## Estimating Distance While PaddlingPractical Geometry for Kayakers & Canoeists by Wes Kisting However flat the earth may look, we all know that it's actually round. Because of this fact, distant objects vanish over the horizon. When we paddle toward them, we see the tops of those objects emerge into view before we see their bottoms. The same is true as those objects move away from us: their bottoms vanish over the horizon before their tops. Kayakers who are attentive to this phenomenon (and willing to do some simple math) can use this knowledge to gauge distance more accurately. ## Curvature of the Earth: DistanceThe three pictures below show the position of an island in relation to the horizon. In the first picture, the bottom edge of the island is over the horizon. In the second picture, the island is much closer, but its bottom still has not crossed over into view. In the third picture, the bottom of the island has finally crossed over so that its sandy shoreline is now just barely visible. At this point, the kayaker can safely conclude that the island is 1.5 miles away. [Editor's Note: Images are not shown to scale. The island has been enlarged in all three pictures to help make its relative position to the horizon easier to see on a monitor.]
How can we tell that the shore is 1.5 miles away in the third picture? By combining some simple geometry with our knowledge of the earth's curvature. Let's call this The
For a typical paddler sitting upright in a kayak, eye level will be approximately 2 feet above the water. If we take the square root of 2, we find that the horizon should be located 1.4 miles away (call it 1.5 for the sake of convenience). The exact distance will be
Why is it useful to know the horizon distance formula? Well, for one thing, it is now possible to reliably estimate your distance from shores and objects by using the horizon. As the ## Expanding the Distance Rule: Beyond the Horizon
We can expand on the same formula to estimate the distance from the horizon to an object (of known height) which is visible over the horizon, The
Let's say that we are paddling toward an island that is known to be 100 feet tall at its highest point. Distance Rule 2 tells us that the tip of the island will just begin to emerge over the horizon when the island is 10 miles ## Putting it all Together: To the Horizon and Beyond
In case you missed it, we just combined our first two distance rules to calculate the total distance from the paddler to an object emerging over the horizon. Let's call this The
Because we know that the The
The diagram below will help clarify this principle. Notice that because of the curvature of the earth, the paddler is
Distance Rule 1 gives us the paddler's "Distance to Horizon" (shown in green). Distance Rule 2 gives us the horizon's "Distance to Emerging Object" (shown in orange). Distance Rule 3 simply adds the results of these two rules together to calculate "Total Distance" (shown in blue).
Returning to our earlier example of the 100-foot tall island, we added the square root of the island's height (10) to the square root of the paddler's height at eye level (1.5) to discover that a 100-foot tall island must be approximately 11.5 miles away when its tip first begins to emerge over the horizon (as shown in the diagram). By extension, if Obviously, to apply these rules you need to know an approximate height for the object you are paddling toward. Fortunately, many nautical charts and topographical maps provide this information. A prudent navigator will also research this information prior to the trip and make height annotations on the map for later reference. ## Estimating Distance by Guessing Height
Even if the |
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