This is probably the question I get asked most:
How do you plan your crossings?
I’m very happy that people ask this because I think it’s a great exercise: try to come up with your own universal definition of something that feels intuitive.
Most people will begin with: “Well maybe cross it across the width of the lake”. Of course that makes sense for a nice rectagular lake, but try to apply it to the Vierwaldstättersee… it doesn’t really much sense anymore, does it? So really: try to come up with your own definition and then apply it to different lakes, does it work well? If it’s missing to this post, send it to me and I’ll add it.
I was a bit obsessed with sticking to a single definition before starting crossing lakes, but I soon realised, like anyone who has tried to apply mathematics to the real world, that there is a significant gap between theory and practice. In particular, not all the shore is accessible and some places are dangerous to swim in (especially in reservoirs), how do you fit this into your exact definition? Despite that, I still find it fun to come up with definitions, so here is a selection of lake crossing definitions that people have proposed.
Notation and assumptions
Let $L$ be a polygon (a lake) with cyclically ordered verices $V = {v_0, v_1, …, v_n}$. A crossing is a chord: a segment in $L$ that links two point on the perimeter of $L$.
Definitions
🐠 Nemo
Shortest segment going through the center of the largest inscribed circle (otherwise known as the point of inaccessibility, or, for the oceans: the nemo point).
The above is actually the definition I decided to stick with. It is simple enough and seems to capture the essence of a crossing. Also, this point often overlaps with the deepest point of the lake, which makes it kind of exciting and scary at the same time.
🎯 Through the center
Shortest segment contained in L going through the centroid of $L$.
🚧 Lowest upper bound
For every angle, find the longest chord aligned with that angle. Then, chose the angle with the minimal longest chord.
📐 Perpendicular
Find the longest internal chord or longest geodesic $c_g$. Take the longest chord perpendicular to $c_g$.
🚰 Source to sink
Find a perpendicular chord to the line between the largest tributary of the lake and its outlet.
It is worth mentionning, not all lakes have outlets…
⚖️ Balanced
Choose any chord that divides the lake in two parts of equal area.
↔️ Equal lengths
Starting from any point, you have to get to another shore point where the shore length clockwise and counterclockwise back to your starting point is the same.
📉 Gradient Descent
Define a function that returns the closest point to the shore from any point of the lake. This function is concave. From the global maxima, do gradient descent in one direction either shallowest or steepest path. Then do gradient descent in the other direction.
🎰 Monte Carlo 1
Take n random points in the lake and compute the shortest distance to a shoreline. Average all the distances to get a minimal crossing length.
🃏 Monte Carlo 2
Take $n$ pairs of random points on the perimeter of $L$. Compute the average distance between all pairs of points to get a minimal crossing length.
👁️ Vantage point
Select a trajectory that contains at some point a direct line of sight to all points on the perimeter of $L$.
👀 Scenic
Select the most beautiful lake crossing.
🏙️ Sightseeing
Link two personally or generally relevant points (e.g. cities, monument, lido, …) on that lake.
These last two ones were suggested by my father, who clearly does not have a scientific background, and a friend who grew up on the side of the Léman. Though I did not like these definitions at first, as you have lakes litterally in the middle of nowhere, this is often a tie-breaker when I am not sure about the trajectory. Aestetics and symbolism, though harder to define, have been a significant driver of this project.