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<divclass="arithmatex">$$|m| + |n| = \text{max}(|m + n|, |m - n|).$$</div>

<p>To prove this, we just need to analyze the signs of<spanclass="arithmatex">$m$</span> and<spanclass="arithmatex">$n$</span>. And it's left as an exercise.</p>

<p>We may apply this equation to the Manhattan distance formula to find out that</p>

<divclass="arithmatex">$$d((x_1, y_1), (x_2, y_2)) = |x_1 - x_2| + |y_1 - y_2| = \text{max}(|(x_1 + y_1) - (x_2 + y_2)|, |(x_1 -y_1) - (x_2 -y_2)|).$$</div>

<p>The last expression in the previous equation is the<ahref="https://en.wikipedia.org/wiki/Chebyshev_distance">Chebyshev distance</a> of the points<spanclass="arithmatex">$(x_1 + y_1,x_1 -y_1)$</span> and<spanclass="arithmatex">$(x_2 + y_2,x_2 -y_2)$</span>. This means that, after applying the transformation</p>

<divclass="arithmatex">$$\alpha : (x, y) \to (x + y,x -y),$$</div>

<divclass="arithmatex">$$d((x_1, y_1), (x_2, y_2)) = |x_1 - x_2| + |y_1 - y_2| = \text{max}(|(x_1 + y_1) - (x_2 + y_2)|, |(y_1 -x_1) - (y_2 -x_2)|).$$</div>

<p>The last expression in the previous equation is the<ahref="https://en.wikipedia.org/wiki/Chebyshev_distance">Chebyshev distance</a> of the points<spanclass="arithmatex">$(x_1 + y_1,y_1 -x_1)$</span> and<spanclass="arithmatex">$(x_2 + y_2,y_2 -x_2)$</span>. This means that, after applying the transformation</p>

<divclass="arithmatex">$$\alpha : (x, y) \to (x + y,y -x),$$</div>

<p>the Manhattan distance between the points<spanclass="arithmatex">$p$</span> and<spanclass="arithmatex">$q$</span> turns into the Chebyshev distance between<spanclass="arithmatex">$\alpha(p)$</span> and<spanclass="arithmatex">$\alpha(q)$</span>.</p>

<p>Also, we may realize that<spanclass="arithmatex">$\alpha$</span> is a<ahref="https://en.wikipedia.org/wiki/Spiral_similarity">spiral similarity</a> (rotation of the plane followed by a dilation about a center<spanclass="arithmatex">$O$</span>) with center<spanclass="arithmatex">$(0, 0)$</span>, rotation angle of<spanclass="arithmatex">$45^{\circ}$</span> in clockwise direction and dilation by<spanclass="arithmatex">$\sqrt{2}$</span>.</p>

<p>Here's an image to help visualizing the transformation:</p>

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