# 'Linear aproximation' of a set of points?

1. Jul 15, 2006

Let there be a set of points $$T=\left\{T_{i}=(x_{i}, y_{i}) \in \textbf{R}^2, i=1, \dots, n : x > 0 \wedge y >0 \right\}$$, a line $$Ax + By + C = 0$$, and another set $$D=\left\{d_{i}=\frac{\left|Ax_{i}+By_{i}+C\right|}{\sqrt{A^2+B^2}} \in \textbf{R}, i=1, \dots, n\right\}$$. Next, let's form a sum $$S = \sum_{i=1}^n \frac{\left|Ax_{i}+By_{i}+C\right|}{\sqrt{A^2+B^2}}$$ which is required to have a minimum. Are $$\frac{\partial S}{\partial A} = 0$$, $$\frac{\partial S}{\partial B} = 0$$, and $$\frac{\partial S}{\partial C} = 0$$, the only requirements we need to 'place' the line as near to the dots as possible (if possible?) ?