becomes if the sea shoreline is a circle, what tools will help to solve the problem now?

Figure

caub
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  • I don't understand the quoted question. If there one fire or two? If there are two, over which fire are you to release the water? – MvG Aug 04 '14 at 22:18
  • One fire, one plane at an initial location, on straight sea line that the place must each before getting to the fire – caub Aug 05 '14 at 13:51
  • Geometrically, what you need to find is the smallest ellipse whose foci is placed at the initial position of the plane and location of fire and the ellipse requires to touch the circle. – achille hui Aug 05 '14 at 14:12
  • @achillehui: I was thinking along the same lines, but in the straight-line case the ellipse formulation doesn't appear to give any directly usable insight into how to obtain the touching point. So in the circular case it might well be that there is a way to construct that point without finding touching conics. – MvG Aug 05 '14 at 14:17
  • @MvG I suspect if one look at the locus of the point on the ellipse whose normal line passing through the center of the sea, we will get some manageable algebraic curve. – achille hui Aug 05 '14 at 14:30
  • @achillehui nice intuition, I could also have put a more generalized elliptic sea line, and the idea is maybe to find the largest ellipse that fits in the other – caub Aug 05 '14 at 14:41
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    The locus should be a cubic curve. The derivation of it should be very similar to my answer to a similar question which construct an ellipse tangent to an external circle. The point you want will be the intersection of the cubic curve and the circular sea-shore. – achille hui Aug 05 '14 at 14:53
  • Thinking about it, if the plane is initially inside the sea, there is no point to fly to the shore. So the problem is actually equivalent to the other problem in previous comment. – achille hui Aug 05 '14 at 15:06
  • yes in particular cases, a straight line works – caub Aug 05 '14 at 16:52

1 Answers1

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Your sketch is an excellent hint. Locally, the best point is where the two marked angles to the tangent line are equal. This is like seeing an image in a concave mirror. Globally, there might be a shorter path. Think if the radius of the circle were small enough that a point between the red and blue $X$s were on the circle. You could still have equal angles where you draw them and that path would be shorter than the path through any nearby point on the circle, but there is a shorter path using the other side of the circle.

Ross Millikan
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  • it can be the longer path or shorter path, I think. If my assumption is right, the other parallel tangent below the circle is the longer path possible – caub Aug 05 '14 at 14:36
  • It can be max or min. The other "parallel" tangent, which need not be parallel, will also be a local minimum-shorter than nearby curves. I think there will be two points more, near the points the line between the crosses hits the circle, where the angles are equal that will be local maxima. – Ross Millikan Aug 05 '14 at 22:46