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January 7th, 2017, 02:42 AM   #1
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Distance between two parallel lines


I have two parallel lines. All I know is the coordinates of their start and end points. How can I calculate the distance between these lines?

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January 7th, 2017, 04:22 AM   #2
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The most important thing to remember is that the shortest distance between two parallel lines/segments occurs perpendicular to the lines/segments.

Now on to this question - How can lines have starting and ending points? A line is infinitely long...

If however you meant "the start and end points of the line segments", or "I know two points that lie on each line", then I would do this:

1. Work out the equation of one of the lines/segments.

2. Work out the equation of the normal to that line/segment at one of your known points.

3. Work out the point of intersection of the normal and the other line/segment.

4. Work out the distance between that point and the starting point. This will be the shortest distance between the two parallel lines/segments.
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January 7th, 2017, 04:45 AM   #3
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If they are line segments rather than lines, the meaning of "the distance between these lines" isn't clear. If they are unbounded lines (without any starting point or ending point), "the distance between these lines" still isn't clear, but I would assume that "the perpendicular distance between these lines" is intended.
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January 7th, 2017, 04:59 AM   #4
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That is a very good point skipjack. Perhaps the OP should post the ENTIRE question, and possibly draw a diagram of the situation. See if it's possible to do a perpendicular distance between the two lines/segments...
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January 8th, 2017, 07:46 AM   #5
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If the problems refers to line segments, there isn't enough information. Take the equations y = x and y = x + 2. If you look at both equations from x = 0 to x = 5, the distance between the line segments is 2. However, if you look at y = x from x = 0 to x = 5 and y = x + 2 from x = 10 to x = 20, it's impossible to draw a line that is perpendicular to both line segments and will intersect both of them.
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January 8th, 2017, 06:08 PM   #6
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Let us call both of them (line segments) as $\overline{a}$ and $\overline{b}$. Let me assume coordinates of one point of $\overline{a}$ to be $x_1$ and $y_1$ and the coordinates of corresponding point of $\overline{b}$ to be $x_2$ and $y_2$. If the line segments are both vertical, then $\overline{a}$ would be towards $y$-axis and distance between $\overline{a}$ and $\overline{b}$ would be $x_2 - x_1$, but if both line segments are horizontal, then $\overline{a}$ would be towards the $x$-axis and distance between $\overline{a}$ and $\overline{b}$ would be $y_2 - y_1$.

Since both have start and end points, both are line segments and not lines.
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