My favorite things - analytical and sorta useless. However, it is interesting that moving the figurative to the literal could be deriving a single dimension metric and applying simple trig.
That plot is deceiving, just because the x-axis representing the ratio goes so large so quickly. I'd say roughly 95% of normally encountered triangles have edge ratios of less than 1:20, which represents 2% of the x axis.<p>Glancing at that graph you might wrongly conclude "no point cutting corners unless the triangle is reaaaally close to 1:1".
> This is why my preferred method for crossing intersections with 4-way stop signs is to go straight through the middle.<p><a href="https://www.youtube.com/watch?v=IJNR2EpS0jw" rel="nofollow">https://www.youtube.com/watch?v=IJNR2EpS0jw</a>
I have a feeling that this would be much easier to understand using trigonometry. For a unit hypothenuse, you're trying to maximise sin(θ) + cos(θ), which is the case when θ = τ/8, giving √2.<p>Given that we cut a corner, the longest possible value for the original problem is √2.<p>If we want the proportion cut, we get (1 - 1/√2) ≅ 0.3