While returning from a hike, you see from a distance that your tent is on fire. Luckily, you're holding a bucket and are near a river. Where along the river should you fill the bucket with water to minimize the total distance back to your tent?
In the interactive model below, point A represents one location along the river where you might run. Drag point A along the river's edge and note the corresponding distance measurement. This represents the total distance you need to travel to go from 'you,' to the river, to the tent.
Can you find the best location for point A? When you have, move the location of either 'you' or the tent and try again. What observations can you make? What happens when 'you' and the tent are both equally distant from the river?
How can you find the best spot to run without taking measurements? Here's an approach that makes use of an ellipse. The interactive model below is the same as above, only now it includes an ellipse with focal points at 'you' and the tent that passes through point A.
What is true about the points on this ellipse? Well, one definition of an ellipse is: "the set of points such that the sum of the distances from each point to two fixed points (the foci) is constant." This means that running from 'you,' to any point on the ellipse, to the tent is the same distance.
Consider those locations along the river that lie outside the ellipse. Are these better or worse places to run than point A? And how about those locations along the river that lie inside the ellipse? Are these better or worse locations to run than point A? Is there another location along the river that's no better or worse than running to point A?
Drag point A. You'll see that the ellipse's focal points remain the same, but the curve adjusts to pass through point A's new location. How do you know when you've found its optimal location?
You and a friend are relaxing in a circular pool. You decide to swim to the pool's edge, remove your sunglasses, and then swim directly to your friend. Where along the edge should you swim to minimize your total distance?
In the interactive model below, point A represents one location along the edge of the pool where you might swim. Drag point A around the pool and observe the different possible swimming paths.
The focal points of the ellipse are 'you' and your friend, and the ellipse passes through point A. All other points on the ellipse are equally good (or perhaps bad) as point A in terms of distance. Drag point A around the circle and look for some interesting ellipse behavior.
Based on the ellipse's position relative to the circle, how can you tell when you've found the best location for point A?
How can you tell when you've found the absolute worst location for point A (the location with the longest swimming distance)?
There are several locations for point A where the ellipse is tangent to the circle, yet point A is neither the best nor worst place to swim. What's going on there?
Move the location of 'you' and your friend to solve the problem for new placements of the two points.
A cowgirl wants to give her horse some food and water before returning to her tent. She starts at point C and decides to travel to the green pasture first, then the blue river, and then back to her tent, point T. What path should she take to minimize her riding distance?
This problem is typically solved by using a reflection argument. Several years ago, a student in my NYU geometry course devised the very clever proof below using the same ellipse technique as above.
For starters, points A and B are two random locations along the pasture and river, respectively. We don't know if these points are the best places to go, but they'll serve as our initial guesses.
Assume for the moment that point A is fixed and consider whether point B's location might be improved. Press the 'Show' button. The ellipse that appears has focal points at A and T, and passes through B. The picture below shows that B's location is not optimal: the ellipse crosses the river twice. So move point B until the ellipse is tangent to the river. Then scroll down.
To continue, let's take point B's location as fixed and consider point A. Press the 'Show More' button to reveal a second ellipse with foci at B and C passing through point A. Again, since the ellipse is not tangent to the pasture, point A needs adjusting. Drag point A until the ellipse is tangent to the pasture's edge. But oh! Now the first ellipse is no longer tangent to the river at point B. Drat!
The situation has improved, however. As long as either ellipse is not tangent, this refinement process can be continued, alternating between adjusting A and B, each time shortening the overall distance. When both ellipses are tangent simultaneously, you've found the best locations for points A and B.
Move the location of the cowgirl and the tent to solve the problem for new placements of the two points.