Distance between two parallel lines: Difference between revisions

Revision as of 18:12, 22 December 2017

Revision as of 18:17, 22 December 2017

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Because the lines are parallel, the perpendicular distance between them is a constant, so it does not matter which point is chosen to measure the distance. Given the equations of two non-vertical parallel lines

:<math>y = mx+q_1b_1\,</math>

:<math>y = mx+q_2b_2\,,</math>

the distance between the two lines is the distance between the two intersection points of these lines with the perpendicular line

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:<math>\begin{cases}

y = mx+q_1b_1 \\

y = -x/m \, ,

\end{cases}</math>

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:<math>\begin{cases}

y = mx+q_2b_2 \\

y = -x/m \, ,

\end{cases}</math>

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to get the coordinates of the intersection points. The solutions to the linear systems are the points

:<math>\left( x_1,y_1 \right)\ = \left( \frac{-q_1mb_1m}{m^2+1},\frac{q_1b_1}{m^2+1} \right)\, ,</math>

and

:<math>\left( x_2,y_2 \right)\ = \left( \frac{-q_2mb_2m}{m^2+1},\frac{q_2b_2}{m^2+1} \right)\, .</math>

The distance between the points is

:<math>d = \sqrt{\left(\frac{q_1mb_1m-q_2mb_2m}{m^2+1}\right)^2 + \left(\frac{q_2b_2-q_1b_1}{m^2+1}\right)^2}\,,</math>

which reduces to

:<math>d = \frac{|q_2b_2-q_1b_1|}{\sqrt{m^2+1}}\,.</math>

When the lines are given by