Distance method

The distance method finds a knee by searching for the point which is as far as possible from the line connecting the first coordinate pari and the last coordinate pair.

Since knarrow scales the input to fit inside a square \(\left[0, 1\right]\times\left[0, 1\right]\), this effectively means finding a point which is furthest from the line \(y=x\).

If you plugin a point \((x_0, y_0)\) in the equation, and play with the numbers letters, you’ll arrive at a conclusion that the perpendicular distance from a point to a line is proportional to the vertical distance with a factor of \(\sqrt{2}\).

knarrow makes use of the fact that the scaling factor does not matter when finding the maximum distance, provided that the factor is strictyl positive, which \(\sqrt{2}\) surely is. Therefore, the implementation searches for the maximum of \(y-x\).