Distance from Point to Line

• \[ \overrightarrow{PQ} = \overrightarrow{QA}-t~\overrightarrow{AB} = \begin{pmatrix} x_a-x+t(x_a-x_b) \\ y_a-y+t(y_a-y_b) \end{pmatrix} \]
• \[ d(t)=d(P,Q) = |x_a-x+t(x_a-x_b)| + |y_a-y+t(y_a-y_b)| \]
• Minimums:
  • $x_a \neq x_b$: \[t_x=\frac{x-x_a}{x_a-x_b}\] \[d_x=\left|y_a-y+\frac{x-x_a}{x_a-x_b}(y_a-y)\right|=\] \[|y_a-y|\left| \frac{x_a-x_b+x-x_a}{x_a-x_b} \right|\] \[\left| \frac{(x-x_b)(y-y_a)}{x_b-x_a} \right|\]
  • $y_a \neq y_b$: \[t_y=\frac{y-y_a}{y_a-y_b}\] \[d_y=\left|x_a-x+\frac{y-y_a}{y_a-y_b}(x_a-x)\right|=\] \[\left|\frac{(x_a-x)(y_b-y)}{y_b-y_a}\right|\]
• Projection Candidates:
  • $x_a \neq x_b$: \[ Q_x= \begin{pmatrix} x\\ \frac{ y_a(x_b-a)+y_b(x-x_a) }{ x_b-x_a } \end{pmatrix} \]
  • $y_a \neq y_b$: \[ Q_y= \begin{pmatrix} \frac{ x_a(y_b-y)+x_b(y-y_a) }{ y_b-y_a } \\y \end{pmatrix} \]